Congruences satisfied by Stark units

Anthony Hayward

Ph.D. Thesis King’s College London

Abstract

“Refined abelian Stark conjectures” are special value conjectures for L-functions associated to abelian extensions of global fields at the origin. We review the conjectures of Rubin-Stark and Gross, and describe a common refinement formulated by Burns. This refined conjecture is the main object of study in this thesis.

Having formulated the conjecture, we study its behaviour with respect to variation of the data on which it depends. Combining these results with ideas of Rubin and Gross yields a complete proof in the case of extensions of degree two.

We also look at the case of base fields which have at most one infinite place, where we axiomatise the explicit Stark units known over such fields, and verify the axioms by analytic constructions in each context. This then leads to a proof of the conjecture in a wide range of cases, for example, odd-degree extensions of ℚ.

The conjecture of Burns provides interpretation of conjectures of Darmon and Gross as accounting for its behaviour under quadratic extension of the base field. We state and prove these interpretations.

Acknowledgements

I would like to thank David Burns for his help and tireless encouragement, and for suggesting this topic of study. I also wish to thank Radan Ku čera, whose meticulous reading of much of this material led to a large number of corrections, and Werner Bley, Manuel Breuning, Cornelius Greither, William Hart and David Solomon, who all gave me helpful suggestions and advice.

I received funding during the work on this thesis from the Engineering and Physical Sciences Research Council, to whom I am very grateful.

Contents

1 Introduction 6

2 Notation 10 1 Global fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2 Abelian extensions of global fields . . . . . . . . . . . . . . . . . . . 12 3 L-functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4 Torsion in unit groups . . . . . . . . . . . . . . . . . . . . . . . . . 18 5 Exterior powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 6 Constructions in group rings . . . . . . . . . . . . . . . . . . . . . . 21

3 Special values of L-functions 23 1 Stark units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2 The analytic class number formula . . . . . . . . . . . . . . . . . . 25 3 Gross’s conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4 The Rubin-Stark conjecture . . . . . . . . . . . . . . . . . . . . . . 27 5 Burns’s conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4 Properties of the conjectures 30 1 Well-statedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2 Dependence on fields and T . . . . . . . . . . . . . . . . . . . . . . 30 3 Changing S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5 Proof in quadratic extensions 35 1 Cohomology of U_{K,S,T} . . . . . . . . . . . . . . . . . . . . . . . . . 35 2 The case d = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3 The case d > 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

6 Explicit Stark units 42 1 Systems of special units . . . . . . . . . . . . . . . . . . . . . . . . 42 2 Cyclotomic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3 Function fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4 Robert’s functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 5 Elliptic units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 7 Evidence from special units 58

1 Minors of matrices of zero row-sum . . . . . . . . . . . . . . . . . . 58 2 First congruence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3 Reciprocity symbol calculations . . . . . . . . . . . . . . . . . . . . 66 4 Relation to Burns’s conjecture . . . . . . . . . . . . . . . . . . . . 69

8 Minus-unit conjectures 72 1 Quadratic extension of the base field for L-functions . . . . . . . . 73 2 Unit groups in quadratic extensions . . . . . . . . . . . . . . . . . 74 3 Darmon’s circular unit . . . . . . . . . . . . . . . . . . . . . . . . . 78 4 Calculation of η . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 5 Darmon’s conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . 81 6 Factorisation of the regulator . . . . . . . . . . . . . . . . . . . . . 85 7 Base change for the congruence . . . . . . . . . . . . . . . . . . . . 87 8 Gross’s conjecture on the L-functions of tori . . . . . . . . . . . . . 88 9 Base change via the tori conjecture . . . . . . . . . . . . . . . . . . 91

9 The augmentation filtration 93 1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

Bibliography 97

Chapter 1

Introduction

An algebraic object such as an extension of number fields possesses an array of invariants, which may be crudely divided into two kinds: the analytic and the algebraic. Algebraic invariants come from (loosely-speaking) looking at the object “one bit at a time”, while analytic invariants correspond to combining infinitely many of these parts via a limiting process. Many of the greatest advances in the history of number theory have come from comparing invariants of these two different natures – one striking example (amongst many possibilities) would be the original proof of the prime number theorem. The analytic invariant involved in this is the Riemann zeta-function, the beginning point for the theory of L-functions in general, which in their various guises have proved to be analytic invariants which encode information about the original object with supreme subtlety. It is surely unnecessary to argue the importance of L-functions to the history of number theory.

The objects of study of this thesis are special value conjectures (at the origin) for Dirichlet L-functions associated to abelian extensions of global fields. Two classical results give tantalising indications of relations between such analytic invariants and algebraic ones. Firstly, the analytic class number formula relates the leading term at 0 of a zeta-function to two algebraic invariants of the field: the class number and the unit group. Secondly, the value at 0 of the derivative of an L-function associated to an even Dirichlet character χ mod m may be expressed in terms of absolute values of a cyclotomic element (1 − ζ_{m})(1 − ζ_{m}^{−1}). Interpreting χ as a character of the Galois group of the abelian extension ℚ(ζ_{m})^{+}/ℚ, we have a relation between a special value of the L-function of a Galois character at s = 0 and an integer of restricted support (“S-unit”) of the top field. That such relations should in fact be a general phenomenon is the core idea of Stark’s conjectures, developed in a series of papers ([42] – [45]) in the 1970s, which led to a proliferation of special value conjectures in the following years. The S-units appearing also have applications to the problem of explicit generation of class fields (Hilbert’s twelfth problem), a fascinating question which however we will not consider further, except to note that the computer system PARI [30] makes use of the assertion of Stark’s conjecture for exactly this purpose. Stark also considered this application of his conjecture in [44].

The “refined abelian Stark conjecture”, as it came to be known, is formulated with respect to an abelian extension K/k of global fields (with Galois group G), and a set S of places of the base field k. The set S is required to contain all infinite primes, and those ramified in K/k; in addition, the decomposition groups of elements of S control the order of vanishing of the L-functions of the characters of G at 0. Thus the fact that the conjecture concerns the first derivative of the Struncated L-function at 0 restricts the class of K/k for which it is meaningful: for example, if k has two complex places then the first derivative, and the conjecture, are trivial. The conjecture of Rubin [39] removes this condition by allowing for any number r of totally split places in the set S, and specialises to Stark’s conjecture in the case r = 1. However the new “Stark units” for the case r > 1 may no longer be taken to be S-units, but rather are combinations of them, in fact elements of (a vector space generated by) the r-th exterior power of the ℤ[G]-module of S-units. At this (great) expense, Rubin produces a more flexible conjecture, which as well as recovering Stark’s conjecture (and hence pertaining to the result about cyclotomic fields mentioned above), also finds a use in a special case for the analytic class number formula [39, Prop. 3.1]. Hence we see progressing conjectures capturing increasingly the analytic-algebraic relations which are to be found.

Subsequent to Rubin’s conjecture, Popescu [34] formulated a slightly weaker version (still recovering Stark’s conjecture) which had the advantage of a pleasant base-change property in a special case. For the purposes of the work discussed in this thesis, Rubin’s conjecture will be the most applicable of the two. Rubin’s conjecture makes sense also for r = 0, where it gives integrality properties for values of L-functions known to be true by work of Deligne, Ribet and Weil. In [18], Gross had formulated a conjecture for these values of a different kind: beginning from the integrality, he proposed a congruence in ℤ[G] between an equivariant combination of L-function values and the product of a class number by a certain “algebraic regulator”. This regulator lies naturally in the (#S − 1)-th power of the augmentation ideal of ℤ[G], and the congruence is modulo the next power of this ideal. Gross’s conjecture was inspired in part by the work of Mazur and Tate on the arithmetic of modular curves. Most noteworthy for our present discussion is that Gross’s conjecture involves a class number, and uses the S-unit group of k in its formulation of an algebraic regulator – both invariants which we noted to be related to special values by the analytic class number formula, but absent from the conjectures discussed so far. Unfortunately, the conjectures of Gross and Rubin exhibit a certain “orthogonality”: passing from r = 0 to r = 1, Gross’s conjecture becomes trivial (since the L-values are now all 0) while Rubin’s conjecture goes from a known result to a somewhat inaccessible one (namely, Stark’s conjecture). These two conjectures together therefore represent an unsatisfactory conjectural generalisation of the two classical results noted above, although each catches some aspects of them. It is clearly desirable to formulate a conjecture which captures both, and so gives a clearer picture of the analytic-algebraic relations which exist.

The conjectures of Rubin-Stark and Gross are both interesting, difficult and much studied (see e.g. [8]). But the above discussion shows that neither deserves to be seen as a satisfactory “big picture”. The state-of-the-art answer to the question of what does deserve this title is the “equivariant Tamagawa number conjecture”. This conjecture has its genesis in work of Bloch and Kato, was further developed by Fontaine and Perrin-Riou, and reaches its most general form in the formulation due to Burns and Flach [5]. Its intent fits neatly into the idea of relating analytic to algebraic invariants; the object of study being in general a motive, the analytic object a special value of a motivic L-function (at an integer), and the algebraic invariant a virtual object arising from motivic cohomology. In our context of abelian extensions of number fields, which correspond to “Tate motives”, we recover Dirichlet L-functions, and there exists a pleasant “arithmetic” interpretation of the motivic cohomology (see [4, Prop. 3.1]). Indeed, under certain hypotheses it is possible to reformulate the equivariant Tamagawa number conjecture in terms of the existence of a set of units satisfying certain relations [3, Thm. 3.2]. But the important thing to note in the context of the present discussion is that the equivariant Tamagawa number conjecture is very general and much less ad hoc in its formulation than the conjectures mentioned previously. Its interpretation in a special case therefore carries a certain momentum behind it; indeed, it provides us with another special value conjecture which has the appealing property of being the specialisation of a much bigger philosophical framework, rather than an analogy, or generalisation of a special case. Recent work of Burns [6] seeks to explore the relationship between the specialisation of the equivariant Tamagawa number conjecture and various refined abelian Stark conjectures which have been postulated, including those explained above. The results are striking: the equivariant Tamagawa number conjecture recovers at least the essential aspects, and in many cases the whole statement, of all these conjectures. The results are given in [7], and some of them are mentioned below.

The equivariant Tamagawa number conjecture thus provides illumination and unification for the disparate collection of conjectures which constituted the field at the end of the 20th century. Its interpretation in terms of explicit units, moreover, led Burns to formulate a less “high-tech” conjecture directly unifying the conjectures of Gross and Rubin described above. This conjecture (Conjecture 3.5.1 below) arose following discussions with the author attempting to understand a conjecture of Darmon [12], a “refinement” of Gross’s, in the context of Rubin’s conjecture. Such an interpretation (Theorem 8.7.1 below) was subsequently proved by the author, as well as a number of other properties and special cases of Conjecture 3.5.1, in work which first appeared in [22] (although we note that Conjecture 3.5.1 is there only stated up to sign, a deficiency which is here removed). The conjecture adds to Rubin’s conjecture a congruence in the style of Gross’s, thereby overcoming the unfortunate “orthogonality” mentioned above. Indeed it comes close (bar a small-looking kernel) to recovering the analytic class number formula (Theorem 2.1), as well as capturing the abelian Stark conjecture. It therefore represents progress towards a theory which accounts for both of the classical phenomena noted above. The conjecture is proved in the special case of quadratic extensions below (Chapter 5); its formulation in this case gives an explanation of the power-of-two factor appearing in the formula for the Stark unit (5.4), and its proof provides a new proof of Gross’s conjecture for quadratic extensions. This new proof is considerably smoother than the detailed consideration of special cases necessary in [18], due to the fact that Conjecture 3.5.1 allows S and r to be increased, which makes it possible to neuter an awkward class group. The mixture of Gross’s and Rubin’s ideas also sheds light on Gross’s “conjecture on tori” [18, Conj. 8.8] and the conjecture of Darmon mentioned above, in that both can be interpreted as expressing the change in the statement of Conjecture 3.5.1 caused by replacing k by a quadratic extension, replacing K by its compositum with this extension, and increasing r appropriately (Theorems 8.7.1 and 8.9.2). This shows that the new conjecture provides both theoretical and practical advantages in the area.

In the second classical result mentioned at the start of this introduction, a cyclotomic element appeared as the Stark unit for a cyclotomic field. It is wellknown that cyclotomic elements satisfy certain distribution properties with respect to norms, and indeed this is not restricted to this one context; in fact analogous units are known to exist in abelian extensions of two other kinds of fields: imaginary quadratic fields and global function fields. These are then all cases of global fields which have at most one archimedean place. We systematise the distribution relations and prove them in each context, the units being defined in quite different ways: via special values of elliptic functions for the imaginary quadratic fields, and division points on rank-one sign-normalised Drinfeld modules for function fields. In the case of elliptic units, we need a new result to show that the elliptic units used by Bley in [2] fit into this picture. After establishing that the special units satisfy the desired relations in all cases, and are all L-function evaluators, we proceed to use these relations to establish Conjecture 3.5.1 for a wide class of extensions of these three kinds of fields. The proof requires a study of the combinatorics of minors of matrices of zero row sum – we develop this theory in Chapter 7, §1 below – and also an integer identity dating back to Euler. This proof, as well as establishing the conjecture in new cases, gives a view of the conjecture and its various constructions in a very explicit setting. We note that this combinatorial approach was first suggested by work of Greither and Ku čera [17].

Gross’s conjecture, and by extension Conjecture 3.5.1 below, involve the augmentation filtration of the group ring ℤ[G] for a finite abelian group G. For the study of these conjectures, it is helpful to understand the structure of this filtration. The structure is well understood when for each p the p-part of G is either cyclic or elementary abelian, but for other G it is rather tricky to deal with. The author carried out a computer investigation into this structure for certain small G not satisfying these simplifying conditions. The results are described in Chapter 9 below. In particular, we show that the conjectures cannot be reduced to statements only concerning cyclic subextensions of the given extension.

Notation

1 Global fields

By a global field F we mean either a number field (i.e. a finite extension of the rational field ℚ) or a global function field (a finite extension of the field of rational functions 𝔽(T), where 𝔽 is a finite field). A place v of F is an equivalence class of absolute values on F, where two absolute values are equivalent if they define the same topology on F. We distinguish between archimedean (also called infinite) places and non-archimedean (finite) ones. Only number fields F have infinite places; we identify them with equivalence classes of embeddings F → ℂ under complex conjugation. If the image of an embedding is contained in ℝ, the corresponding infinite place is called real and the associated absolute value | · |_{v} is given by composing this embedding with the absolute value on ℝ; if not, the place is complex, and the absolute value is given by composing this with the square of the absolute value^{1} on ℂ. The sum of the number of real places and twice the number of complex places is equal to the degree [k : ℚ] of the number field, and is therefore finite.

If v is a finite place, it is represented by an absolute value |x|_{v} = α^{v(x)}, where α is any positive real number less than 1 and x ↦ v(x) is a uniquely determined surjective homomorphism F^{×} → ℤ which is a discrete valuation on F. If F is a number field, nontrivial discrete valuations v are in bijective correspondence with prime ideals 𝔭 of the ring of integers, and if F is a global function field, they correspond to primes 𝔭 of the field; we write v_{𝔭} for the discrete valuation associated to 𝔭. We choose a representative absolute value of the place by setting |x|_{v} = N_{v}^{−v(x)} for all x in F^{×}, where N_{v} denotes the number of elements of the residue field at v. We also write N_{𝔭} := N_{v_{𝔭}}.

Suppose S is a finite nonempty set of places of a global field F, such that S contains all infinite places. The ring of S-integers of F is the subset O_{F,S} = {x ∈ F : v(x) ≥ 0 for all v ∉ S}. If F is a number field and S is the set of infinite places only, this is O_{F}, the ring of integers of F. The multiplicative group O_{F,S}^{×} will be denoted U_{F,S} and called the S-units of F. The following theorem of Dirichlet describes the abelian group structure of U_{F,S}:

1This does not strictly give a field absolute value, due to the square. Removing this square does give an absolute value in the strict sense, which induces an equivalent metric on F.

Theorem 1.1 (Dirichlet’s unit theorem). The abelian group U_{F,S} is the direct product of its torsion subgroup (a finite cyclic group) with a free subgroup of rank #S − 1.

The torsion subgroup of U_{F,S} is independent of S and equal to the torsion subgroup of the multiplicative group F^{×}; we denote it by μ_{F}, and its order by w_{F}.

The free abelian group on the finite places of F is called the divisor group of F, denoted I_{F}. Those divisors all of whose coefficients are non-negative are called effective divisors, and these form a submonoid I_{F}^{+} of I_{F}. A divisor is called coprime to S if no primes in S have nonzero coefficient; the group of such divisors is naturally isomorphic to the group of fractional ideals of O_{F,S}, and denoted I_{F,S}. There is a natural truncation map I_{F} → I_{F,S} defined by setting the coefficients of the primes which lie in S to 0. Any nonzero element x of F has an associated divisor (x) given by its valuations, and the group of these principal divisors will be denoted P_{F}, with P_{F,S} the subgroup of the truncations at S of the principal divisors. The quotient I_{F,S}/P_{F,S} is defined to be the S-class group of F, denoted A_{F,S}; it is the ideal class group of the ring O_{F,S}. This group is finite since S is assumed to be nonempty. To summarise, we have an exact sequence

0 ⟶ U_{F,S} ⟶ F^{×} ⟶ I_{F,S} ⟶ A_{F,S} ⟶ 0.  (2.1)

Let 𝔱 be an ideal of O_{F,S}, so that its support Supp(𝔱) is a finite collection of finite places not in S. An element x ∈ k is said to satisfy x ≡ 1 (mod 𝔱) if v(x−1) ≥ v(𝔱) for all v ∈ Supp(𝔱). We denote U_{F,S,𝔱} = {x ∈ U_{F,S} : x ≡ 1 (mod 𝔱)} the group of S-units congruent to 1 modulo 𝔱. We let μ_{F,𝔱} be the subgroup of μ_{F} consisting of roots of unity congruent to 1 modulo 𝔱, and denote its size by w_{F,𝔱}, so that w_{F,𝔱} = #(U_{F,S,𝔱})_{tors}. Let P_{F,S,𝔱} denote the subgroup of P_{F,S} consisting of divisors (x) which have a generator x congruent to 1 modulo 𝔱, and note that this definition implies that P_{F,S,𝔱} is contained in I_{F,S∪Supp(𝔱)}. Define A_{F,S,𝔱} = I_{F,S∪Supp(𝔱)}/P_{F,S,𝔱}, the (S, 𝔱)-class group of F.

The class groups fit into an exact sequence

0 ⟶ U_{F,S,𝔱} ⟶ U_{F,S} ⟶ (O_{F,S}/𝔱)^{×} ⟶ A_{F,S,𝔱} ⟶ A_{F,S} ⟶ 0,  (2.2)

The third arrow here is the reduction map, and the fourth takes a residue to the class of the principal ideal generated by an S-integer with that residue, which exists by the weak approximation theorem. The exactness is clear.

The sizes of the class groups define the S-class number h_{F,S} = #A_{F,S} and (S, 𝔱)-class number h_{F,S,𝔱} = #A_{F,S,𝔱}. Sequence (2.2) gives exact sequences of finite groups 0 → U_{F,S}/U_{F,S,𝔱} → (O_{F,S}/𝔱)^{×} → A_{F,S,𝔱} → A_{F,S} → 0 and 0 → (O_{F,S}/𝔱)^{×}/Ũ_{F,S} → A_{F,S,𝔱} → A_{F,S} → 0, where Ũ_{F,S} denotes the image of U_{F,S} in the residue ring. Since in such sequences the product of the sizes of the evenindexed terms is equal to that of the odd-indexed terms, we obtain the following formulas relating the class numbers:

h_{F,S,𝔱} = h_{F,S} \frac{#(O_{F,S}/𝔱)^{×}}{(U_{F,S} : U_{F,S,𝔱})} = h_{F,S}((O_{F,S}/𝔱)^{×} : Ũ_{F,S}).

For the most part we will take 𝔱 to be a prime ideal or a square-free product of prime ideals. For consistency with much of the literature, when we do this we will set T = Supp(𝔱) and may write T in place of 𝔱 in the subscripts. We use the expression “x congruent to 1 modulo T ” to mean that v(x − 1) > 0 for all v ∈ T. Since for these square-free 𝔱 we have O_{F,S}/𝔱 ≅ ∏_{v∈T} 𝔽_{v}, where 𝔽_{v} denotes the residue field of v, the last displayed equation gives

h_{F,S,T} = h_{F,S} \frac{∏_{v∈T} (N_{v} − 1)}{(U_{F,S} : U_{F,S,T})}.  (2.3)

When F is a number field, we sometimes omit the subscripts S on the various objects defined in this section, on the understanding that this means S is to be taken to be the set of all archimedean places of F. Hence we write O_{F} (the ring of integers), U_{F} (the units), I_{F} and P_{F} (agreeing with our previous definitions), and A_{F} and h_{F} for the class group and class number.

2 Abelian extensions of global fields

Let K/k be an extension of global fields, and let S and T be sets of places of k as described in §1. Then we denote by S_{K} and T_{K} the sets of places of K lying above those in S and T respectively. We will usually denote places of k by v and places of K by w.

In this thesis we will always denote by K/k a finite abelian extension of global fields having group G. We will write the conjugate by an element g of an element x of K as x^{g}, and then since G is abelian we have x^{gh} = (x^{g})^{h} for all g, h ∈ G. The Galois group G acts on places w of K by the following transformation of absolute values:

|x|_{gw} = |x^{g^{−1}}|_{w},

where g ∈ G and x is any element of K. The action of G is transitive on the set of places lying above each place v of k. The stabiliser of a place w of K is the decomposition group G_{w} of the place, and we note that, since G is abelian, it only depends upon the place v of k induced by w and may therefore be written G_{v} := G_{w}.

Recall that we normalise the absolute value corresponding to a place w of K as follows:

|x|_{w} = \begin{cases}|x| & if K_{w} = ℝ \\ |x|^{2} & if K_{w} = ℂ \\ N_{w}^{−w(x)} & if w is a finite place, \end{cases}

where a finite place w is considered as a discrete valuation normalised so that its image is ℤ. With these normalisations, the product rule for absolute values states that we have

∏_{w place of K} |x|_{w} = 1,  (2.4)

as in [23, Chap. V, §1].

For a finite place w of K lying over a place v of k, the inertia group I_{w} is defined to be the kernel of the quotient map G_{w} → Gal(𝔽_{w}/𝔽_{v}), where 𝔽_{w}/𝔽_{v} is the residue field extension associated to w/v. If I_{w} is trivial, the place v is called unramified. Almost all places are unramified – a finite place can only ramify if it divides the discriminant of the extension K/k.

The Galois group Gal(𝔽_{w}/𝔽_{v}) is cyclic, generated by the (arithmetic) Frobenius automorphism

\begin{aligned}σ_{w} : & 𝔽_{w} & ⟶ & 𝔽_{w} \\ & x & ⟼ & x^{N_{v}}. \end{aligned}

The Frobenius may also be considered as an element of G_{w}/I_{w}. If I_{w} is trivial, i.e. w/v is unramified, then G_{w} is cyclic, generated by the Frobenius σ_{w}. Given v, the objects σ_{w} and I_{w} only depend on w up to conjugation in G, and so, since G is abelian, we may write σ_{v} and I_{v} instead. The choice of Frobenius also leads to the definition of the local reciprocity map (see [41, Ch. XIII, §4]), which is a homomorphism (−, K_{w}/k_{v}) : k_{v}^{×} → G_{v} (independent of choice of w above v). This has the properties that (x, K_{w}/k_{v}) ∈ I_{v} if v(x) = 0 and (x, K_{w}/k_{v}) = σ_{v}^{v(x)} if w/v is unramified.

We may also make the definitions of the previous two paragraphs for an extension of infinite places w/v of K. If w is real or v is complex, then G_{w} is trivial and so σ_{v} := 1. Otherwise w is complex and v is real, so that G_{w} has order 2. The nontrivial element of this group is taken for σ_{v}, and I_{w} is taken to be trivial. We have (x, K_{w}/k_{v}) = 1 if x is positive under the real embedding corresponding to v, and (x, K_{w}/k_{v}) = σ_{v} otherwise.

Let us consider the local reciprocity symbol (x, K_{w}/k_{v}) for x ∈ k^{×} fixed and v varying, choosing for each v a place w of K above v. For almost all v, it is equal to 1 – it can only be otherwise if v is ramified, or v is infinite, or v(x) ≠ 0. Therefore the product across all v of these symbols is finite. In fact, we have the following product rule for local reciprocity maps:

∏_{v place of k}(x, K_{w}/k_{v}) = 1.

This is a central fact of global class field theory – see e.g. [41, Ch. XIV (Appendix)].

For every finite place v of k which is unramified in K, we have defined the Frobenius element σ_{v} ∈ G. Since these places form a basis of the free abelian group I_{k,S}, we obtain a homomorphism (Artin map)

\begin{matrix}I_{k,S} & ⟶ & G \\ 𝔞 & ⟼ & σ_{𝔞} \end{matrix}

extending v ↦ σ_{v}.

Let Y_{S_{K}} = {∑_{w∈S_{K}} n_{w}w : n_{w} ∈ ℤ} be the free abelian group on S_{K}, and X_{S_{K}} = {∑_{w∈S_{K}} n_{w}w ∈ Y_{S_{K}} : ∑_{w∈S_{K}} n_{w} = 0} its augmentation subgroup. These are modules for ℤ[G], the integral group ring of G. For any commutative ring R and ℤ[G]-module M, we let RM denote the R[G] module R ⊗_{ℤ} M obtained by scalar extension. Then ℝY_{S_{K}} is the real vector space with basis S_{K} and associated G-action. The abstract ℝ[G] structure of ℝY_{S_{K}} is easily seen to be

ℝY_{S_{K}} ≅ ⨁_{v∈S} ℝ[G/G_{v}].

Using these normalisations, we may define the Dirichlet logarithmic regulator with respect to S_{K}:

\begin{aligned}λ_{S_{K}} : & U_{K,S} & ⟶ & ℝX_{S_{K}} \\ & x & ⟼ & − ∑_{w∈S_{K}} ln |x|_{w}w. \end{aligned}

The product formula (2.4) shows that the image of this map is indeed contained in the ℝ[G]-submodule ℝX_{S_{K}} of ℝY_{S_{K}}. The Dirichlet logarithmic regulator is easily seen to be a G-equivariant map. The map induced by λ_{S_{K}} on ℝU_{K,S} by scalar extension (which will be denoted by the same symbol) is an ℝ-vector space isomorphism. This is part of the proof of Dirichlet’s unit theorem, and is proved for example as the Unit Theorem in [23, Chap. V, §1]. It follows that

λ_{S_{K}} : ℝU_{K,S} ⟶ ℝX_{S_{K}}

is an isomorphism of ℝ[G]-modules.

Note that the fact that λ_{S_{K}} is an ℝ[G]-isomorphism implies the following

Proposition 2.1. The ℚ[G]-modules ℚU_{K,S} = ℚU_{K,S,T} and ℚX_{S_{K}} are isomorphic.

Proof. This is explained in Tate [46, Ch. 1, §4].

Finally, we look at a comparison of the maps λ_{S_{K}} and λ_{S_{k}}. We define a map

\begin{aligned}i_{K/k} : & Y_{S_{k}} & ⟶ & Y_{S_{K}} \\ & v & ⟼ & N_{K/k}w, \end{aligned}

where w is a place of K chosen arbitrarily above the place v of k, and N_{K/k} = N_{G} is the trace element of ℤ[G]. With these maps, the following diagram is commutative:

\begin{matrix}U_{K,S,T} & ⟶^{λ_{S_{K}}} & ℝX_{S_{K}} \\ ↑ & & ↑^{i_{K/k}} \\ U_{k,S,T} & ⟶^{λ_{S_{k}}} & ℝX_{S_{k}}. \end{matrix}  (2.5)

This follows from the relation |x|_{w} = |x|_{v}^{#G_{v}} between our normalised absolute values (for w/v an extension of places in K/k and x ∈ k).

3 L-functions

If G is a finite abelian group, the complex group ring ℂ[G] decomposes by the irreducible characters of the group into one-dimensional submodules. More precisely, if χ is an irreducible complex character of G, define an element of ℂ[G] by

e_{χ} = \frac{1}{#G} ∑_{g∈G} χ(g)g^{−1}.

Then ℂ[G]e_{χ} is a representation of G with character χ, that is, a one-dimensional ℂ-vector space on which g ∈ G acts by multiplication by χ(g). The elements e_{χ} satisfy:

e_{χ}^{2} = e_{χ};  e_{χ}e_{χ′} = 0 for χ ≠ χ′;  χ(e_{χ}) = 1;  χ′(e_{χ}) = 0 for χ ≠ χ′;

hence they form an orthogonal system of idempotents. We have an isomorphism of ℂ[G]-modules

ℂ[G] ≅ ⨁_{g∈G} ℂ[G]e_{χ}.

The submodule ℂ[G]e_{χ} will be referred to as the χ-component of ℂ[G]. It is a one-dimensional ℂ-vector space on which g ∈ G acts by multiplication by χ(g).

Now let K/k be an abelian extension with Galois group G, χ a complex character of G, and S and T finite sets of places of k as in §1. Assume that S contains all primes of k which ramify in K. To this data is associated an S-truncated, T -modified Dirichlet L-function, which is a meromorphic function on ℂ. The function is defined for s ∈ ℂ with Re s > 1 by the convergent Euler product

L_{K/k,S,T} (s, χ) = ∏_{v∈T}(1 − N_{v}^{1−s}χ(σ_{v})) · ∏_{v∉S}(1 − N_{v}^{−s}χ(σ_{v}))^{−1}.  (2.6)

For empty T we will also write L_{K/k,S}(s, χ) := L_{K/k,S,∅}(s, χ). Note that the factors for the ramified primes have been removed by the condition that these primes lie in S, so that σ_{v} is a well-defined element of G_{v} for each v. The number N_{v}^{−s} in the formula is the complex power of a positive real number N_{v} defined in the usual way, that is N_{v}^{−s} := exp(−s ln N_{v}). Expanding the Euler product gives the usual Dirichlet series expansion

L_{K/k,S}(s, χ) = ∑_{𝔞∈I_{k,S}^{+}} χ(σ_{𝔞})N_{𝔞}^{−s} for Re s > 1.  (2.7)

All this generalises to the case where G is not abelian; the L functions are replaced by Artin L-functions, which correspond to the Dirichlet L-functions when G is abelian.

It is well known that these L-functions may be meromorphically continued to all of ℂ (see for example [28, Ch. VII, Cor. 8.6]). The L-function (as defined in (2.6) above) for the trivial character χ_{0} is called the S-truncated, T -modified zeta-function of k, written ζ_{k,S,T} (s). We may factorise the zeta-function of the top field K in terms of the L-functions of all the characters χ of G as follows:

ζ_{K,S,T} (s) = ∏_{χ} L_{K/k,S,T} (s, χ).  (2.8)

(This follows from [46, Ch. 0 §4 (1)]).

So far we have looked at the L-function associated to a character. We now introduce a way of looking at all these L-functions at the same time, by introducing the equivariant L-function associated to the abelian extension K/k and the sets S and T. Let χ^{−1} denote the conjugate character to χ, so that

χ^{−1}(g) = \overline{χ(g)} = χ(g^{−1}) for all g ∈ G,

where the bar denotes complex conjugation. Then the equivariant L-function is defined by

Θ_{K/k,S,T} (s) = ∑_{χ} L_{K/k,S,T} (s, χ^{−1})e_{χ}.

Thus, if we consider a character χ as a ring homomorphism ℂ[G] → ℂ in the natural way, we have χ(Θ_{K/k,S,T} (s)) = L_{K/k,S,T} (s, χ^{−1}). It is also possible to express the function without mentioning characters, as in [46, Ch. IV, Prop. 1.6]:

Θ_{K/k,S,T} (s) = ∏_{v∈T}(1 − σ_{v}^{−1}N_{v}^{1−s}) ∏_{v∉S}(1 − σ_{v}^{−1}N_{v}^{−s})^{−1} for s > 1.

This has the advantage of splitting off the dependence on T as the finite product. The value of this product at s = 0 will be written

δ_{T} = ∏_{v∈T}(1 − σ_{v}^{−1}N_{v}).

We will be interested in the behaviour of these L-functions at s = 0. We write r_{S}(χ) for the order of L_{K/k,S,T} (s, χ) at s = 0; it is clear that this depends on S but not on T. We have r_{S}(χ^{−1}) = r_{S}(χ) because the corresponding L-functions are mutually conjugate along the real line. A formula for r_{S}(χ) is given in terms of S and χ in [46, Ch. 1, Prop. 3.4], and in the abelian case it gives the following result.

Proposition 3.1. If χ is the trivial character, we have r_{S}(χ) = #S−1. Otherwise r_{S}(χ) is equal to the number of v ∈ S such that χ is trivial on G_{v}.

The result is proved by using the well-known fact that L_{K/k,S}(s, χ) has order 0 at s = 1 for χ nontrivial and order −1 for χ trivial, and then using the functional equation for the L-functions.

We let r_{S} denote the number of totally split places in S, or r_{S} = #S − 1 if all places are totally split. That is, r_{S} = min(#{v ∈ S : G_{v} = 0}, #S − 1). It is clear from Proposition 3.1 that all the L_{K/k,S,T} (s, χ) vanish to order at least r_{S} at the point s = 0, and hence Θ_{K/k,S,T} (s) does as well. If we consider the Taylor expansion of the L-functions at the point s = 0, the terms in s^{0}, s^{1}, …, s^{r_{S}−1} are all identically 0. We define L_{K/k,S,T}^{n} (0, χ) to be the n-th term in the Taylor expansion of L_{K/k,S,T} (s, χ) at s = 0, and Θ_{K/k,S,T}^{n} (0) similarly for the equivariant function, so that

Θ_{K/k,S,T}^{n} (0) = ∑_{χ} L_{K/k,S,T}^{n} (0, χ^{−1})e_{χ},

where χ runs through all irreducible characters of G. Since L_{K/k,S,T} (s, χ^{−1}) and L_{K/k,S,T} (s, χ) are complex conjugates for s real, each term Θ_{K/k,S,T}^{n} (0) is actually an element of ℝ[G]. We may also express it as Θ_{K/k,S,T}^{n} (0) = \frac{1}{n!} \frac{d^{n}}{ds^{n}} Θ_{K/k,S,T} (s)|_{s=0}.

We set e_{r_{S}} := ∑_{χ:r_{S}(χ)=r_{S}} e_{χ} to be the idempotent corresponding to the characters where the order of vanishing of the associated L-function is exactly r_{S}. Then Θ_{K/k,S,T}^{r_{S}} (0) lies in the subspace ℝ[G]e_{r_{S}} and so is stable under multiplication by e_{r_{S}}.

We may also define Θ_{K/k,S}(s) in terms of partial zeta-functions. Let σ ∈ G. Since S contains all ramifying places, we may define

ζ_{K/k,S}(s, σ) = ∑_{\begin{matrix}𝔞∈I_{k,S}^{+}: \\ σ_{𝔞}=σ \end{matrix}} N_{𝔞}^{−s}  (2.9)

for Re s > 1, where 𝔞 runs through all effective divisors of k coprime to S with Artin symbol σ. Comparing this to (2.7) and using the orthogonality properties of characters, we have the relations

L_{K/k,S}(s, χ) = ∑_{σ∈G} χ(σ)ζ_{K/k,S}(s, σ); ζ_{K/k,S}(s, σ) = \frac{1}{#G} ∑_{χ} χ(σ)L_{K/k,S}(s, χ^{−1}),

for Re s > 1, and hence ζ_{K/k,S}(s, σ) may be meromorphically continued to the whole plane. The relations also show that

Θ_{K/k,S}(s) = ∑_{σ∈G} ζ_{K/k,S}(s, σ)σ^{−1}.  (2.10)

Proposition 3.2. Let L/K/k be a tower of global fields with L/k and K/k abelian. Let S be a set of places of k containing all infinite places and all places ramifying in L/k. Then

ζ_{K/k,S}(s, σ) = ∑_{\begin{matrix}θ∈Gal(L/k): \\ θ=σ on K \end{matrix}} ζ_{L/k,S}(s, θ).

Proof. This is immediate from the definition (2.9).

Corollary 3.3. The composition of Θ_{L/k,S,T} with the natural projection map ℂ[Gal(L/k)] → ℂ[Gal(K/k)] is Θ_{K/k,S,T}.

Proof. This follows from the proposition and (2.10).

4 Torsion in unit groups

We keep the notation K/k, G, S, T from the previous sections, so that S is a nonempty set of places of k containing all infinite places and those ramifying in K/k, and T is a finite disjoint set of primes. The purpose of T is the following hypothesis:

Hypothesis 4.1. The set T is such that U_{K,S,T} is torsion free.

Another way of stating the hypothesis is that we should have w_{K,T} = 1. The natural exact sequence of ℤ[G]-modules

0 ⟶ μ_{K,T} ⟶ μ_{K} ⟶ ∏_{v∈T} 𝔽_{v}^{×}

shows that, if Hypothesis 4.1 is satisfied, the element δ_{T} = ∏_{v∈T} (1 − σ_{v}^{−1}N_{v}) ∈ ℤ[G] defined in §3 annihilates μ_{K}.

We will use the following lemma to see some possible choices for T.

Lemma 4.2. Let K be a global field with w_{K} roots of unity. Let v be a discrete valuation on K. Write w_{K,v} for the number of roots of unity ζ in K which satisfy v(ζ − 1) > 0. Then

1. If K is a function field, w_{K,v} = 1.

2. If K is a number field and w_{K,v} > 1, then v(w_{K}) > 0 and the ramification index for v in the extension K/ℚ is at least p − 1, where p is the residue characteristic of v.

3. If K is a number field and v′ another discrete valuation on K of residue characteristic coprime to that of v, then there are no nontrivial roots of unity ζ in K satisfying v(ζ − 1) > 0 and v′(ζ − 1) > 0.

Proof. In the function field case, for any root of unity ζ we have that ζ − 1 is a constant and so it is either a root of unity or zero. This shows the first assertion.

Now assume K is a number field. The valuation v corresponds to a nonzero prime ideal 𝔭 in the ring of integers O_{K} of K. If ζ is a root of unity of order n in K satisfying v(ζ − 1) > 0 then ζ − 1 ∈ 𝔭 and so 𝔭 | (ζ − 1). If n has more than one prime factor then ζ − 1 is a global unit [47, Ch. 2, Prop. 2.8] and this is a contradiction to 𝔭 | (ζ − 1). Therefore n must be a prime power q^{a}; since 𝔭 | p and (ζ − 1) | q we must have q = p. Hence n = p^{a}, and ζ is a primitive p^{a}-th root of unity. So p | w_{K}, and this shows v(w_{K}) > 0 as required by the second assertion. Now (p) = (1 − ζ)^{p^{a−1}(p−1)} and so the assertion about ramification holds as well.

The third assertion is clear by this argument since v′(p) = 0, and so it is not possible that the prime ideal of O_{K} corresponding to v′ divides (ζ − 1), since the latter ideal divides p.

If K is a global function field with constant field of size 2, Hypothesis 4.1 is satisfied for any T. For all other function fields, the lemma shows that T need only be non-empty to satisfy Hypothesis 4.1. For a number field K, we may take T containing a single prime of sufficiently large residue characteristic, or two primes of coprime residue characteristic, and Hypothesis 4.1 will be satisfied.

Hypothesis 4.1 has the following happy consequence for a quotient of (S, T)- unit groups.

Lemma 4.3. For any K/k, S, T such that K/k is Galois and U_{K,S,T} is torsion free, the quotient U_{K,S,T} /U_{k,S,T} is also torsion free.

Proof. Suppose u ∈ U_{K,S,T} is such that u^{n} ∈ U_{k,S,T} for some n > 0. This means that for all σ ∈ Gal(K/k), we have (u^{n})^{σ−1} = 1. But this is (u^{σ−1})^{n}. Hence for all σ ∈ Gal(K/k), u^{σ−1} is a torsion element of U_{K,S,T} and so is 1. Therefore u ∈ k as required.

This lemma will allow us to extend a basis of the free abelian group U_{k,S,T} to a basis of U_{K,S,T}.

5 Exterior powers

In the remainder of this chapter we deal with some algebra which will be required for the later formulation of conjectures.

Let R be a commutative ring with 1, and M an R-module. Then for any integer n ≥ 0, the n-th exterior power of M over R is defined as in [24, Ch. XIX], and denoted ∧_{R}^{n}M. An element m_{1} ∧ ⋯ ∧ m_{n} of this will be referred to as the wedge product of m_{1}, …, m_{n}. If ϕ : M → R is a map of R-modules, it naturally induces an R-module homomorphism

\begin{aligned}ϕ : & ∧_{R}^{n}M & ⟶ & ∧_{R}^{n−1}M \\ & m_{1} ∧ ⋯ ∧ m_{n} & ⟼ & ∑_{i=1}^{n}(−1)^{i+1}ϕ(m_{i})m_{1} ∧ ⋯ ∧ m_{i−1} ∧ m_{i+1} ∧ ⋯ m_{n}. \end{aligned}

Iterating this process gives an R-linear map

\begin{matrix}∧_{R}^{r}Hom_{R}(M, R) & ⟶ & Hom_{R}(∧_{R}^{n}M, ∧_{R}^{n−r}M) \\ ϕ_{1} ∧ ⋯ ∧ ϕ_{r} & ⟼ & ϕ_{1} ∘ ⋯ ∘ ϕ_{r}. \end{matrix}

If Φ = ϕ_{1} ∧ ⋯ ∧ ϕ_{r} is in the domain, and m = m_{1} ∧ ⋯ ∧ m_{n} ∈ ∧_{R}^{n}M, we have the following formula for Φ(m), in which S_{n} denotes the symmetric group on n letters:

Lemma 5.1. We have

Φ(m) = ∑_{\begin{matrix}σ∈S_{n} \\ σ(r+1)<...<σ(n) \end{matrix}} sign(σ)( ∏_{j=1}^{r} ϕ_{j}(m_{σ(j)}))m_{σ(r+1)} ∧ ⋯ ∧ m_{σ(n)},

Proof. Necessarily by induction on r.

r = 1: We want to show

∑_{i=1}^{n}(−1)^{i+1}ϕ(m_{i})m_{1} ∧ ⋯ ∧ m̂_{i} ∧ ⋯ ∧ m_{r} =

∑\begin{matrix}σ∈S_{n} \\ σ(2)<...<σ(n) \end{matrix} sign(σ)ϕ_{j}(m_{σ(1)})m_{σ(2)} ∧ ⋯ ∧ m_{σ(n)},

where ˆ means that this part of the wedge should be omitted.

We match the i-th term on the left with the term on the right corresponding to the permutation σ_{i} ∈ S_{n} which takes 1 to i and satisfies σ_{i}(2) < … < σ_{i}(n). The relations

σ_{1} = id, σ_{i+1} = (i i + 1)σ_{i}

show that sign(σ_{i}) = (−1)^{i+1} as required.

r > 1: By induction we have

Φ(m_{1} ∧ ⋯ ∧ m_{n}) = ∑_{\begin{matrix}τ∈S_{n} \\ τ(r)<...<τ(n) \end{matrix}} sign(τ)( ∏_{j=1}^{r−1} ϕ_{j}(m_{τ(j)})) ·

∑_{i=0}^{n−r}(−1)^{i}ϕ_{n}(m_{τ(r+i)})m_{τ(r)} ∧ ⋯ ∧ { \atop m_{τ(r+i)}} ∧ ⋯ ∧ m_{τ(n)}.

We wish to compare this to

∑_{\begin{matrix}σ∈S_{n} \\ σ(r+1)<...<σ(n) \end{matrix}} sign(σ)( ∏_{j=1}^{r} ϕ_{j}(m_{σ(j)})) · m_{σ(r+1)} ∧ ⋯ ∧ m_{σ(n)}.

We match the term corresponding to a pair (τ, i) to σ, where σ satisfies

σ(r) = τ(r + i), σ(r + 1) = τ(r), σ(r + 2) = τ(r + 1), …, σ(r + i) = τ(r + i − 1),

and is otherwise equal to τ. That is,

σ = τ(r + i r + i − 1 … r + 1 r).

Therefore sign(σ) = (−1)^{i}sign(τ), and we see the terms match up.

Definition 5.2. Let r ≤ n be nonnegative integers. Define a set of permutations

\left[{n \atop r}\right] := \{σ ∈ S_{n} : σ(1) < σ(2) < … < σ(r) and σ(r + 1) < σ(r + 2) < … < σ(n)\}.

Note that the cardinality of this set is the binomial coefficient \binom{n}{r}. Each element corresponds to choosing a subset of r elements from {1, …, n}, and associates to it a sign, sign(σ).

Lemma 5.3. Using the notation of the previous lemma, and Definition 5.2, we have

Φ(m) = ∑_{σ∈\left[{n \atop r}\right]} sign(σ) det(ϕ_{j}(m_{σ(i)}))_{i,j}m_{σ(r+1)} ∧ ⋯ ∧ m_{σ(n)}

Proof. Fix such a σ as appears in the sum in the statement of the lemma. Consider the coefficient of m_{σ(r+1)} ∧⋯∧m_{σ(n)} in Φ(m) as expanded in the previous lemma. It is:

∑_{\begin{matrix}τ∈S_{n}: \\ τ(j)=σ(j) \\ for j=r+1,...,n \end{matrix}} sign(τ) ∏_{j=1}^{r} ϕ_{j}(m_{τ(j)})

which is

sign(σ) ∑_{θ∈S_{r}} sign(θ) ∏_{j=1}^{r} ϕ_{j}(m_{σθ(j)}),

as required.

These results show that if r = n, we have

Φ(m) = det(ϕ_{j}(m_{i}))_{i,j}.

6 Constructions in group rings

For any group G, we let aug : ℤ[G] → ℤ denote the augmentation homomorphism, that is, the ring homomorphism which maps every g ∈ G to 1 ∈ ℤ. The kernel of this map is called the augmentation ideal and denoted I_{G}. It is the two-sided ideal generated as an abelian group by elements of the form g − 1 for g ∈ G. The powers of this ideal form a decreasing chain of ideals in the group ring.

The quotient ℤ[G]/I_{G} is obviously isomorphic to ℤ. The next quotient I_{G}/I_{G}^{2} is isomorphic to the abelianisation of G, and hence to G itself if G is abelian. For a finite group, it is the case that #G annihilates each of the quotients I_{G}^{n}/I_{G}^{n+1} for n > 0, and these quotients are also finite (this follows from the surjectivity of the canonical map from the n-th symmetric power of G to I_{G}^{n}/I_{G}^{n+1} studied in [32]). An investigation into the structure of the filtration for several small finite abelian groups is given in Chapter 9 below.

Certain manipulations will be commonly made in these augmentation quotients. We summarise these in the following lemmas.

Lemma 6.1. Suppose x ∈ ℤ[G] and y ∈ I_{G}^{n}. Then xy ≡ aug(x)y (mod I_{G}^{n+1}).

Proof. This is because x − aug(x) has augmentation 0.

Lemma 6.2. The map

\begin{matrix}G & ⟶ & I_{G}/I_{G}^{2} \\ g & ⟼ & g − 1 \end{matrix}

is an isomorphism. Hence, if g ∈ G and t ∈ ℤ, we have g^{t}−1 ≡ t(g−1) (mod I_{G}^{2}).

Proof. This is well-known. A proof may be found for example in [9, p. 190].

Moreover, this shows:

Corollary 6.3. If g ∈ G, t ∈ ℤ and y ∈ I_{G}^{n}, then (g^{t}−1)y ≡ t(g−1)y (mod I_{G}^{n+1}).

Corollary 6.4. Let a_{1}, …, a_{n} and b_{1}, …, b_{n} be 2n elements of G, and r a positive integer. Suppose a_{i}^{r} = b_{i}^{r} for all i. Then r(a_{1} −1) ⋯ (a_{n} −1) ≡ r(b_{1} −1) ⋯ (b_{n} −1) (mod I_{G}^{n+1}).

Proof. We simply apply Lemmas 6.1 and 6.2 to each factor in turn, replacing r(a_{1} − 1) by r(b_{1} − 1).

We now turn to a discussion of group ring-linear duals. If G is a finite abelian group and M is a ℤ[G]-module, the ℤ[G]-dual of M is the ℤ[G]-module Hom_{ℤ[G]}(M, ℤ[G]), where the G-action is induced by that on M.

Lemma 6.5. There is a canonical isomorphism of abelian groups

\begin{matrix}Hom_{ℤ}(M, ℤ) & ⟶ & Hom_{ℤ[G]}(M, ℤ[G]) \\ ϕ^{1} & ⟼ & (m ↦ ∑_{g∈G} ϕ^{1}(g^{−1}m)g). \end{matrix}

Proof. This is easily seen to be a homomorphism. An inverse is given by the composition with the map ℤ[G] → ℤ given by taking the coefficient of 1 ∈ G.

We define a ℤ[G]-action on Hom_{ℤ}(M, ℤ) by (gϕ^{1})(m) = ϕ^{1}(gm), and then the map of Lemma 6.5 is an isomorphism of ℤ[G]-modules – indeed, we have defined a natural equivalence of ℤ[G]-module-valued functors from Hom_{ℤ}(−, ℤ) to Hom_{ℤ[G]}(−, ℤ[G]).

Corollary 6.6. The ℤ[G]-linear dual functor Hom_{ℤ[G]}(−, ℤ[G]) is exact on short exact sequences of ℤ-torsion free ℤ[G]-modules.

Proof. Such a sequence consists of free ℤ-modules.

We write N_{G} = ∑_{g∈G} g ∈ ℤ[G] for the norm (or trace) element of ℤ[G]. It is clear that if m ∈ M is fixed by every element of G then ϕ(m) = ϕ^{1}(m)N_{G} for maps ϕ and ϕ^{1} corresponding to each other as in Lemma 6.5.

If we now look at an element Φ ∈ ∧_{ℤ[G]}^{r}Hom_{ℤ[G]}(M, ℤ[G]), this corresponds by this isomorphism to a unique element of ∧_{ℤ[G]}^{r}Hom_{ℤ}(M, ℤ). Then the natural inclusion M^{G} → M gives us an element Φ̃ ∈ ∧_{ℤ}^{r}Hom_{ℤ}(M^{G}, ℤ). Hence we obtain a homomorphism

˜ : ∧_{ℤ[G]}^{r}Hom_{ℤ[G]}(M, ℤ[G])) ⟶ ∧_{ℤ}^{r}Hom_{ℤ}(M^{G}, ℤ).

Special values of L-functions

Throughout this chapter, we continue with the notation K/k, G, S, T developed in Chapter 2, so K/k is an abelian extension of global fields with Galois group G, S is a finite nonempty set of places of k containing all archimedean ones and all those that ramify in K/k, and T is a finite set of places of k disjoint from S. Recall that r_{S} is defined to be the number of places in S which split completely in K/k, or #S − 1, whichever is smaller. We define nonnegative integers n and d by the equations #S = n + 1 = r_{S} + d + 1.

1 Stark units

We first consider exterior powers of the module X_{S_{K}} defined in Chapter 2, §2. There are r_{S} places in S which are totally split in K, and we choose places w_{1}, …, w_{r_{S}} in S_{K} above each of these. We define an element e_{r_{S}} ∈ ℝ[G] as in Chapter 2, §3. Then the following is proved by Rubin in [39, Lem. 2.6ii].

Lemma 1.1. Let w be a place of S_{K} which is not conjugate to any w_{i}, and define x = (w_{1} − w) ∧ ⋯ ∧ (w_{r_{S}} − w) ∈ ∧_{ℤ[G]}^{r_{S}}X_{S_{K}}. Then

e_{r_{S}} ∧_{ℤ[G]}^{r_{S}} X_{S_{K}} = e_{r_{S}}ℤ[G]x.

Proof. We need to show that e_{r_{S}}x generates e_{r_{S}} ∧_{ℤ[G]}^{r_{S}}X_{S_{K}} as a ℤ[G]-module. Any element of X_{S_{K}} can be written as

∑_{i=1}^{r_{S}} α_{i}(w_{i} − w) + β  (3.1)

CHAPTER 3. SPECIAL VALUES OF L-FUNCTIONS 24

with α_{i} ∈ ℤ[G] and β = ∑_{w′} β_{w′}w′, where the w′ are not conjugate to any w_{i}, and β_{w′} lies in ℤ[G]. Let χ be a character of G; we will show that e_{χ}e_{r_{S}}β = 0. If χ is nontrivial on the decomposition group G_{w′} for such a w′, then e_{χ}w′ = 0 (since the sum of values of a nontrivial character of G_{w′} is zero). Otherwise, if χ is not the trivial character, Lemma 2.3.1 shows that e_{χ}e_{r_{S}} = 0. If χ is the trivial character χ_{0}, the element e_{χ}e_{r_{S}} is 0 if #S > r_{S} + 1 (by Lemma 2.3.1) and it is e_{χ_{0}} otherwise. Hence we have

e_{χ}e_{r_{S}}w′ = \begin{cases}e_{χ_{0}}w′ & if χ = χ_{0} and #S = r_{S} + 1; \\ 0 & otherwise. \end{cases}

For a nontrivial character χ we have e_{χ}e_{r}β = 0, and this is also true for the trivial character if #S > r_{S} + 1. If χ is the trivial character and #S = r_{S} + 1 then β can be written as β_{w}w, and must satisfy e_{χ_{0}}β_{w} = 0 since e_{χ_{0}} annihilates every element of X_{S_{K}}. Therefore in all cases e_{χ}e_{r_{S}}β = 0. Now an arbitrary element of ∧_{ℤ[G]}^{r_{S}}X_{S_{K}} is a wedge product of elements of the form (3.1), and this gives the result.

Let T be a finite set of primes of k disjoint from S. We know that

∧^{r_{S}}λ_{S_{K}} : ∧_{ℝ[G]}^{r_{S}}ℝU_{K,S,T} ⟶ ∧_{ℝ[G]}^{r_{S}}ℝX_{S_{K}}

is an isomorphism, and so this restricts to an isomorphism

∧^{r_{S}}λ_{S_{K}} : e_{r_{S}} ∧_{ℝ[G]}^{r_{S}} ℝU_{K,S,T} ⟶ e_{r_{S}} ∧_{ℝ[G]}^{r_{S}} ℝX_{S_{K}} = e_{r_{S}}ℝ[G]x.

Definition 1.2. The preimage under this map of the special value Θ_{K/k,S,T}^{r_{S}} (0)x will be called the Stark unit corresponding to the data K/k, S, T, x, and denoted η (adorned with as many subscripts taken from the data as is necessary for clarity in each context). That is, η is the unique

η ∈ e_{r_{S}} ∧_{ℝ[G]}^{r_{S}} ℝU_{K,S,T}

satisfying

(∧^{r_{S}}λ_{S_{K}})(η) = Θ_{K/k,S,T}^{r_{S}} (0)x.

Let us examine the dependence of η on x = (w_{1}−w)∧⋯∧(w_{r_{S}} −w). Note that e_{r_{S}}(w^{σ}−w) = 0 for w not conjugate to any of w_{1}, …, w_{r_{S}} and σ ∈ G, which follows from the preceding proof for the component idempotents of e_{r_{S}} corresponding to nontrivial characters, and is obvious for the trivial character. This shows that replacing a w_{i} by a Galois conjugate changes e_{r_{S}}x to a Galois conjugate of itself, and also that changing w to w^{σ} in x leaves e_{r_{S}}x unchanged. Changing w to a place which is not conjugate to w also has no effect on e_{r_{S}}x, since in this case #S > r_{S} + 1 and the preceding proof shows that e_{r_{S}}w = 0 for any allowable w. If r_{S} > 1 we may reorder the w_{i} and this can induce a change of sign in e_{r_{S}}x. In the special case where all places in S split in K, we may also exchange w with a w_{i}. It is easily checked that this has the effect of negating x.

To summarise, choosing a different x will change e_{r_{S}}x by a Galois element, and possibly induce a sign change if r_{S} > 1. This dependency of η on x is straightforward, and so we will choose x freely in the sequel.

In the case r_{S} = 0 we have x = 1 and η = Θ_{K/k,S,T} (0) equal to the special value itself. In the case r_{S} = 1, the Stark unit η lies in e_{r_{S}}ℝU_{K,S,T} and so may be written in the form x ⊗ u where x ∈ ℝ and u ∈ U_{K,S,T}. In fact, according to the Rubin-Stark conjecture below, it should be possible to take x = 1 (for almost all T), and so the Stark unit is really an element of U_{K,S,T}. In general, η will be a sum of wedge products of (S, T)-units with real coefficients, which it should be possible to make rational numbers with “small” denominator. The Stark unit η and its integrality and arithmetic properties will be the object of study of the conjectures which follow.

2 The analytic class number formula

We use all the notation of §1. If K = k we have the classical analytic class number formula, which we now recall and translate into the language of η. We order the places of S as v_{1}, …, v_{n+1}, and also choose a basis u_{1}, …, u_{n} for U_{k,S,T} modulo torsion (we do not need to assume Hypothesis 2.4.1 to state the theorem). Then setting x = (v_{1} − v_{n+1}) ∧ ⋯ ∧ (v_{n} − v_{n+1}), there is a unique nonzero R ∈ ℝ such that

(∧^{n}λ_{S})(u_{1} ∧ ⋯ ∧ u_{n}) = R · x.

Explicitly, we have

R = det(− ln |u_{i}|_{v_{j}})_{i,j=1}^{n}.  (3.2)

The (S, T)-regulator R_{k,S,T} is defined to be the absolute value of R. Note that the sign of R depends upon the ordering of the elements of S and the choice of basis u_{i}, while R_{k,S,T} is independent of these.

The analytic class number formula reads as follows (see [35, (1.2)]):

Theorem 2.1. The zeta function ζ_{k,S,T} = L_{k/k,S,T} (s, χ_{0}) has exact order of vanishing #S − 1 at s = 0, and its leading coefficient there is

ζ_{k,S,T}^{n} (0) = (−1)^{#T +1} \frac{h_{k,S,T}}{w_{k,T}} R_{k,S,T} .  (3.3)

Note that the factor (−1)^{#T} is missing from the statement of this theorem in [18, (1.6)]. We define a sign

ξ = (−1)^{#T +1}sign(R),  (3.4)

which depends upon the set T, the ordering of the set S and the choice of basis of U_{k,S,T}. Thus the right-hand side of (3.3) is ξ(h_{k,S,T} /w_{k,T})R.

We may restate Theorem 2.1 in the language of §1. Let K be an abelian extension of k in which every place in S splits, for example K = k. Let G = Gal(K/k). We have in this situation that

Θ_{K/k,S,T}^{n} (0) = ζ_{k,S,T}^{n} (0)e_{χ_{0}}.

Hence the following proposition follows directly from the definitions.

Proposition 2.2. [39, Prop. 3.1] Let K/k be an abelian extension with group G such that every place in S splits completely in K. Theorem 2.1 is equivalent to the statement that

η^{ξ} = \frac{1}{#G^{n}} \frac{h_{k,S,T}}{w_{k,T}} u_{1} ∧ ⋯ ∧ u_{n}.

Note that the rational coefficient of the Stark unit here may have a fairly large denominator.

3 Gross’s conjecture

In [18] (1988), Gross formulated a conjecture bearing a certain resemblance to the analytic class number formula, which we now describe.

As before, we let K/k be an abelian extension of global fields with group G, and sets S and T. We assume Hypothesis 2.4.1 for the rest of this section. Then Θ_{K/k,S,T} (0) ∈ ℝ[G] has the following integrality property:

Theorem 3.1 (Deligne, Ribet, Weil). We have Θ_{K/k,S,T} (0) ∈ ℤ[G].

Proof. The proof is given in [18, Prop. 3.7].

Gross’s conjecture looks at this element relative to the augmentation filtration. Composing the local reciprocity map for a place v of S with the natural iso-

morphism of Lemma 2.6.2 gives a map

\begin{aligned}f_{v} : & k_{v}^{×} & ⟶ & I_{G}/I_{G}^{2} \\ & x & ⟼ & (x, K_{w}/k_{v}) − 1. \end{aligned}

Let v_{1}, …, v_{m} be an ordering of a subset of S of size m. We may combine the maps f_{v_{j}} to obtain an algebraic regulator homomorphism

\begin{aligned}Reg_{G}^{(v_{1},...,v_{m})} : & ∧_{ℤ}^{m}k^{×} & ⟶ & I_{G}^{m}/I_{G}^{m+1} \\ & x_{1} ∧ ⋯ ∧ x_{m} & ⟼ & det(f_{v_{j}}(x_{i}))_{i,j=1}^{m}. \end{aligned}

Note that if any of the places v_{j} is totally split in the extension K/k, then G_{v_{j}} is trivial and hence f_{v_{j}} is the zero map and so is Reg_{G}^{(v_{1},...,v_{m})}.

Let us fix an ordering v_{1}, …, v_{#S} of S. Since we are assuming Hypothesis 2.4.1, the group U_{k,S,T} is a free ℤ-module of rank n, and we fix a basis u_{1}, …, u_{n} of this module. This determines the regulator R_{k,S,T} ∈ ℝ as defined in §2. We set Reg_{G} := Reg_{G}^{(v_{1},...,v_{#S−1})} and u := u_{1} ∧ ⋯ ∧ u_{n}, so that Reg_{G}(u) is the algebraic regulator corresponding to R_{k,S,T}. Then [18, Conj. 4.1] states the following:

Conjecture 3.2 (Gross). We have

Θ_{K/k,S,T} (0) ≡ ξh_{K,S,T} Reg_{G}(u) (mod I_{G}^{#S}).

For comparison with Gross’s notation in [18], we note that choosing the function “f” of loc. cit. to be the composition of the Artin reciprocity map with inversion (so that Gross’s “g(𝔭)” is our σ_{𝔭}^{−1}), the element “θ_{G}” of loc. cit. is the same as our Θ_{K/k,S,T} (0). The element “det_{G} λ” appearing in [18, Conj. 4.1] is congruent to (−1)^{#S−1}Reg_{G}(u) (mod I_{G}^{#S}), due to the choice of f and using Lemma 2.6.2. On the other hand, the differing signs in [18, (1.1)] and the definition of λ_{S_{K}} above give that the integer “m” appearing in [18, Conj. 4.1] is equal to (−1)^{#S−1}ξh_{k,S,T}. That Conjecture 3.2 is equivalent to [18, Conj. 4.1] is now clear.

The conjecture is not altered by changing the choice or ordering of places or units. This will be proved below for a generalisation of the conjecture.

Note that Conjecture 3.2 says 0 ≡ 0 in the case r_{S} > 0. So another way to express this conjecture is that, if r_{S} = 0, then we should have

η ≡ ξh_{k,S,T} Reg_{G}(u) (mod I_{G}^{#S}).

4 The Rubin-Stark conjecture

We use all the notation of §1 and assume Hypothesis 2.4.1. Considering the element η ∈ e_{r_{S}}ℝ ∧_{ℤ[G]}^{r_{S}} U_{K,S,T}, Rubin first asserts a rationality conjecture [39, Conj. A′] which is equivalent to “principal Stark conjecture” [46, Conj. I.5.1] for the extension K/k and those characters χ satisfying r(χ) = r_{S} [39, Prop. 2.3]:

Conjecture 4.1. η ∈ ℚ ∧_{ℤ[G]}^{r_{S}} U_{K,S,T}.

Stark formulated a finer conjecture in the case of abelian extensions, the “refined abelian Stark conjecture” [46, Ch. IV, §2]. In the notation developed so far, it reads as follows:

Conjecture 4.2. If r_{S} = 1 then, for all acceptable T, we have η ∈ U_{K,S,T}.

We refer to [39, Prop. 2.5] for the comparison with the formulation in [46]. In fact, it is sufficient to check this conjecture for almost all acceptable singleton sets T (by [46, Prop. IV.1.2]).

The Rubin-Stark conjecture generalises Conjecture 4.2 to allow other values of r_{S}. The most obvious generalisation, to expect that for all r_{S} we have η ∈ ∧_{ℤ[G]}^{r_{S}}U_{K,S,T}, does not hold. Rubin gives counterexamples in [39, §4]. Instead, the integrality property is expressed using a dual construction. By ℝ-linear extension, an element Φ ∈ ∧^{r_{S}}Hom_{ℤ[G]}(U_{K,S,T} , ℤ[G]) induces an ℝ[G]-homomorphism ℝ ∧_{ℤ[G]}^{r_{S}}U_{K,S,T} → ℝ[G]. The Rubin-Stark conjecture [39, Conj. B’] is the following statement about the image of η under such maps:

Conjecture 4.3 (Rubin). If Φ ∈ ∧_{ℤ[G]}^{r_{S}}Hom_{ℤ[G]}(U_{K,S,T} , ℤ[G]), then we have

Φ(η) ∈ ℤ[G].

This conjecture specialises to Conjecture 4.2 in the case r_{S} = 1 since {x ∈ ℝU_{K,S,T} : ϕ(x) ∈ ℤ[G] for all ϕ ∈ Hom_{ℤ[G]}(U_{K,S,T} , ℤ[G])} = U_{K,S,T}, as is easy to see by, for example, picking a basis of the unit group.

An alternative generalisation of Conjecture 4.2 was proposed by Popescu in [34]. His version is weaker than Rubin’s in the case where the torsion μ_{K} is not G-cohomologically trivial, but has the advantage of a particular base-change property. However, because of its use of group rings, Conjecture 4.3 will be the object of study here.

Note that in the case r_{S} = 0, Conjecture 4.3 says that η ∈ ℤ[G], which is Theorem 3.1 above. So Rubin’s conjecture is known in this case. On the other hand, for higher r_{S}, Rubin’s conjecture is open while Gross’s conjecture (Conjecture 3.2) becomes trivial. These conjectures however allow a common refinement which brings congruences in the style of Gross’s conjecture to the more general context of Rubin’s. This refinement, formulated by Burns, is described in the next section and will be the main object of study of this thesis.

5 Burns’s conjecture

We use the notation of the preceding sections, and assume Hypothesis 2.4.1. According to Conjecture 4.3, the Stark unit η satisfies Φ(η) ∈ ℤ[G] for all Φ ∈ ∧_{ℤ[G]}^{r_{S}}Hom_{ℤ[G]}(U_{K,S,T} , ℤ[G]). The conjecture which will be described, formulated by Burns, proposes that this should hold, and furthermore Φ(η) should satisfy a congruence with respect to the augmentation filtration of ℤ[G] in the style of Conjecture 3.2.

Order the elements of S as v_{1}, …, v_{n+1} such that v_{1}, …, v_{r_{S}} are totally split in K. We define the map Reg_{G} := Reg_{G}^{(v_{r_{S} +1},...,v_{n})} in the notation of §3. An element Φ ∈ ∧_{ℤ[G]}^{r_{S}}Hom_{ℤ[G]}(U_{K,S,T} , ℤ[G]) gives us Φ̃ ∈ ∧_{ℤ}^{r_{S}}Hom_{ℤ}(U_{k,S,T} , ℤ) as in Chapter 2, §6. Combining these gives us a map

Reg_{G} ∘ Φ̃ : ∧_{ℤ}^{n}U_{k,S,T} ⟶ ℤ[G]/I_{G}^{d+1}.

The map may be written down explicitly as follows, using the results of Chapter 2, §5. We have

(Reg_{G} ∘ Φ̃)(u_{1} ∧ ⋯ ∧ u_{n})

= ∑_{σ∈\left[{n \atop r_{S}}\right]} sign(σ) det(ϕ_{j}^{1}(u_{σ(i)}))Reg_{G}(u_{σ(r_{S}+1)} ∧ ⋯ ∧ u_{σ(n)})

= | \begin{matrix}ϕ_{1}^{1}(u_{1}) & ⋯ & ϕ_{r_{S}}^{1}(u_{1}) & f_{v_{r_{S} +1}}(u_{1}) & ⋯ & f_{v_{n}}(u_{1}) \\ ⋮ & ... & ⋮ & ⋮ & ... & ⋮ \\ ϕ_{1}^{1}(u_{n}) & ⋯ & ϕ_{r_{S}}^{1}(u_{n}) & f_{v_{r_{S} +1}}(u_{n}) & ⋯ & f_{v_{n}}(u_{n}) \end{matrix} | .

Conjecture 5.1 (Burns). Let S be ordered v_{1}, …, v_{#S} so that v_{1}, …, v_{r_{S}} are totally split in K, and choose an integral basis u_{1}, …, u_{n} of U_{k,S,T}. Conjecture 4.3 holds, and furthermore we have

Φ(η) ≡ ξh_{k,S,T} · (Reg_{G} ∘ Φ̃)(u_{1} ∧ ⋯ ∧ u_{n}) (mod I_{G}^{d+1})

for all Φ ∈ ∧_{ℤ[G]}^{r_{S}}Hom_{ℤ[G]}(U_{K,S,T} , ℤ[G]), where ξ = ±1 is the sign defined in (3.4) with respect to the choices and orderings of v_{j} and u_{i}.

Note that by the form of the regulator term, this suggests that in Conjecture 4.3, the element Φ(η) ∈ ℤ[G] actually lies in the submodule I_{G}^{d}.

It is not necessarily the case that every Φ ∈ ∧_{ℤ[G]}^{r_{S}}Hom_{ℤ[G]}(U_{K,S,T} , ℤ[G]) can be written as a primitive tensor ϕ_{1} ∧ ⋯ ∧ ϕ_{r_{S}}. However, by the linear form of the conjecture it is sufficient to consider only these. For r_{S} = 0 it is sufficient to consider the generator Φ = 1 ∈ ℤ[G].

We will show below (Chapter 4, §1) that Conjecture 5.1 is independent of the choices made in its statement.

Proposition 5.2. Conjecture 5.1 is a common refinement of Conjectures 4.3 and 3.2.

Proof. It is obvious from the formulation that Conjecture 5.1 implies Conjecture 4.3.

If r_{S} > 0, taking Φ = 1, Conjecture 3.2 states that 0 ≡ 0 and so is a consequence of Conjecture 5.1. Otherwise, the statements of the conjectures are the same.

Proposition 5.3. If all places in S split in K, then Conjecture 5.1 is true.

Proof. We have r_{S} = n, and use Proposition 2.2 to see that

η^{ξ} = \frac{1}{#G^{n}} h_{k,S,T} · u_{1} ∧ ⋯ ∧ u_{n}.

We apply Φ = ϕ_{1} ∧ ⋯ ∧ ϕ_{n} ∈ ∧_{ℤ[G]}^{n}Hom_{ℤ[G]}(U_{K,S,T} , ℤ[G]) to this equation. Note that, since u_{i} ∈ k, we have ϕ_{j}(u_{i}) = ϕ_{j}^{1}(u_{i})N_{G}, where N_{G} = ∑_{g∈G} g ∈ ℤ[G]. We obtain

Φ(η) = ξ \frac{1}{#G^{n}} h_{k,S,T} N_{G}^{n} det(ϕ_{j}^{1}(u_{i})) = ξ \frac{1}{#G}h_{k,S,T} N_{G} det(ϕ_{j}^{1}(u_{i})).

Since S contains all ramifying places, the extension K/k is unramified and K is a subfield of the Hilbert class field H_{k} of k. The Frobenius elements for the v_{i} in the extension H_{k}/k restrict to 1 in K/k, and this shows that #G | h_{k,S} | h_{k,S,T}, as noted by Rubin in [39, Prop. 3.1, proof]. Therefore reducing the equality modulo I_{G} gives Conjecture 5.1.

Corollary 5.4. If K = k, then Conjecture 5.1 is true.

We note that the proof of Proposition 5.3 shows that the statement is simply the statement of the analytic class number formula (Proposition 2.2) hit by all possible Φ and reduced modulo I_{G}.

Properties of the conjectures

1 Well-statedness

We first show that the statement of Conjecture 3.5.1 does not depend on the various choices of bases and places. Firstly, it is clear that any change to the basis of U_{k,S,T} will equally affect the signs ξ and u_{1} ∧ ⋯ ∧ u_{n}, so that this does not change the statement of Conjecture 3.5.1.

Let us now consider a reordering of the places v_{1}, …, v_{#S}, so that the new ordering is given by a σ ∈ S_{#S}. It is either the case that all the places are split in K/k, or exactly r_{S} of them are. If all places are split, the conjecture is true (Proposition 5.3). Otherwise, it is necessary that σ restricts to a permutation σ_{1} on {1, …, r_{S}}. Call the restriction of σ to the complement of this set σ_{2}. Then in the congruence of Conjecture 3.5.1, reordering the places by σ induces sign changes by sign(σ_{1}) in Φ(η), sign(σ) in R_{k,S,T} and sign(σ_{2}) in Reg_{G}. Since sign(σ_{1}) = sign(σ)sign(σ_{2}), this does not alter the statement. Therefore Conjecture 3.5.1 is well stated.

We now study the dependence of Conjecture 3.5.1 on the data K/k, S, T, x used in Chapter 3. Note that the dependence of η on x was dealt with after Definition 1.2. Aside from the possibility of changing the sign of η by reordering the places appearing in x (the effect of which on Conjecture 3.5.1 has already been considered), changes to x may replace η with a Galois conjugate. The form of Conjecture 3.5.1 combined with Lemma 2.6.1 shows that this does not alter the statement of the conjecture.

2 Dependence on fields and T

We next show that we may lower the top field in Conjecture 3.5.1.

Proposition 2.1. Let L/K/k be a tower of finite extensions, with L/k and K/k abelian with groups Γ and G = Γ/H respectively. Suppose the only primes in S totally split in K/k are those totally split in L/k. Then, if Conjecture 3.5.1 holds for L/k, it also holds for K/k.

Proof. By the commutative diagram (2.5) for the extension L/K and Proposition 2.3.3, we see that

(∧^{r_{S}}λ_{S_{L}})(η_{K/k}) = (∧^{r_{S}}(i_{L/K} ∘ λ_{S_{K}}))(η_{K/k}) = Θ_{L/k,S,T}^{r_{S}} (0)(∧^{r_{S}}N_{L/K})x,

and hence that η_{K/k} = (∧^{r_{S}}N_{L/K})η_{L/k}.

The inclusion U_{K,S,T} ↪ U_{L,S,T} and the ℤ[G]-module isomorphism

\begin{matrix}ℤ[Γ]^{H} & ⟶^{∼} & ℤ[G] \\ N_{L/K} & ⟼ & 1 \end{matrix}

induce a surjective map

Hom_{ℤ[Γ]}(U_{L,S,T} , ℤ[Γ]) ⟶ Hom_{ℤ[G]}(U_{K,S,T} , ℤ[G])

whereby each ϕ in the second group can be lifted to a ϕ̂ in the first in such a way that the projection of ϕ̂(u) to ℤ[G] is ϕ(N_{L/K}u) for all u ∈ U_{L,S,T}. To see this, we apply the functor Hom_{ℤ[Γ]}(−, ℤ[Γ]) to the short exact sequence

0 ⟶ U_{K,S,T} ⟶ U_{L,S,T} ⟶ U_{L,S,T} /U_{K,S,T} ⟶ 0,

which consists of torsion-free groups by Lemma 2.4.3, and note Corollary 2.6.6, to see that the restriction map Hom_{ℤ[Γ]}(U_{L,S,T} , ℤ[Γ]) → Hom_{ℤ[Γ]}(U_{K,S,T} , ℤ[Γ]) is surjective. Elements ϕ of the domain group fit into the following commutative diagram [39, (16)]:

U_{L,S,T} ⟶^{ϕ} ℤ[Γ]  ℤ[Γ]
N_{L/K} ↓  ^{N_{L/K}} ↓  ↓^{canon.}
U_{K,S,T} ⟶^{ϕ|_{K}} ℤ[Γ]^{H} ⟶^{∼} ℤ[G].

This proves the claim.

Now we take Φ ∈ ∧_{ℤ[G]}^{r_{S}}Hom_{G}(U_{K,S,T} , ℤ[G]) and apply this lifting process componentwise to obtain Φ̂ ∈ ∧_{ℤ[Γ]}^{r_{S}}Hom_{ℤ[Γ]}(U_{L,S,T} , ℤ[Γ]). We have

Φ̂(η_{L/k}) ≡ ξh_{k,S,T} · Reg_{Γ}(˜Φ̂(u)) (mod I_{Γ}^{d+1}),

where u = u_{1} ∧ ⋯ ∧ u_{n}, u_{i} a ℤ-basis of U_{k,S,T}. Passing to the quotient in this congruence, and noting that ˜ Φ̂ = Φ̃, we get

Φ(η_{K/k}) ≡ ξh_{k,S,T} · Reg_{G}(Φ̃(u)) (mod I_{G}^{d+1}),

as required.

We now look at what happens if we enlarge T. Recall from Chapter 2, §4 that it is always sufficient for the condition of Hypothesis 2.4.1 to take T to consist of at most two places, or one place of sufficiently large residue characteristic. The next proposition shows that these cases imply the cases for larger T.

Proposition 2.2. Suppose Conjecture 3.5.1 holds for K/k, S, T, and v is a place of k not in S or T. Set T′ = T ∪ {v}. Then Conjecture 3.5.1 holds for K/k, S, T′.

Proof. The definition of Θ_{K/k,S,T} shows that η_{T′} = η_{T}^{1−N_{v}σ_{v}^{−1}}.

We will adapt Popescu’s argument in [34, Prop. 5.3.1, proof]. Let ϕ′_{1}, …, ϕ′_{r} be elements of Hom_{ℤ[G]}(U_{K,S,T′}, ℤ[G]) and set Φ′ = ϕ′_{1} ∧ ⋯ ∧ ϕ′_{r}. Popescu proves that there exists ϕ_{0} ∈ Hom_{ℤ[G]}(U_{K,S,T′}, ℤ[G]), and (for each i = 1, …, r) ϕ_{i} ∈ Hom_{ℤ[G]}(U_{K,S,T} , ℤ[G]) and α_{i} ∈ ℤ[G], such that ϕ′_{i} = ϕ_{i} + α_{i}ϕ_{0} for all i = 1, …, r. Let δ_{v} = 1 − N_{v}σ_{v}^{−1}; it is then clear that the map δ_{v}ϕ_{0} : x ↦ ϕ_{0}(δ_{v}x) is in Hom_{ℤ[G]}(U_{K,S,T} , ℤ[G]). Popescu shows that

Φ′(η_{T′}) = Ψ(η_{T}),  (4.1)

where Ψ ∈ ∧_{ℤ[G]}^{r_{S}}Hom_{ℤ[G]}(U_{K,S,T} , ℤ[G]) is given by

δ_{v}(ϕ_{1} ∧ ⋯ ∧ ϕ_{r_{S}}) + ∑_{i=1}^{r_{S}} α_{i}ϕ_{1} ∧ ⋯ ∧ ϕ_{i−1} ∧ δ_{v}ϕ_{0} ∧ ϕ_{i+1} ∧ ⋯ ∧ ϕ_{r_{S}}.  (4.2)

Now let u = u_{1} ∧ ⋯ ∧ u_{n} be the wedge of a basis of U_{k,S,T} and u′ = u′_{1} ∧ ⋯ ∧ u′_{n} similarly for U_{k,S,T′}. Adjusting u_{1} by inversion if necessary, we have (U_{k,S,T} : U_{k,S,T′})u = u′ in ∧_{ℤ}^{n}U_{k,S,T}. Apply Ψ̃ to both sides of this equality. We note that (δ_{v}ϕ_{0})^{1}(u′_{i}) = aug(δ_{v})ϕ_{0}^{1}(u′_{i}) since u′_{i} ∈ U_{k,S,T′}. Note also that aug(δ_{v}) = −(N_{v}−1). By the form of (4.2), this shows that

(U_{k,S,T} : U_{k,S,T′})Ψ̃(u) = −(N_{v} − 1)Φ̃′(u′) in ∧_{ℤ}^{d}U_{k,S,T}.

But (U_{k,S,T} : U_{k,S,T′}) divides (N_{v} −1). Since the group in which the equality holds is torsion-free, we may cancel (U_{k,S,T} : U_{k,S,T′}) from both sides. Furthermore, by (2.3),

h_{k,S,T′} = \frac{(N_{v} − 1)}{(U_{k,S,T} : U_{k,S,T′})}h_{k,S,T} .

So we have h_{k,S,T} Ψ̃(u) = −h_{k,S,T′} Φ̃′(u′) in ∧_{ℤ}^{d}U_{k,S,T}. Note that the sign corresponds to the factor (−1)^{#T +1} in (3.4). Applying Reg_{G} to this and using (4.1) gives the result.

3 Changing S

We next look at changing S. We will use:

Lemma 3.1. Let k, S, T be such that U_{k,S,T} is torsion free. Suppose u_{1}, …, u_{n} is a ℤ-basis for U_{k,S,T}. Let v be a place of k not in S or T. Take u′ ∈ U_{k,S∪{v},T} such that v(u′) is minimal positive. Then u_{1}, …, u_{n}, u′ is a basis for U_{k,S∪{v},T}.

Proof. Let u ∈ U_{k,S∪{v},T}. Then there exists a ∈ ℤ such that v(u) = av(u′). Thus u/u′^{a} ∈ U_{k,S,T}, so we see u_{1}, …, u_{n}, u′ generates U_{k,S∪{v},T}. Linear independence follows from considering the valuations at v.

Lemma 3.2. Using the notation of the previous Lemma, we have

h_{k,S∪{v},T} · v(u′) = h_{k,S,T} .

Proof. Write S′ = S ∪ {v} for short. The result follows from the analytic class number formula (Theorem 3.2.1) as follows. If n = #S − 1, we have

ζ_{k,S′,T} (s) ≡ (−1)^{#T +1}h_{k,S′,T} R_{k,S′,T} s^{n+1} (mod s^{n+2}),
ζ_{k,S,T} (s) ≡ (−1)^{#T +1}h_{k,S,T} R_{k,S,T} s^{n} (mod s^{n+1}),

and the leading terms are related by

h_{k,S′,T} R_{k,S′,T} = lim_{s→0} \frac{1 − N_{v}^{−s}}{s} h_{k,S,T} R_{k,S,T} .

Hence h_{k,S′,T} R_{k,S′,T} = (ln N_{v})h_{k,S,T} R_{k,S,T}. On the other hand, the definition of the regulator, and the fact that v(u_{i}) = 0 for i = 1, …, n shows that R_{k,S′,T} = ln |u′|_{v}R_{k,S,T} = v(u′)(ln N_{v})R_{k,S,T}. This gives the result.

Proposition 3.3. Let v be a place of k not in S or T and which splits completely in K/k. Set S′ = S ∪ {v}. If Conjecture 3.5.1 holds for K/k, S′, T, then it holds for K/k, S, T.

Proof. Choose bases as in Lemma 3.1. We order the set S′ = {v, v_{1}, v_{2}, …, v_{n+1}}, and order the basis of U_{k,S′,T} as u′, u_{1}, …, u_{n}. We choose w above v to go into the definition of x′, which corresponds to S′ as x does to S. By [39, Prop. 5.2], we have η_{S} = w̌(η_{S′}), where w̌ ∈ Hom_{ℤ[G]}(U_{K,S′,T} , ℤ[G]) is defined by

w̌(u) = ∑_{g∈G} w(g^{−1}u)g.

Take Φ ∈ ∧_{G}^{r_{S}}Hom_{ℤ[G]}(U_{K,S,T} , ℤ[G]). The inclusion map U_{K,S,T} → U_{K,S′,T} is an injective map of ℤ-torsion-free ℤ[G]-modules with ℤ-torsion-free cokernel, and therefore by Corollary 2.6.6 the map Hom_{ℤ[G]}(U_{K,S′,T} , ℤ[G]) → Hom_{ℤ[G]}(U_{K,S,T} , ℤ[G]) (restriction) is surjective. So we may lift Φ componentwise to an element Φ′ ∈ ∧_{G}^{r}Hom_{ℤ[G]}(U_{K,S′,T} , ℤ[G]). Then

Φ(η_{S}) = (Φ′ ∘ w̌)(η_{S′}) ≡ ξh_{k,S′,T} Reg_{G}(Φ̃( ˜w̌(u′ ∧ u_{1} ∧ ⋯ ∧ u_{n}))) (mod I_{G}^{d+1}),

and ˜ w̌(u′∧u_{1}∧⋯∧u_{n+1}) = w(u′)u_{1}∧⋯∧u_{n}+∑_{i=1}^{n+1}(−1)^{i}w(u_{i})u′∧u_{1}∧⋯∧u_{i−1}∧ u_{i+1} ∧ ⋯ ∧ u_{n}. But u_{1}, …, u_{n} are S-units so this collapses to w(u′)u_{1} ∧ ⋯ ∧ u_{n}. Now because v splits completely in K/k, we have w(u′) = v(u′). Hence

Φ(η_{S}) ≡ ξv(u′)h_{k,S′,T} Reg_{G}(Φ̃(u_{1} ∧ ⋯ ∧ u_{n})) (mod I_{G}^{d+1}),

which by Lemma 3.2, and noting that ξ is unchanged, is what we want.

Proposition 3.4. Suppose Conjecture 3.5.1 holds for K/k, S, T. Let v be a place of k not in S or T and which does not split completely in K, and set S′ = S ∪{v}. Then Conjecture 3.5.1 holds for K/k, S′, T.

Proof. We define u_{n+1} = u′, with u′ as in Lemma 3.1. We note that, because S contains all places ramifying in K/k, v is unramified in K/k. Therefore the Artin symbol at v can be calculated by the formula

f_{v}(x) = (σ_{v})^{v(x)},

for all x in k^{×}. The definition of Θ_{K/k,S,T} shows that η_{S′} = η_{S}^{1−σ_{v}^{−1}}. Take Φ = ϕ_{1} ∧⋯∧ϕ_{r_{S}} ∈ ∧_{ℤ[G]}^{r_{S}}Hom_{ℤ[G]}(U_{K,S′,T} , ℤ[G]), then Conjecture 3.5.1 for S′ asks for:

Φ(η_{S′}) ≡ ξh_{k,S′,T} Reg_{G}(Φ̃(u_{1} ∧ ⋯ ∧ u_{r+d+1})) (mod I_{G}^{d+2}).

We order the elements of S′ as v′_{i}, where v′_{i} = v_{i} for i = 1, …, n, v′_{n+1} = v and v′_{n+2} = v_{n+1}. That is, we insert the new place v before the last place in S. The effect of passing from S to S′ on the real number R defined in §2 is multiplication by v(u′) ln N_{v}, and hence the changes have no effect on ξ.

For the algebraic regulator, we have

Reg_{G}(Φ̃(u_{1} ∧ ⋯ ∧ u_{n+1})) = ∑_{σ∈\left[{n+1 \atop r_{S}}\right]} sign(σ) det (ϕ_{j}^{1}(u_{σ(i)})) det (f_{v_{j}}(u_{σ(r_{S}+i)})).

Let us consider two cases of σ. If n + 1 ∈ {σ(1), …, σ(r_{S})} then in the corresponding term, the column in the second determinant corresponding to v is all zeros, since u_{1}, …, u_{r+d} are S-units. The other possibility is that n+1 = σ(n+1). Then this same column is all zeros apart from the bottom-right entry, which is

f_{v}(u_{n+1}) − 1 = (σ_{v})^{v(u′)} − 1 ≡ v(u′)(σ_{v} − 1) (mod I_{G}^{2}).

Hence Reg_{K/k,S′}(Φ̃(u_{1} ∧ ⋯ ∧ u_{n+1})) = v(u′)(σ_{v} − 1)Reg_{K/k,S}(Φ̃(u_{1} ∧ ⋯ ∧ u_{n})), where in the second expression we may consider Φ as being restricted to U_{K,S,T}. So Conjecture 3.5.1 for S′ now reads, using Lemma 3.2,

(1 − σ_{v}^{−1})Φ(η_{S}) ≡ ξ(σ_{v} − 1)h_{k,S,T} Reg_{G}(Φ̃(u_{1} ∧ ⋯ ∧ u_{r+d})) (mod I_{G}^{d+2}).

Using Corollary 2.6.3, we may replace (σ_{v} − 1) in this congruence by (1 − σ_{v}^{−1}). Therefore if Conjecture 3.5.1 holds for S, it holds for S′.

On the other hand, the proof shows that we have the following, possibly weaker, implication in the other direction.

Proposition 3.5. Suppose Conjecture 3.5.1 holds for K/k, S ∪ {v}, T, where v is a place of k not in S or T. Then we have, for the data K/k, S, T,

(1 − σ_{v}^{−1})Φ(η) ≡ (1 − σ_{v}^{−1})(ξh_{k,S,T} Reg_{G}^{Φ}) (mod I_{G}^{d+1}),

in the notation of Conjecture 3.5.1. That is, we obtain the image of the congruence in the next level of the augmentation filtration under multiplication by (1 − σ_{v}^{−1}).

Proof in quadratic extensions

We use the notation of Chapter 3, assume that Hypothesis 2.4.1 holds, and furthermore that the abelian extension K/k is quadratic, i.e. of degree 2. Thus G is a group of order 2, generated by its nontrivial element which we call τ. The purpose of this chapter is to prove Conjecture 3.5.1 in this case. By Proposition 3.5.2, this also establishes Conjectures 3.4.3 and 3.3.2 – indeed, the calculation of η using quotients of zeta-functions is a central part of Rubin’s proof of his conjecture in this case. Gross also gives a proof of Conjecture 3.3.2 for quadratic extensions in [18, §6] and, while we use ideas from this proof in Lemma 3.4 below, the proof given here is simpler and shorter than that of loc. cit., in that it avoids the technicalities and special cases which must be considered there. This is because the extra flexibility of Conjecture 3.5.1 with respect to increase of S (Propositions 4.3.3 and 4.3.5 above) make it possible to assume the strong hypothesis on a class group used in Lemma 3.4 below.

For comparison, note that §2 below corresponds to the known case “n = 0” of Gross’s conjecture [18, (4.3)], and that in §3 below the sign of the regulator is irrelevant.

We establish the conjecture in §§1-3. Noting Proposition 3.5.3, we may assume that not all places in S split in K/k, and so there are exactly r_{S} places which are split and all the rest are inert or ramified. Hence #S_{K} = n + r_{S} + 1. Throughout this chapter we abbreviate N = N_{K/k} for the norm map, so Nx = x^{1+τ} for x ∈ K.

1 Cohomology of U_{K,S,T}

Let u_{1}, …, u_{n+r_{S}} be a basis of U_{K,S,T} such that u_{1}, …, u_{n} is a basis of U_{k,S,T}, which is possible by Lemma 2.4.3. The relevant structure of this basis is closely related to the Galois cohomology of the G-module U_{K,S,T}. We will use Galois cohomology, and Tate’s modified cohomology for finite groups, as described in [41].

Lemma 1.1. Without assuming Hypothesis 2.4.1, we have

\frac{#Ĥ^{0}(G, U_{K,S,T})}{#H^{1}(G, U_{K,S,T})} = 2^{d}.

Proof. Note the left-hand side is the Herbrand quotient h(U_{K,S,T}) of the ℤ[G]- module U_{K,S,T} in the sense of [41, Ch. VIII, §4]. The composite isomorphism of ℚ[G]-modules (Proposition 2.2.1)

ℚU_{K,S,T} ≅ ℚX_{S_{K}} ≅ ℚ[G]^{r_{S}} ⊕ ℚ^{d}

implies that there is a ℤ[G]-homomorphism from U_{K,S,T} to ℤ[G]^{r_{S}} ⊕ℤ^{d} with finite kernel and cokernel. Then h(U_{K,S,T}) = h(ℤ[G])^{r_{S}}h(ℤ)^{d} = 2^{d} as required.

The following result appears in [39, Thm. 3.5, proof].

Lemma 1.2. If H^{1}(G, U_{K,S,T}) ≠ 0 then we can assume N_{K/k}u_{n+1} = 1.

Proof. Take u ∈ U_{K,S,T} representing a nontrivial element of H^{1}(G, U_{K,S,T}) = U_{K,S,T}^{−} /U_{K,S,T}^{1−τ}, where U_{K,S,T}^{−} is the set of (S,T)-units of norm 1. Write u = ϵ ∏_{i} u_{n+i}^{α_{i}}, where ϵ ∈ U_{k,S,T}, and write ϵ_{i} for the norm u_{n+i}^{1+τ} ∈ U_{k,S,T}. Then u_{n+i}^{1−τ} = u_{n+i}^{2}ϵ_{i}^{−1}. Therefore we can assume each α_{i} is 0 or 1. But they cannot all be 0, since U_{k,S,T} ∩ U_{K,S,T}^{−} = {1}. Hence u can go into a basis of U_{K,S,T}.

Lemma 1.3. If H^{1}(G, U_{K,S,T}) = 0, we can assume that u_{i} = N_{K/k}u_{n+i} for i = 1, …, r_{S}.

Proof. Write Nu_{n+j} = ∏_{i=1}^{n} u_{i}^{α_{ji}} for j = 1, …, r_{S}. We may perform the following operations on the r_{S} × n matrix (α_{ji}): elementary column operations, which correspond to swapping and multiplying the units in the basis of U_{k,S,T}, and elementary row operations, which correspond to swapping and multiplying the units u_{n+1}, …, u_{n+r_{S}}.

Thus we can put (α_{ji}) into diagonal form with integers a_{1}, …, a_{r_{S}} on the diagonal. Now suppose some a_{i} is even, so Nu_{n+i} = ϵ^{2} for some ϵ in U_{k,S,T}. Then u_{n+i}ϵ^{−1} is a unit in K − k with norm 1. But the group U_{K,S,T}^{1−τ} ⊆ U_{k,S,T} U_{K,S,T}^{2} . So u_{n+i}ϵ^{−1} represents a nontrivial element of

U_{K,S,T}^{−} /U_{K,S,T}^{1−τ} = H^{1}(G, U_{K,S,T}).

This is a contradiction to our assumption. We conclude that each a_{i} is odd. Now replacing u_{n+i} by u_{n+i}u_{i}^{−[a_{i}/2]} gives us the result.

In our situation, noting that S contains a place not splitting in K, [39, Lem. 3.4] states:

Lemma 1.4 (Rubin). 1. h_{k,S,T} | h_{K,S,T}

2. #H^{1}(G, U_{K,S,T}) | h_{k,S,T} 3. If Ĥ^{0}(G, U_{K,S,T}) and H^{1}(G, U_{K,S,T}) are trivial then h_{k,S,T} ≡ h_{K,S,T} /h_{k,S,T}

(mod 2).

We write ϵ_{−} = u_{n+1} ∧ ⋯ ∧ u_{n+r_{S}}. In proving his conjecture for quadratic extensions [39, proof of Prop. 3.5], Rubin uses the analytic class number formula to express η in terms of this element; we make extensive use of his formulas, which are quoted below. We note that these results may also be obtained using the calculation (8.10) below.

2 The case d = 0

We assume d = 0, that is, #S = r_{S} + 1 and r_{S} = n, and show that the congruence in Conjecture 3.5.1 holds. We require:

Φ(η) ≡ ξh_{k,S,T} | \begin{matrix}ϕ_{1}^{1}(u_{1}) & ⋯ & ϕ_{n}^{1}(u_{1}) \\ ⋮ & ... & ⋮ \\ ϕ_{1}^{1}(u_{n}) & ⋯ & ϕ_{n}^{1}(u_{n}) \end{matrix} | (mod I_{G}).  (5.1)

Note that this is an equality in ℤ[G]/I_{G} ≅ ℤ. We observe that Lemma 1.1 gives that #Ĥ^{0}(G, U_{K,S,T}) = #H^{1}(G, U_{K,S,T}) =: H.

As in [39, proof of Prop. 3.5], we choose ū_{1}, …, ū_{n} ∈ U_{K,S,T} which are such that Nū_{1}, …, Nū_{n} is a basis for NU_{K,S,T}, and set ε_{+} = ū_{1} ∧ ⋯ ∧ ū_{n}. The index [U_{k,S,T} : NU_{K,S,T}] = H, and so, inverting ū_{1} if necessary to get the correct sign, we have

(∧^{n}N)ε_{+} = H · u_{1} ∧ ⋯ ∧ u_{n} ∈ ∧_{ℤ[G]}^{n}U_{k,S,T} .  (5.2)

Observe that, by (2.5), λ_{S_{K}}(ε_{+})e_{χ_{0}} = λ_{S}((∧^{n}N)ε_{+}) under the natural map

ℚY_{S} ⟶^{×e_{χ_{0}}} ℚY_{S_{K}}. Therefore, by the analytic class number formula (Chapter 2, §2), we have e_{χ_{0}}η = \frac{ξ}{H} h_{k,S,T} e_{χ_{0}}ε_{+}. Rubin [39, proof of Prop. 3.5] calculates the part of η corresponding to the nontrivial character, and using this gives

η = \frac{ξ}{H} e_{χ_{0}}h_{k,S,T} ε_{+} ± \frac{h_{K,S,T}}{h_{k,S,T}} e_{χ}ε_{−}.  (5.3)

Assume first that H = 1. Then we may assume u_{n+i}^{1+τ} = u_{i} for i = 1, …, n by Lemma 1.3, so that ε_{+} = ε_{−} (inverting u_{n+1} if necessary to maintain (5.2)). Hence

η = ξ (h_{k,S,T} \frac{(1 + τ)}{2} ± \frac{h_{K,S,T}}{h_{k,S,T}} \frac{(1 − τ)}{2}) ϵ_{−}.

Lemma 1.4 shows the ℚ[G]-factor here lies in ℤ[G], and we note its augmentation is ξh_{k,S,T}. Furthermore

ϕ(u_{n+i}) = ϕ^{1}(u_{n+i}) − τϕ^{1}(u_{n+i}) + τϕ^{1}(u_{i}) ≡ ϕ^{1}(u_{i}) (mod I_{G}).

Hence (5.1) is satisfied in this case.

Now assume H ≠ 1. Then by Lemma 1.2, we may assume N_{K/k}u_{n+1} = 1. A similar argument to the proof of Lemma 1.2, but working with Ĥ^{0}(G, U_{K,S,T}), shows that we may also assume ū_{1} ∈ U_{k,S,T}. Therefore e_{χ}ε_{+} = e_{χ_{0}}ε_{−} = 0. Hence (5.3) becomes

η = \frac{ξ}{H} h_{k,S,T} ε_{+} ± \frac{h_{K,S,T}}{h_{k,S,T}} ε_{−}.

Note that Lemma 1.4 again gives that the ℚ[G]-factors here are in ℤ[G]. By the assumption on Nu_{n+1}, we have

ϕ_{j}(u_{n+1}) ≡ ϕ_{j}^{1}(N_{K/k}u_{n+1}) = 0 (mod I_{G}),

so Φ(ε_{−}) ≡ 0 (mod I_{G}). Now we note that Φ(ε_{+}) ≡ Φ̃((∧^{n}N)ε_{+}) (mod I_{G}), and so using (5.2) completes the proof of (5.1) in this case.

This verifies Conjecture 3.5.1 in the case d = 0.

3 The case d > 0

Now we assume d > 0, ie. #S > r_{S} + 1. For d > 0, I_{G}^{d}/I_{G}^{d+1} is a group of order 2, so the congruence statement in Conjecture 3.5.1 only concerns in which power of the augmentation ideal the terms lie. We have (1 − τ)^{m} = 2^{m−1}(1 − τ) for m > 0. Note that the map

\begin{matrix}I_{G}^{d}/I_{G}^{d+1} & ⟶ & I_{G}^{d+1}/I_{G}^{d+2} \\ x & ⟼ & (1 − τ)x \end{matrix}

is a bijection. We have the following freedom to increase S:

Lemma 3.1. Let [K : k] = 2. Assume d = #S − r_{S} − 1 > 0. Let v be a place of k not in S or T, and set S′ = S ∪ {v}. Then if Conjecture 3.5.1 holds for K/k, S′, T, it also holds for K/k, S, T.

Proof. The place v is either split or inert in K/k. If it is split, the result follows from Proposition 4.3.3. If v is inert, we have σ_{v} = τ and Proposition 4.3.5, combined with the stability of the augmentation quotients, shows the result.

Rubin shows in [39, Prop. 3.5, proof] that for the case K/k quadratic and d > 0 we have

η = ±2^{d} \frac{h_{K,S,T}}{h_{k,S,T}} \frac{(1 − τ)}{2}ϵ_{−}.  (5.4)

We note that 2^{d−1}(1 − τ) ∈ I_{G}^{d}.

It turns out that we only need to consider the congruence statement under the following cohomological assumption^{1}.

1In fact it is easy to show that both sides of the congruence vanish if this assumption is not satisfied.

Lemma 3.2. Suppose d > 0 and H^{1}(G, U_{K,S,T}) = 0. Then the congruence of Conjecture 3.5.1 is implied by the following statement:

2^{d−1}(1 − τ)\frac{h_{K,S,T}}{h_{k,S,T}} ≡ h_{k,S,T} Reg_{G}(u_{r_{S}+1} ∧ ⋯ ∧ u_{n}) (mod I_{G}^{d+1}).  (5.5)

Proof. We apply Lemma 1.3 to assume that u_{i} = Nu_{n+i} for i = 1, … r_{S}. Then ϕ_{j}(u_{n+i}) ≡ ϕ_{j}^{1}(u_{i}) (mod I_{G}). Thus

Φ(η) ≡ 2^{d−1}(1 − τ)\frac{h_{K,S,T}}{h_{k,S,T}} | \begin{matrix}ϕ_{1}^{1}(u_{1}) & ⋯ & ϕ_{r_{S}}^{1}(u_{1}) \\ ⋮ & ... & ⋮ \\ ϕ_{1}^{1}(u_{r_{S}}) & ⋯ & ϕ_{r_{S}}^{1}(u_{r_{S}}) \end{matrix} | (mod I_{G}^{d+1})

For the right-hand side of the congruence in Conjecture 3.5.1, we note that if any u_{i}, 1 ≤ i ≤ r_{S}, appears in the argument of Reg_{G}, then the corresponding term is 0 (because u_{i} is a norm from K, and therefore in the kernel of all the local reciprocity maps). So the right-hand side collapses to a single term:

ξh_{k,S,T} | \begin{matrix}ϕ_{1}^{1}(u_{1}) & ⋯ & ϕ_{r_{S}}^{1}(u_{1}) \\ ⋮ & ... & ⋮ \\ ϕ_{1}^{1}(u_{r_{S}}) & ⋯ & ϕ_{r_{S}}^{1}(u_{r_{S}}) \end{matrix} | Reg_{G}(u_{r_{S}+1} ∧ ⋯ ∧ u_{n}).

This gives the result since multiplication by ξ = ±1 is the identity on I_{G}^{d}/I_{G}^{d+1}.

Next we identify a condition for the non-vanishing of the regulator. We need an auxiliary lemma. For Tate’s theory of the group cohomology of finite cyclic groups, we refer to [41, Ch. VIII, §4].

Lemma 3.3. Suppose h_{k,S} = 1. Then there is an isomorphism

\frac{U_{k,S} ∩ NK^{×}}{NU_{K,S}} ≅ A_{K,S}^{G}.

Proof. (cf. [18, p191]) We have the exact sequence (from (2.1)):

0 ⟶ K^{×}/U_{K,S} ⟶ ⨁_{𝔓∉S_{K}} 𝔓^{ℤ} ⟶ A_{K,S} ⟶ 0.

Considering the decompositions of primes shows that H^{1}(G, ⨁_{𝔓∉S_{K}} 𝔓^{ℤ}) = 0. Then taking cohomology gives an exact sequence

0 ⟶ H^{0}(G, K^{×}/U_{K,S}) ⟶ ⨁_{𝔭∉S_{k}} 𝔭^{ℤ} ⟶^{0} A_{K,S}^{G} ⟶ H^{1}(G, K^{×}/U_{K,S}) ⟶ 0.

Observe that k^{×}/U_{k,S} injects into H^{0}(G, K^{×}/U_{K,S}), and (2.1) for k shows that it surjects onto ⨁_{𝔭∉S_{k}} 𝔭^{ℤ}, because the S-class group of k is trivial. This is why the map marked 0 is zero. Therefore we have A_{K,S}^{G} ≅ H^{1}(G, K^{×}/U_{K,S}).

On the other hand, applying Hilbert’s Theorem 90 [41, Ch. X, Prop. 2] and Tate cohomology to the short exact sequence

0 ⟶ U_{K,S} ⟶ K^{×} ⟶ K^{×}/U_{K,S} ⟶ 0

gives an exact sequence

0 ⟶ H^{1}(G, K^{×}/U_{K,S}) ⟶ Ĥ^{0}(G, U_{K,S}) ⟶ Ĥ^{0}(G, K^{×}).

Therefore

H^{1}(G, K^{×}/U_{K,S}) ≅ ker(U_{k,S}/NU_{K,S} ⟶ k^{×}/NK^{×}) = U_{k,S} ∩ NK^{×}/NU_{K,S}.

This completes the proof.

Lemma 3.4. Suppose h_{K,S,T} = 1. Then Reg_{G}(u_{r_{S}+1} ∧ ⋯ ∧ u_{n}) ≡ / 0 (mod I_{G}^{d+1}).

Proof. By (2.3) and Lemma 1.4 (part (i)), we have that h_{k,S} = 1, and then [36, Cor. 2] shows that H^{1}(G, U_{K,S}) = 0. Therefore Ĥ^{0}(G, U_{K,S}) is a 2-torsion group with 2^{d} elements by Lemma 1.1, and hence is isomorphic to (ℤ/2ℤ)^{d}.

Similarly to [18, p191], we define a homomorphism

f : U_{k,S} ⟶ G^{S−S_{1}} ≅ (ℤ/2ℤ)^{d+1}

by the local reciprocity maps f_{v}. Then, by the product formula,

im f ⊆ V := {(g_{v})_{v∈S−S_{1}} : ∏ g_{v} = 1} ≅ (ℤ/2ℤ)^{d}.

Now u ∈ U_{k,S} is in ker f if and only if u is a local norm at all the places in S − S_{1}, and we note it is automatically a norm at all other places. Since K/k is cyclic, u is a local norm everywhere if and only if it is a global norm. So ker f = U_{k,S} ∩NK^{×}. Since h_{K,S,T} = 1, we have A_{K,S}^{G} = 0 and so Lemma 3.3 shows that ker f = NU_{K,S}. On the other hand, Ĥ^{0}(G, U_{K,S}) = U_{k,S}/NU_{K,S} ≅ (ℤ/2ℤ)^{d}, so we have f(U_{k,S}) = V.

We note that the form of our regulator and the choice of our unit basis show that the non-vanishing of our regulator

Reg_{G}(u_{r_{S}+1} ∧ ⋯ ∧ u_{n}) ≡ / 0 (mod I_{G}^{d+1})

is equivalent to saying f(U_{k,S,T}) = V.

The reduction map U_{K,S} ⟶ ∏_{𝔓∈T_{K}} 𝔽_{𝔓}^{×} is surjective, by (2.2) for K and the assumption that h_{K,S,T} = 1. Also, the norm in an extension of finite fields is surjective. Hence ker f = NU_{K,S} surjects onto ∏_{𝔭∈T} 𝔽_{𝔭}^{×}. The sequence (2.2) for k shows that this latter is isomorphic by the reduction map to U_{k,S}/U_{k,S,T}. So every element of U_{k,S} can be written as the product of something in ker f by something in U_{k,S,T}, which gives our result.

Now consider A_{K,S,T}, the S_{K} ray class group modulo T_{K}. Choose a set S′_{K} of primes of O_{K,S} coprime to T_{K} whose classes generate this group. Set S′ to be the set of places of k lying below these. If S contains S′, then h_{K,S,T} = 1. On the other hand:

Lemma 3.5. If h_{K,S,T} = 1, then the congruence of Conjecture 3.5.1 holds.

Proof. By Lemma 1.4 (part (i)) we have that h_{k,S,T} = 1, which also shows by part (ii) of the same lemma that H^{1}(G, U_{K,S,T}) = 0. So by Lemma 3.2 it is sufficient to show (5.5) holds. Lemma 3.4 shows that the right-hand side of (5.5) is not zero. On the other hand h_{K,S,T} /h_{k,S,T} = 1 so the left-hand side is not zero either.

Now by Lemma 3.1, we may assume S′ ⊆ S. Then h_{K,S,T} = 1, so Lemma 3.5 implies that the congruence of Conjecture 3.5.1 holds.

Therefore we have established the following theorem in all cases.

Theorem 3.6. Suppose [K : k] = 2. Then Conjecture 3.5.1 holds.

Explicit Stark units

Let k be a global field which has at most one archimedean place. This means that k is either the rational field ℚ, an imaginary quadratic field, or a global function field. In the first two cases we let ∞ denote the unique archimedean place, and for k a function field we choose a place freely and call it ∞. If K/k is an abelian extension of k in which ∞ splits completely, we have explicit analytic constructions of the Stark units η ∈ U_{K,S,T}. It is sufficient to consider the ray class fields totally split at ∞, since given K and S, K is contained in such a field H(𝔪), where 𝔪 is an ideal of the ∞-integers of k supported above the primes in S. The Stark unit for K/k is then the norm of the Stark unit for H(𝔪)/k, as in the proof of Proposition 4.2.1. This Proposition also shows that it is sufficient to prove Conjecture 3.5.1 for H(𝔪)/k to know it for K/k. In this chapter we describe the ray class fields H(𝔪) and the special elements which lie in them.

1 Systems of special units

We fix a global field k and a place ∞ of k such that all other places of k are finite. Then the ring of ∞-integers O := O_{k,∞} (as defined in Chapter 2, §1) is a Dedekind domain whose prime ideals correspond to the other places of k.

The idèle group J_{k} of k is the subgroup of elements i = (i_{v})_{v} ∈ ∏_{v} k_{v}^{×} (where v runs over all places of k) such that v(i_{v}) = 0 for almost all v, considered as a topological group with respect to the topology defined in [23, Ch. VII, §§1,3]. The open subgroups of J_{k} containing the diagonally-embedded subgroup k^{×} correspond to the finite abelian extensions of k via global class field theory (see e.g. [23, Ch. IX–XI]). In particular, for a nonzero ideal 𝔪 of O_{k,∞}, define S = Supp(𝔪) ∪ {∞} and

U(𝔪) = {i ∈ J_{k} : v(i_{v}) = 0 if v ∉ S
v(i_{v} − 1) ≥ v(𝔪) if v ≠ ∞ and v | 𝔪},

the group of idèles which are local units away from Supp(𝔪) and ∞ and congruent to 1 modulo 𝔪. The group k^{×}U(𝔪) is then an open subgroup of J_{k} of finite index, and the corresponding field H(𝔪) is the ray class field modulo 𝔪 totally split at ∞. There is a canonical identification

Gal(H(𝔪)/k) ≅ J_{k}/k^{×}U(𝔪) ≅ A_{k,S,𝔪}.

(One reference for the second isomorphism here is [23, Ch. VII, §4]). Therefore the exact sequence (2.2) with S = {∞} and 𝔱 = 𝔪 becomes

0 ⟶ μ_{k,𝔪} ⟶ μ_{k} ⟶ (O/𝔪)^{×} ⟶ Gal(H(𝔪)/k) ⟶ A_{k,∞} ⟶ 0.  (6.1)

Note that the class group A_{k,∞} of O is the Galois group of the subextension H/k, where H is the Hilbert class field totally split at ∞ (i.e. the maximal unramified extension of k totally split at ∞).

If 𝔞 and 𝔟 are coprime ideals of O, the restriction map Gal(H(𝔞𝔟)/H) → Gal(H(𝔞)/H) fits into the exact commutative diagram

0  0  0
↓  ↓  ↓
0 ⟶ μ_{k,𝔞}/μ_{k,𝔞𝔟} ⟶ (O/𝔟)^{×} ⟶ Gal(H(𝔞𝔟)/H(𝔞)) ⟶ 0
↓  ↓  ↓
0 ⟶ μ_{k}/μ_{k,𝔞𝔟} ⟶ (O/𝔞𝔟)^{×} ⟶ Gal(H(𝔞𝔟)/H) ⟶ 0
↓  ↓  ↓
0 ⟶ μ_{k}/μ_{k,𝔞} ⟶ (O/𝔞)^{×} ⟶ Gal(H(𝔞)/H) ⟶ 0
↓  ↓  ↓
0  0  0.

(6.2)

We now introduce some notation which will make it easier to talk about primepower factors of ideals. If 𝔪 is an ideal of O, it may be written in the form

𝔪 = 𝔭_{1}^{e_{1}}𝔭_{2}^{e_{2}} ⋯ 𝔭_{n}^{e_{n}}  (6.3)

for primes 𝔭_{i} of O and positive integers e_{i}. Write B = {1, …, n} for the set of indices. If 1 ≤ i ≤ n, we write 𝔪_{i} = 𝔭_{i}^{e_{i}}, and 𝔪_{I} = ∏_{i∈I} 𝔪_{i}, S_{I} = {p_{i} : i ∈ I} relative to a subset I ⊆ B. We abbreviate w_{I} = w_{k,𝔪_{I}}, the number of roots of unity of k congruent to 1 modulo 𝔪_{I}.

Let us fix 𝔪 and also a prime 𝔱 of O which is coprime to 3𝔪w_{H(𝔪)}.

Proposition 1.1. There is a system of elements λ̂_{I} of the maximal abelian extension of k, indexed by nonempty subsets I ⊆ {1, …, n}, such that for each prime 𝔱 of O coprime to 3𝔪w_{H(𝔪)}, the elements λ̂_{I}^{δ_{𝔱}} =: λ_{I} satisfy the following properties.

1. λ_{I} ∈ U_{H(𝔪_{I}),S_{I},𝔱}.

2. Distribution relations.

(a) If i ∉ I, then N_{H(𝔪_{I∪{i}})/H(𝔪_{I})}(λ_{I∪{i}}^{w_{I}/w_{I∪{i}}}) = λ_{I}^{1−σ_{𝔭_{i}}^{−1}}.

(b) N_{H(𝔪_{i})/H}(λ_{i}^{w_{k}/w_{i}}) = α_{i}, where α_{i}O_{H} = 𝔭_{i}^{1−N_{𝔱}}O_{H} (O_{H} denoting the integral closure of O in H).

3. L-function evaluation. There is a place ∞′ of H(𝔪) above ∞ such that we

have

w_{I}Θ′_{H(𝔪_{I})/k,S_{I},𝔱}(0)∞′ = λ_{∞′}(λ_{I}).

It follows from part 3 that we have

Proposition 1.2.

η_{H(𝔪)/k,S,𝔱}^{w_{I}} = λ_{I}.

Proposition 1.3. To verify property 3 of Proposition 1.1, it is sufficient to check that there is a place ∞′ of H(𝔪_{I})/k such that

w_{I}(ζ′_{H(𝔪_{I})/k,S_{I}} (0, σ) − N_{𝔱}ζ′_{H(𝔪_{I})/k,S_{I}} (0, σσ_{𝔱}^{−1})) = − ln |λ_{I}^{σ}|_{∞′}

for all σ ∈ Gal(H(𝔪_{I})/k).

Proof. Expanding the left-hand side of the equality in property 3 using (2.10) shows that the desired equality is

w_{I}( ∑_{σ} (ζ′_{H(𝔪_{I})/k,S_{I}} (0, σ) − N_{𝔱}ζ′_{H(𝔪_{I})/k,S_{I}} (0, σσ_{𝔱}^{−1}))σ^{−1})∞′

= ( − ∑_{σ} ln |λ_{I}^{σ}|_{∞′}σ^{−1})∞′.

Remark 1.4. 1. The w-factors in the distribution relations are the sizes of the kernels arising in the top row of (6.2). This shows that the relations appear more neatly when we consider norms with respect to the groups (O/𝔟)^{×}, which surject onto the relevant Galois groups. These formulations appear in Lemma 2.1 below.

2. The w-factor in the L-function evaluation property can be accounted for as follows. The partial zeta-function is defined for ℜ(s) > 1 by

ζ_{H(𝔪)/k,S}(s, σ) = ∑_{I∈𝔠}N_{I}^{−s},

where I runs through all ideals of O in the congruence class 𝔠 corresponding to the automorphism σ by global class field theory. If I, I′ are two elements of 𝔠 then I′/I = (x) for some x ∈ k^{×}, x ≡ 1 (mod 𝔪). Thus fixing one element I of 𝔠 allows us to define a similar function

z_{𝔪}(s, 𝔠) = ∑_{\begin{matrix}x∈I^{−1}: \\ x≡1(mod 𝔪) \end{matrix}}N_{xI}^{−s}.

If μ injects into (O/𝔪)^{×} then the x and the I are in bijection, so the functions are the same. In general there are w_{𝔪} of the x to each one of the I. So z_{𝔪}(s, 𝔠) = w_{𝔪}ζ_{𝔪}(s, σ). The more natural infinite-series form of the function z_{𝔪}(s, 𝔠) means it is amenable to analytic methods of evaluation (see [45, §2] or [21, p236]).

3. It is interesting to contrast the approach of this chapter with that of Gross and Rosen in [19]. They consider a way of constructing special units corresponding to L-function values via Fourier analysis. The resulting units differ from those defined above due to our use of the “smoothing” factors δ_{𝔱} to ensure that our units lie in the desired ray class field; by contrast, in loc. cit. this is achieved by raising to powers, with the required power possibly depending on the conductor (see [19, (7.14)]). This dependence corresponds to less satisfactory distribution relations. It is likely that their approach could be adapted to provide a more unified construction than the piecemeal approach of §§2–5 for our particular choice of elements.

The verification of the axioms will give a proof of the refined abelian Stark conjecture:

Corollary 1.5. Let K/k be a finite abelian extension, where k is a field of the type discussed above. Let S be a set as in Chapter 2 such that r_{S} = 1 and ∞ ∈ S splits completely in K. Then Conjecture 3.4.2 is true.

Proof. We first remark that it is sufficient to verify the conjecture for almost all T = {𝔱}, and also sufficient to verify it for an abelian extension L/k satisfying the conditions of Proposition 4.2.1 relative to K/k. Since #S ≥ r_{S} + 1 = 2, there are n ≥ 1 finite (i.e. different from ∞) primes 𝔭_{1}, …, 𝔭_{n} in S. By global class field theory and the finiteness of the group of roots of unity of k, there exist integers e_{1}, …, e_{n} such that, if 𝔪 is defined by (6.3), then K ⊆ H(𝔪) and w_{i} = 1 for all i. Taking L = H(𝔪), we see that for almost all 𝔱 we have the membership relation η_{L/k,S,𝔱} ∈ U_{L,S,𝔱}, as required.

Remark 1.6. Explicit units are used to prove the refined abelian Stark conjecture in [46, Prop. 3.9] (k = ℚ), [45] (k imaginary quadratic) and [21] (k a function field). We note that the use of Rubin’s reformulation of Stark’s conjecture produces a unified approach to this method of proof, and that in particular the use of elliptic units is somewhat neater than in [45].

In the rest of this chapter we prove Proposition 1.1 for the three different kinds of k.

2 Cyclotomic fields

We begin with the most familiar case, that of k = ℚ, which is also in many ways the simplest because ℚ has class number one. We have H = k = ℚ, O = ℤ, and write 𝔪 = (m), 𝔭_{i} = (p_{i}) and 𝔱 = (t) in the notation of §1, for positive integers m, p_{i}, t. Here t is required to be an odd prime not dividing m. We define positive integers m_{i}, m_{I} for each I ⊆ {1, …, n} ∋ i by (m_{i}) = 𝔪_{i}, (m_{I}) = 𝔪_{I}. For a positive integer r, let ζ_{r} = e^{2πi/r}. Define

λ̂_{I} = (1 − ζ_{m_{I}}) λ_{I} = λ̂_{I}^{δ_{t}}.

Then λ̂_{I} generates the narrow ray class field associated to the divisor m_{I}∞, with Galois group over ℚ equal to (ℤ/m_{I}Z)^{×} ≅ I_{ℚ,S}/P_{ℚ,S,m_{I}∞}, where P_{ℚ,S,m_{I}∞} is the group of principal fractional ideals with a positive generator congruent to 1 mod m_{I}. The generator of the decomposition group at infinity is complex conjugation, corresponding to −1 ∈ (ℤ/m_{I}ℤ)^{×}. This acts on λ̂_{I} by multiplication by a 2m-th root of unity. Since the 2m-th roots of unity are annihilated by δ_{t} (using Lemma 2.4.2(2) to see that Hypothesis 2.4.1 is satisfied by ℚ(ζ_{m})/ℚ), this means that complex conjugation acts trivially on λ_{I}, which is therefore real and so lies in the ray class field H(m_{I}). By the definition of the Frobenius map, λ_{I} satisfies the congruence condition at primes above 𝔱. So we have established Proposition 1.1 (1) for the cyclotomic elements.

Consider the factorisation in the polynomial ring ℂ[X]

X^{m_{i}} − 1 = ∏_{j=1}^{m_{i}}(X − ζ_{m_{i}}^{j}) = [ ∏_{j∈(ℤ/m_{i}ℤ)^{×}}(X − ζ_{m_{i}}^{j})](X^{m_{i}/p_{i}} − 1).  (6.4)

Let i ∉ I. Choose integers a and b with am_{i} + bm_{I} = 1, so that ζ_{m_{I∪{i}}} = ζ_{m_{i}}^{b}ζ_{m_{I}}^{a}. Substitute X = ζ_{m_{I}}^{−a} into (6.4) and multiply through by ζ_{m_{I}}^{am_{i}}:

1 − ζ_{m_{I}}^{am_{i}} = [ ∏_{j∈(ℤ/m_{i}ℤ)^{×}}(1 − ζ_{m_{I}}^{a} ζ_{m_{i}}^{j})](1 − ζ_{m_{I}}^{am_{i}/p_{i}})

Note that ζ_{I}^{σ_{(p_{i})}} = ζ_{I}^{p_{i}} and m_{i} = p_{i}^{e_{i}}. Applying δ_{t}, we obtain

λ_{m_{I}}^{1−σ_{(p_{i})}^{−1}} = ∏_{j∈(ℤ/m_{i}ℤ)^{×}} λ_{I∪{i}}^{σ_{(r_{j})}},

where for each j, we pick an integer r_{j} congruent to 1 modulo m_{I} and to j modulo m_{i}. Now take 𝔞 = (m_{I}) and 𝔟 = (m_{i}) in (6.2). We note that w_{i}/w_{I∪{i}} is the size of the kernel of the map (ℤ/m_{i}ℤ)^{×} ↠ Gal(H(m_{I∪{i}})/H(m_{I})), and so our calculations verify assertion (2a) of Proposition 1.1.

Now from (6.4) we obtain

∑_{n=0}^{p_{i}−1} X^{m_{i}n/p_{i}} = ∏_{j∈(ℤ/m_{i}ℤ)^{×}}(X − ζ_{m_{i}}^{j}).

Setting X = 1 in this equality gives p_{i} = ∏_{j∈(ℤ/m_{i}ℤ)^{×}}(1 − ζ_{m_{i}}^{j}). Applying δ_{t}, we

get p_{i}^{1−N_{t}} = N_{H(m_{i})/H}λ_{i}^{w_{k}/w_{i}}. This shows assertion (2b).

As for L-function evaluation, [45, (3)] shows that we have, for σ the element of Gal(H(m_{I})/ℚ) corresponding to a ∈ ℤ → (ℤ/mℤ)^{×}/{±1}

w_{ℚ,m_{I}}ζ′_{H(m_{I})/ℚ,S_{I}}(0, σ) = z′_{m_{I}} (0, a) = − ln | 1 − e^{2πia/m_{I}} |,  (6.5)

where

z_{m_{I}} (s, a) = ∑_{n≡a mod m_{I}} |n|^{−s}

is the function mentioned in Remark 1.4(2) above. Therefore, by Proposition 1.3, we obtain

w_{I}Θ′_{H(m_{I})/k,S_{I},𝔱}(0)∞′ = λ_{∞′}(λ_{I}),

where ∞′ is the place of H(m_{I}) induced by our choice of ζ_{m_{I}} ∈ ℂ. This verifies Proposition 1.1 part (3).

3 Function fields

The special units in this case will come from division points on sign-normalised rank-1 Drinfeld modules. This theory was developed by Hayes (see [20] and [21]) and is neatly exposited in [15, §7].

Let k be a global function field with finite constant field 𝔽. The field H(𝔪) in this context is that denoted k(𝔪)^{+} in [15, Defn. 7.5.10]. As in the cyclotomic case, analytically-constructed elements λ̂_{I}, which generate “narrow” ray class fields, will map under δ_{𝔱} to give the system λ_{I}. We may choose 𝔱 to be any prime of O not dividing 𝔪.

The chosen place ∞ of k is, of course, finite, and so it has a residue field 𝔽_{∞} which is a finite extension of 𝔽. This field 𝔽_{∞} is also the field of constants of the completion k_{∞} of k at ∞. We fix a sign-function sgn: k_{∞}^{×} → 𝔽_{∞}^{×} as in [15, Defn. 7.2.1]. An element x of k_{∞}^{×} will be called positive if sgn(x) = 1; an idèle i ∈ J_{k} will be called positive if sgn(i_{∞}) = 1. By definition, a Drinfeld module with coefficients in ℂ_{∞}, the completion of the algebraic closure of k_{∞}, is an 𝔽-linear homomorphism

\begin{aligned}ρ : & O & ⟶ & ℂ_{∞}[X]_{∘} \\ & x & ⟼ & ρ_{x} \end{aligned}

with values in the composition ring of additive polynomials. We will consider a sgn-normalised rank-1 Drinfeld module (also called a Hayes-module) ρ in the sense of [15, Defn. 7.2.14], so that the leading coefficient of ρ_{x} is sgn(x)^{σ} for all x ∈ O-{0} and some automorphism σ ∈ Gal(𝔽_{∞}/𝔽). The coefficients of any such ρ generate an abelian extension H^{+}/k, called the normalising field with respect to sgn. This field corresponds via class field theory to the idèle subgroup k^{×}U(1)^{+}, where U(1)^{+} is the subgroup of positive idèles in U(1), or equivalently to the ray class group I_{k,∞}/P_{k,∞,∞}, where P_{k,∞,∞} is the subgroup of P_{k,∞} of principal fractional ideals with a positive generator. The extension is ramified only at ∞, and has field of constants 𝔽_{∞}. The decomposition group of ∞ is equal to the inertia group, and isomorphic to 𝔽_{∞}^{×}/𝔽^{×}.

If 𝔞 is a proper nonzero ideal of O, then the 𝔞-torsion points ρ[𝔞] = {x ∈ ℂ_{∞} : ρ_{a}(x) = 0 for all a ∈ 𝔞} of ρ are naturally made via the action of ρ into an Omodule isomorphic to O/𝔞. The elements of ρ[𝔞] which are generators of this cyclic O-module are called primitive 𝔞-division points for ρ. Set S = {∞}∪Supp(𝔞). Any primitive 𝔞-division point λ̂ ∈ ρ[𝔞] is such that the extension H^{+}(λ̂)/k is abelian, and corresponds via class field theory to idèle subgroup k^{×}U(𝔞)^{+} (where U(𝔞)^{+} is the subgroup of positive idèles in U(𝔞)), or accordingly to I_{k,S}/P_{k,S,𝔞∞}, where P_{k,S,𝔞∞} is the group of principal fractional ideals of O with a positive generator x satisfying x ≡ 1 (mod 𝔞). An 𝔞-division point for ρ is thus algebraic over k. Its valuations are zero outside the support of 𝔞 and ∞ [21, Thm. 4.17]. The decomposition group of ∞ in the extension H^{+}(λ̂)/k is equal to the inertia group and isomorphic to 𝔽_{∞}^{×}. The field H(𝔞) is obtained as the fixed field with respect to this group. The constant field of both H(𝔞) and H^{+}(λ̂) is 𝔽_{∞}.

We extend the definition of ρ from elements of O to ideals as follows: if 𝔟 is a nonzero ideal of O, then ρ_{𝔟} is the monic polynomial with roots ρ[𝔟]. This gives an explicit class field theory; for 𝔟 coprime to 𝔞, the Artin symbol for 𝔟 acts on H^{+}(λ̂) via

λ̂^{σ_{𝔟}} = ρ_{𝔟}(λ̂),  (6.6)

(see [15, Prop. 7.5.4 (3)]). The polynomials satisfy an important relation. If 𝔟 and 𝔠 are nonzero ideals of O, then we have

ρ_{𝔟𝔠} = (𝔠 ∗ ρ)_{𝔟}ρ_{𝔠} = (𝔟 ∗ ρ)_{𝔠}ρ_{𝔟},  (6.7)

where 𝔟 ∗ ρ is the sgn-normalised rank one Drinfeld module obtained by letting the class of 𝔟 in I_{k,∞}/P_{k,∞,∞} act on the coefficients of ρ by the reciprocity map.

Note that the fields H^{+}(λ̂) and H(𝔞) are independent of the choices of the sgn-normalised rank one Drinfeld module ρ and the 𝔞-division point λ̂. Indeed, all such Drinfeld modules can be obtained from ρ by the action “∗” of ideals on H^{+}, and the union of their set of 𝔞-division points is permuted transitively by the Galois group of H^{+}(λ̂)/k.

We now proceed to define the system of elements λ_{I} required by Proposition 1.1. Fix a sgn-normalised rank one Drinfeld module ρ and a primitive 𝔪-division point λ̂. Then for I ⊆ {1, …, n} we set

λ̂_{I} = ρ_{I′}(λ̂).

Then (𝔪_{I′} ∗ ρ)_{𝔪_{I}}(λ̂_{I}) = ρ_{𝔪}(λ̂) by (6.7), and this is zero. Hence λ̂_{I} is an 𝔪_{I}- division point for the sgn-normalised rank one Drinfeld module 𝔪_{I′} ∗ ρ. If a ∈ O then (𝔪_{I′} ∗ ρ)_{(a)}(λ̂_{I}) = ρ_{a𝔪_{I′}}(λ̂) and this is zero if and only if a ∈ 𝔪_{I}. Therefore λ̂_{I} is a primitive 𝔪_{I}-division point for 𝔪_{I′} ∗ ρ. The extension H^{+}(λ̂_{I})/H^{+} has Galois group isomorphic to Γ_{𝔪_{I}} = (O/𝔪_{I})^{×}; another such field H^{+}(λ̂_{J}) is linearly disjoint over H^{+} if J is disjoint from I.

There is another characterisation of λ̂_{I}. Using these elements to choose isomorphisms (𝔪_{I′} ∗ ρ)[𝔪_{I}] ≅ O/𝔪_{I} gives us commutative diagrams

ρ[𝔪] ⟶^{ρ_{𝔪_{I′}}} (𝔪_{I′} ∗ ρ)[𝔪_{I}]
≅↓  ≅↓
O/𝔪 ⟶ O/𝔪_{I},

where the map at the bottom is the natural projection. Since O/𝔪 ≅ ∏_{i} O/𝔪_{i}, to know λ̂_{I} it is sufficient to know all its images under ρ_{𝔪_{I}/𝔪_{i}} for each i ∈ I. We

have by definition that ρ_{𝔪/𝔪_{i}}(λ̂) = λ̂_{i}, and so, by (6.6), we have λ̂ = ∑_{i=1}^{n} λ̂_{i}^{σ_{𝔪/𝔪_{i}}^{−1}} . Applying ρ_{𝔪_{I′}} to this gives

λ̂_{I} = ∑_{i∈I} λ̂_{i}^{σ_{𝔪_{I} /𝔪_{i}}^{−1}} .  (6.8)

Recall that Gal(H^{+}(λ̂_{I})/H(𝔪_{I})) is the decomposition group of ∞ in the extension H^{+}(λ̂_{I})/k. If σ is in this group, then σ acts on λ̂_{I} by multiplication by a root of unity in 𝔽_{∞}^{×} ⊆ H(𝔪_{I}) (this follows from [21, Cor. 4.14]). Since δ_{𝔱} annihilates these roots of unity (noting that there are no more roots of unity in H^{+}(λ̂_{I}) than in H(𝔪_{I}), since both have constant field 𝔽_{∞}), the element λ_{I} := λ̂_{I}^{δ_{𝔱}} is fixed by σ and so lies in H(𝔪_{I}).

As in the cyclotomic case, the congruence conditions on λ_{I} above 𝔱 follow directly from the definition, and since λ_{I} satisfy the valuation conditions, we have that Proposition 1.1 (1) holds for these elements. We proceed to check the other properties required by the Proposition.

Let i ∉ I ⊆ {1, …, n}, and set ρ′ = 𝔪_{I′-{i}} ∗ ρ. Then

ρ′_{𝔪_{i}}(X) = ∏_{λ̂:ρ′_{𝔪_{i}}(λ̂)=0}(X − λ̂) = ∏_{σ∈Gal(H^{+}(λ̂_{𝔪_{i}})/H^{+})}(X − λ̃^{σ})ρ′_{𝔪_{i}/𝔭_{i}}(X),  (6.9)

where λ̃ is a primitive 𝔪_{i}-division point for ρ′. Since ρ′_{𝔪_{i}}(λ̂_{i}^{σ_{𝔪_{I}}^{−1}}) = σ_{𝔪_{I}}^{−1}((𝔪/𝔪_{i} ∗

ρ)_{𝔪_{i}}(λ̂_{i})) = 0, we see that λ̃ = −λ̂_{i}^{σ_{𝔪_{I}}^{−1}} (which it is easy to show is primitive) will

do. Assuming I nonempty, we now set X = λ̂_{I}^{σ_{𝔪_{i}}^{−1}}, so that X − λ̃ = λ̂_{I∪{i}} by the characterisation (6.8). Then for any nonzero ideal 𝔞 coprime to 𝔪_{I} we have

ρ′_{𝔞}(X) = σ_{𝔪_{i}}^{−1}((𝔪_{i} ∗ ρ′)_{𝔞}(λ̂_{I})) = σ_{𝔪_{i}}^{−1}((𝔪_{I′} ∗ ρ)_{𝔞}(λ̂_{I})) = σ_{𝔪_{i}}^{−1}σ_{𝔞}(λ̂_{I}).

Hence we obtain from (6.9):

λ̂_{I}^{1−σ_{𝔭_{i}}^{−1}} = ∏_{𝔣} λ̂_{I∪{i}}^{σ_{𝔣}},

where 𝔣 runs through P_{k,∞,𝔪_{I∪{i}}∞}/P_{k,∞,𝔪_{I}∞}. Therefore we have

λ̂_{I}^{1−σ_{𝔭_{i}}^{−1}} = N_{H^{+}(λ̂_{I∪{i}})/H^{+}(λ̂_{I})}λ̂_{I∪{i}}.

We now apply δ_{𝔱} to this equation.

Lemma 3.1. Provided I is not empty, the restriction map

Gal(H^{+}(λ̂_{I∪{i}})/H^{+}(λ̂_{I})) → Gal(H(𝔪_{I∪{i}})/H(𝔪_{I}))

is an isomorphism.

Proof. Note that Gal(H^{+}(λ̂_{𝔞})/H^{+}) ≅ (O/𝔞)^{×} for any primitive 𝔞 division point λ̂_{𝔞} of ρ (see [15, Cor. 7.5.6]), so the domain of the restriction map is naturally isomorphic to (O/𝔪_{i})^{×}. In (6.2), set 𝔞 = 𝔪_{I} and 𝔟 = 𝔪_{i}, then the required restriction map is (O/𝔟)^{×} → Gal(H(𝔞𝔟)/H(𝔞)). Lemma 4.2 shows that w_{𝔞} = 1, hence the kernel of this map is trivial, and so it is an isomorphism.

We obtain assertion (2a) of Proposition 1.1.

We turn now to the proof of assertion (2b). Note the Principal Ideal Theorem which states that for any ideal 𝔞 of O, we have 𝔞O_{H^{+}} = Dρ_{𝔞}O_{H^{+}} ([15, Thm. 7.6.2]). Here D represents taking the coefficient of the linear term of the polynomial. Using this and comparing linear terms in (6.9) for I = ∅ gives the equality Dρ′_{𝔪_{i}} = ±N_{H^{+}(λ̂_{i})/H^{+}}(λ̂_{i})Dρ′_{𝔪_{i}/𝔭_{i}}. We have 𝔪_{i}O_{H^{+}} = Dρ′_{𝔪_{i}}O_{H^{+}} contained in 𝔪_{i}/𝔭_{i}O_{H^{+}} = Dρ′_{𝔪_{i}/𝔭_{i}}O_{H^{+}}, and so Dρ′_{𝔪_{i}/𝔭_{i}} divides Dρ′_{𝔪_{i}} and

N_{H^{+}(λ̂_{i})/H^{+}}(λ̂_{i})O_{H^{+}} = 𝔭_{i}O_{H^{+}}.  (6.10)

From (6.2) we obtain the exact sequence

0 ⟶ μ_{k} ⟶ Gal(H^{+}(λ̂_{i})/H^{+}) ⟶ Gal(H(𝔪_{i})/H) ⟶ 0.

Therefore applying δ_{𝔱} to (6.10) gives

N_{H(𝔪_{i})/H}λ_{i}^{w_{k}}O_{H} = 𝔭_{i}^{1−N_{𝔱}}O_{H}.

This shows Proposition 1.1(2b).

Finally, we turn to assertion (3). Let ∞′ be the place of H^{+}(λ̂) whereby λ̂ corresponds to the element ξ(𝔪)e_{𝔪}(1) ∈ ℂ_{∞} as in [21, (5.5)] where e_{𝔪} is the exponential function associated to the lattice 𝔪 and ξ(𝔪) ∈ ℂ_{∞} is such that ξ(𝔪)𝔪 is a “special” lattice in the sense of [21, §5]. Then λ̂_{I} = ρ_{𝔪_{I′}}(λ̂) corresponds by ∞′ to the element ξ(𝔪_{I})e_{𝔪_{I}}(1) by [21, (5.3)].^{1}

Hayes shows ([21, §6])

ζ′_{H(𝔪_{I})/k,S_{I}} (0, σ) = − ln |ρ_{𝔠}(λ̂_{I})|_{∞′},

1The factor ξ(𝔪_{I}) is only defined up to a root of unity in 𝔽_{∞}^{×}. This choice does not affect the role in L-function evaluation. We simply choose a compatible system of elements ξ(𝔪_{I}).

where 𝔠 is an ideal in a narrow ideal class corresponding to σ. Therefore assertion (3) is satisfied.

We note that it is also possible to phrase the results of §2 in a similar language of polynomials. The operation X ⊞ Y = X + Y − XY makes ℂ-{1} into a ℤmodule. Then n ∈ ℤ acts via X ↦ X^{⊞n} = f_{n}(X) where f_{n}(X) = (1 − (1 − X)^{n}). The cyclotomic elements 1 − ζ_{m_{I}} are primitive m_{I} division points of this module. They satisfy the identities

1−ζ_{m_{I}} = f_{m_{I′}} (1−ζ_{m}); (1−ζ_{m_{I}})^{σ_{a}} = f_{a}(1−ζ_{m_{I}}); 1−ζ_{m_{I}} = ⊞_{i∈I}(1−ζ_{m_{i}})^{σ_{m_{I} /m_{i}}^{−1}} ,

in close analogy with the Drinfeld module case. Finally, in each case the special elements λ_{I} ∈ H(𝔪_{I}) are obtained by “𝔱-modifying” the division points.

4 Robert’s functions

We now move on to the case where k is an imaginary quadratic field. The system of elliptic units we will use is essentially that used by Bley in [2]. One property which we require of these elements is not clear from loc. cit., namely the congruence condition modulo 𝔱 in Proposition 1.1 (1). To establish this, we will interpret Bley’s units as another case of the 𝔱-modifying process we have seen in the previous two sections – that is, we will express the units as λ̂^{δ_{𝔱}} for some λ̂ in the maximal abelian extension of k.

We first review the construction from [38] used in [2]. Let L ⊆ \underline{L} be an inclusion of (full) ℤ-lattices in ℂ with index coprime to 6. Let ω = \binom{w_{1}}{w_{2}} be a column vector whose entries form a ℤ-basis for L, positively oriented in the sense that w_{2}/w_{1} has positive imaginary part, and \underline{ω} = (\begin{matrix}\underline{w} & _{1} \\ \underline{w} & _{2} \end{matrix}) be a positively oriented basis for \underline{L}. There is a matrix B ∈ M_{2}^{+}(ℤ) (meaning 2×2 matrices over ℤ with positive determinant) such that ω = B\underline{ω}. Robert defines ([38, p237])

F̃(t; L; \underline{L}) := ρ(ω, \underline{ω})^{−1} \frac{φ(t; ω)^{det B}}{φ(t; \underline{ω})} .  (6.11)

Here ρ(ω, \underline{ω}) is a twelfth root of unity depending only on B, and φ = φ_{Rob} is the φ-function of the lattice [38, (4)]:

φ_{Rob}(t; w_{1}, w_{2}) = iq^{\frac{1}{2} B_{2}(u)}α^{\frac{u−1}{2}} (1 − αq^{u}) ∏_{n=1}^{∞}(1 − αq^{n+u})(1 − α^{−1}q^{n−u}), (6.12)

where α = e^{2πiv}, q = e^{2πi \frac{w_{2}}{w_{1}}} and B_{2} is the second Bernoulli polynomial B_{2}(X) = X^{2} − X + \frac{1}{6}. Many authors define similar functions, and so it is useful to have a comparison between them. If w_{1}, w_{2} and t are as above, and we write t = uw_{2}+vw_{1} for u, v ∈ ℝ, then we have

φ_{Rob}(t; w_{1}, w_{2}) = ϕ_{St}(u, v, w_{2}/w_{1}) = −iφ_{Sch}(u\frac{w_{2}}{w_{1}} +v|_{1}^{w_{2}/w_{1}}) = −iφ_{Ouk}(t; w_{2}, w_{1}),

(6.13) where φ_{Rob} denotes Robert’s function φ defined in [38], ϕ_{St} is Stark’s function ϕ in [45], φ_{Sch} is the function denoted φ in Schertz’s paper [40], and φ_{Ouk} is Oukhaba’s function φ in [29]. The comparison is by comparing the q-expansions [45, (10)] and [40, (4.3)] to (6.12), and for φ_{Ouk} by [29, §2.2].

To calculate the twelfth root of unity ρ(B), one uses a factor appearing in transformation formulas for the Dedekind η function η(τ) = e^{2πiτ/24} ∏_{n=1}^{∞}(1 − e^{2πinτ}), where τ is an element of the upper half-plane. This function η is a certain 24th root of the usual discriminant function for the lattice ℤ + τℤ. Let a matrix B̃ of positive determinant act on the upper half plane via the usual formula

B̃ = ( {a \atop c} {b \atop d} ) ⇒ B̃τ = \frac{aτ + b}{cτ + d}.

There is a function ε : SL_{2}(ℤ) → μ_{24} which is given by the rule

η(B̃τ) = ε(B̃)\sqrt{cτ + d} η(τ)  (6.14)

for B̃ ∈ SL_{2}(ℤ), where the square root is chosen to have argument θ satisfying −\frac{π}{2} ≤ θ < \frac{π}{2} , and so maps the upper half-plane to the first quadrant. Squaring (6.14), we see that B̃ ↦ ε(B̃)^{2} is a homomorphism SL_{2}(ℤ) → μ_{12} (note that this homomorphism is denoted ω in [45] and ρ^{(2)} in [38]). The group SL_{2}(ℤ) is generated by the matrices ( {1 \atop 0} {1 \atop 1} ) and ( {0 \atop 1} {−1 \atop 0} ) (see eg. [23, Ch. 2, §1, Thm. 1]) and ε^{2} takes values e^{2πi/12} and −i on these matrices, respectively, by [23, Ch. 18, §5, Thm. 6]. This is sufficient to define ε^{2}; in fact, an explicit formula for ε is given in [48, §38, (15)].

The function ε^{2} also gives the transformation formula for the various φ functions in (6.13). The formula [40, (4.2)] states, for B̃ ∈ SL_{2}(ℤ), that

φ_{Sch}((u v)\binom{B̃z}{1}|_{1}^{B̃z}) = ε(B̃)^{2}φ_{Sch}((u v)B̃\binom{z}{1}|_{1}^{z}).  (6.15)

To translate this into a statement for the function φ_{Rob}, we set z = \underline{w}_{2}/\underline{w}_{1} and note that

ω = B\underline{ω}, i.e. \binom{w_{1}}{w_{2}} = B(\begin{matrix}\underline{w} & 1 \\ \underline{w} & 2 \end{matrix}), implies that \frac{w_{2}}{w_{1}} = B̃ \begin{matrix}\underline{w} & 2 \\ \underline{w} & 1 \end{matrix} ,  (6.16)

where B and B̃ are related by

B = ( {d \atop b} {c \atop a} ) ⟺ B̃ = ( {a \atop c} {b \atop d} ) = ( {0 \atop 1} {1 \atop 0} ) B ( {0 \atop 1} {1 \atop 0} ) .  (6.17)

This operation B ↦ B̃ is an automorphism of SL_{2}(ℤ) of order 2, as can be seen by its expression in (6.17) as the restriction of conjugation in GL_{2}(ℤ) by a (selfinverse) matrix. It then follows from (6.15) that

φ_{Rob}(t; B\underline{ω}) = ε(B̃)^{2}φ_{Rob}(t; \underline{ω}).  (6.18)

Since one property of ρ(B) in its definition in [38, §4, Théorèm principal] is that the function (6.11) depends only on the lattices L and \underline{L} and not the choice of bases, choosing L = \underline{L} and using (6.18) shows that we must have ρ(B) = ε(B̃)^{2} for B ∈ SL_{2}(ℤ).

It is natural to ask how ε(B̃)^{2} is related to ε(B)^{2}.

Lemma 4.1. For any B ∈ SL_{2}(ℤ), we have

ε(B̃)^{2} = ε(B)^{−2}.

Proof. Since B̃ ↦ B and ε^{2} are homomorphisms, it is sufficient to check this on generators of SL_{2}(ℤ).

We have

for B̃ = ( {0 \atop 1} {−1 \atop 0} ) , ε(B̃)^{2} = −i; ε(B)^{2} = ε(B̃^{3})^{2} = \frac{1}{−i},

and so it works for this generator. Then

for B̃ = ( {1 \atop 0} {1 \atop 1} ) , ε(B̃)^{2} = e^{2πi/12}; ε(B)^{2} = ε (B̃ ( {0 \atop 1} {−1 \atop 0} ) B̃)^{2} = e^{−2πi/12},

as required.

Hence we have ρ(B) = ε(B)^{−2} for B ∈ SL_{2}(ℤ). We note that this is not stated correctly in [38, (16)] and [38, §11, Rem. 2], since Robert’s ρ^{(2)}(B) is equal to ε(B)^{2} and not its inverse. Possibly Robert has made an error due to confusion between [38, (10)] and the similar equation before [38, (9)] – cf. (6.16) above. Another correction in this regard is that in [38, (11)], “β(ω̄, ω)” should read “β(ω, ω̃)”.

Two further properties of ρ given in [38] suffice to give a formula for it (in terms of ε) for all elements of M_{2}^{+}(ℤ)_{(6)}. Firstly, if B, D ∈ M_{2}^{+}(ℤ)_{(6)}, then [38, (18)] shows that

ρ(BD) = ρ(B)^{det D}ρ(D)  (6.19)

namely, ρ is a 1-cocycle of M_{2}^{+}(ℤ)_{(6)} with values in μ_{12}, where matrices act on roots of unity via their determinants. Secondly, for positive integers c, d coprime to 6, we have by [38, §10, Fait 3]

ρ ( {c \atop 0} {0 \atop d} ) = (−1)^{\frac{d−1}{2}} .

Therefore we obtain the following, which is [38, §11, Rem. 2]:

Lemma 4.2. If B ∈ M_{2}^{+}(ℤ)_{(6)} is expressed in the form

B = E ( {b_{1} \atop 0} {0 \atop b_{2}} ) F,  (6.20)

with E, F ∈ SL_{2}(ℤ), then

ρ(B) = ε(E)^{−b_{1}b_{2}}(−1)^{\frac{b_{2}−1}{2}} ε(F)^{−2}.

We note that by the theory of matrix diagonalisation, any B can be expressed in the form (6.20).

5 Elliptic units

We now construct the system of elliptic units from special values of φ_{Rob}, following [2]. Let k be an imaginary quadratic field, and choose an embedding k → ℂ. If 𝔞 is an ideal of the ring of integers O of k, then 𝔞 is a lattice in ℂ with complex multiplication by O (i.e. an O-module). We define f_{𝔞} to be the smallest positive rational integer contained in 𝔞.

Let L = 𝔞, and choose a positively-oriented basis ω = (w_{1}, w_{2}) for L. By [29, Prop. 2(i)], the complex number iφ_{Rob}(1; ω) lies in the ray class field H(12f_{𝔞}^{2}). One may check that i ∈ H(12f_{𝔞}^{2}) (using, for example, [37, Lem. 5]), and so φ_{Rob}(1; ω) ∈ H(12f_{𝔞}^{2}).

Choosing an ideal 𝔠 of k coprime to 6𝔞, we may ask for the effect of the automorphism corresponding to 𝔠 on φ_{Rob}(1; ω). Choose a positively-oriented basis \underline{ω} = (\underline{w}_{1}, \underline{w}_{2}) for 𝔠^{−1}𝔞, so that ω = B\underline{ω} for some matrix B ∈ M_{2}^{+}(ℤ)_{(6)} with det B = N𝔭.

Proposition 5.1. If 𝔠 = 𝔭 is a prime ideal, N𝔭 = p is a prime and p is unramified in k (i.e. p splits in k) then

φ_{Rob}(1; ω)^{σ_{𝔭}} = ρ(B)φ_{Rob}(1; \underline{ω}).

Proof. This is a consequence of a theorem of Stark [45, Thm. 3], and the main part of this proof will be to relate Robert’s notation to Stark’s using the transformation formulas of §4.

We have by (6.13) that φ_{Rob}(1; w_{1}, w_{2}) = ϕ_{St}(u, v, w_{2}/w_{1}), where 1 = uw_{2} + vw_{1} with u, v ∈ ℚ. The denominators of u and v can be taken to be equal to f_{𝔞}, and so the function ϕ_{St}(u, v, z) is a modular function in z of level 12f_{𝔞}^{2} by [45, (19)], and the coefficients of its q-expansions lie in the cyclotomic field ℚ(ζ_{12f_{𝔞}^{2}}). [45, Thm. 3] then gives

ϕ_{St}(u, v, w_{2}/w_{1})^{σ_{𝔭}} = [ϕ_{St}(u, v, −) ∘ B̃](\underline{w}_{2}/\underline{w}_{1}),

where we note that, if B_{St} denotes the matrix called B in the statement of [45, Thm. 3], then pB_{St}^{−1} = B̃. The “∘” denotes a certain action of matrices on modular forms of level N = 12f_{𝔞}^{2} with coefficients in ℚ(ζ_{12f_{𝔞}^{2}}), which is given explicitly for particular matrices in [45, (23)], as

(f ∘ A)(z) = f(Az) for A ∈ SL_{2}(ℤ) ; (f ∘ A)(z) = f^{σ_{d}}(z) if A = ( {1 \atop 0} {0 \atop d} ),

where σ_{d} acts on the coefficients of f via ζ_{N} ↦ ζ_{N}^{d}.

We write

B̃ = E ( {1 \atop 0} {0 \atop p} ) F,

as in (6.20). Then

ϕ_{St}(u, v, w_{2}/w_{1})^{σ_{𝔭}} = [ϕ_{St}(u, v, −) ∘ B̃] (\begin{matrix}\underline{w} & 2 \\ \underline{w} & 1 \end{matrix} )
= [ϕ_{St}(u, v, −) ∘ E ∘ ( {1 \atop 0} {0 \atop p} ) ∘ F ] (\begin{matrix}\underline{w} & 2 \\ \underline{w} & 1 \end{matrix} ) .

We have [ϕ_{St}(u, v, −) ∘ E](z) = ϕ_{St}(u, v, Ez) = ε(E)^{2}ϕ_{St}((u v)E, z) by (6.15). Therefore

ϕ_{St}(u, v, w_{2}/w_{1})^{σ_{𝔭}} = [(ε(E)^{2}ϕ_{St}((u v)E, −) ∘ ( {1 \atop 0} {0 \atop p} )] (F \begin{matrix}\underline{w} & 2 \\ \underline{w} & 1 \end{matrix} )
= [(ε(E)^{2}ϕ_{St}((u v)E, −))^{σ_{p}}] (F \begin{matrix}\underline{w} & 2 \\ \underline{w} & 1 \end{matrix} ) .  (6.21)

Now we note that (6.13) and (6.12) give

−iφ_{St}(u′, v′, z) = q^{\frac{1}{2} B_{2}(u′)}α^{\frac{u′−1}{2}} (1 − αq^{u′}) ∏_{n=1}^{∞}(1 − αq^{n+u′})(1 − α^{−1}q^{n−u′}),

where α = e^{2πiv′} and q = e^{2πiz}. The effect of σ_{p} on this expression is α ↦ α^{p}, or in other words, v′ gets multiplied by p, and so we have

−i^{p}ϕ_{St}((u v)E, −)^{σ_{p}} = −iϕ_{St} ((u v)E ( {1 \atop 0} {0 \atop p} ) , −) .

Therefore (6.21) becomes

ϕ_{St}(u, v, w_{2}/w_{1})^{σ_{𝔭}} = ε(E)^{2p}ε(F)^{2}i^{1−p}ϕ_{St}((u v)B̃, \underline{w}_{2}/\underline{w}_{1}).

Calculating now ρ(B), we have

B = Ẽ ( {p \atop 0} {0 \atop 1} ) F̃ = (ẼT^{3}) ( {1 \atop 0} {0 \atop p} ) (T F̃), where T = ( {0 \atop 1} {−1 \atop 0} )

and so ρ(B) = ρ(ẼT^{3})^{p}ρ(T F̃) = ε(E)^{2p}ε(T)^{−6p}ε(T)^{−2}ε(F)^{2}. Since ε(T)^{2} = −i, this gives ρ(B) = i^{1−p}ε(E)^{2p}ε(F)^{2}, and so

ϕ_{St}(u, v, w_{2}/w_{1})^{σ_{𝔭}} = ρ(B)ϕ_{St}((u v)B̃, \underline{w}_{2}/\underline{w}_{1}).

Now

(1) = (u v)\binom{w_{2}}{w_{1}} = (u v)B̃(\begin{matrix}\underline{w} & 2 \\ \underline{w} & 1 \end{matrix})

and so by (6.13)

φ_{Rob}(1; w_{1}, w_{2})^{σ_{𝔭}} = ρ(B)φ_{Rob}(1; \underline{w}_{1}, \underline{w}_{2}).

as required.

Corollary 5.2. For any 𝔠 coprime to 6𝔞, we have

φ_{Rob}(1; ω)^{σ_{𝔠}} = ρ(B)φ_{Rob}(1; \underline{ω}).

Proof. We may write 𝔠 = λ𝔭, where λ ∈ k^{×} satisfies λ ≡ 1 (mod 12f_{𝔞}^{2}) and 𝔭 satisfies the hypotheses of Proposition 5.1. That is, we may choose a 𝔭 representing the class of 𝔠 in the ray class group corresponding to the ideal (12f_{𝔞}^{2}). This is possible by Čebotarev’s density theorem. We write λ = \frac{α}{β} with α, β ∈ O, α ≡ β (mod 12f_{𝔞}^{2}) and coprime to 12f_{𝔞}^{2}.

Let \underline{ω}′ = (\underline{w}′_{1}, \underline{w}′_{2}) be the basis for 𝔭^{−1}𝔞 given by \underline{ω}′ = λ\underline{ω} = (λ\underline{w}_{1}, λ\underline{w}_{2}). Let B′ ∈ M_{2}^{+}(ℤ)_{(6)} be the matrix given by ω = B′\underline{ω}′. Then, since σ_{𝔠} = σ_{𝔭} on H(12f_{𝔞}^{2}), by the result of Proposition 5.1 it is sufficient to prove

ρ(B)φ_{Rob}(1; \underline{w}_{1}, \underline{w}_{2}) = ρ(B′)φ_{Rob}(1; \underline{w}′_{1}, \underline{w}′_{2}).

Firstly we claim that ρ(B) = ρ(B′). We have ω = B\underline{ω} = B′\underline{ω}′. Since α\underline{ω} = β\underline{ω}′, this means that Bβ\underline{ω} = B′α\underline{ω}. Let us write θ_{x} for the unique integer 2 × 2 matrix given by the equation x\underline{ω} = θ_{x}\underline{ω}, where x ∈ O. Then we have Bθ_{β} = B′θ_{α}, and hence by (6.19), ρ(B)^{Nα}ρ(θ_{α}) = ρ(B′)^{Nβ}ρ(θ_{β}). Now β = α + 12f_{𝔞}^{2}x for some x ∈ O, and so θ_{α} ≡ θ_{β} (mod 12). Combining Lemma 4.2 with the explicit formulas for ε given in [48, §38, (15)] shows that ρ(B) only depends upon the congruence classes of the entries of B modulo 12. Therefore ρ(θ_{α}) = ρ(θ_{β}), and we conclude that ρ(B)^{Nα} = ρ(B′)^{Nβ} and, since Nα ≡ Nβ (mod 12) and these are coprime to 12, finally that ρ(B) = ρ(B′).

It remains to show that

φ_{Rob}(1; \underline{w}_{1}, \underline{w}_{2}) = φ_{Rob}(1; \underline{w}′_{1}, \underline{w}′_{2}).

Since φ_{Rob}(y; y\underline{w}_{1}, y\underline{w}_{2}) = φ_{Rob}(1; \underline{w}_{1}, \underline{w}_{2}) for any y ∈ ℂ^{×} (as is clear from the definition (6.12)), this is equivalent to

φ_{Rob}(λ; \underline{w}′_{1}, \underline{w}′_{2}) = φ_{Rob}(1; \underline{w}′_{1}, \underline{w}′_{2}).  (6.22)

We now use [40, (4.1)], which states that, for u, v ∈ ℚ such that t = u\underline{w}′_{2} + v\underline{w}′_{1}, and a, b ∈ ℤ, we have the transformation formula

φ_{Rob}(t + a\underline{w}′_{2} + b\underline{w}′_{1}; \underline{w}′_{1}, \underline{w}′_{2}) = (−1)^{ab+a+b}e^{πi(av−bu)}φ_{Rob}(t; \underline{w}′_{1}, \underline{w}′_{2}). (6.23)

Writing 1 = u\underline{w}′_{2} + v\underline{w}′_{1}, and remembering that ℤ\underline{w}′_{1} + ℤ\underline{w}′_{2} = 𝔭^{−1}𝔞, we have that f_{𝔞}u ∈ ℤ and f_{𝔞}v ∈ ℤ. Furthermore, λ − 1 ∈ 𝔭^{−1}12f_{𝔞}^{2}O and so, writing λ − 1 = a\underline{w}′_{2} + b\underline{w}′_{1}, we have that a and b are integers divisible by 12f_{𝔞}. Therefore setting t = 1 in (6.23) shows that (6.22) holds.

We now proceed to define a system of elliptic units, following [2]. We fix an ideal 𝔪 of O expressed as in (6.3). Let 𝔱 be any prime ideal coprime to 6𝔪 such that δ_{𝔱} annihilates μ_{H(𝔪)}. Bley [2, §2] defines

ψ(z; L, 𝔱) = F̃(z; L; 𝔱^{−1}L) = ρ(ω, \underline{ω})^{−1} \frac{φ_{Rob}(z; ω)^{N_{𝔱}}}{φ_{Rob}(z; \underline{ω})} ,

where F̃ is the function defined above in (6.11), ω and \underline{ω} are vectors of positivelyoriented bases of L and 𝔱^{−1}L respectively. We choose L = 𝔪_{I} for each I ⊆ {1, …, n}, and set λ̂_{I} = φ_{Rob}(1, ω) ∈ H(12f_{𝔞}^{2}). We set

λ_{I} := φ_{Rob}(1; ω)^{1−σ_{𝔱}^{−1}N_{𝔱}} = ψ(1; 𝔪_{I}, 𝔱)^{−σ_{𝔱}^{−1}},  (6.24)

using Corollary 5.2 for the last equality.

We now check the statements of Proposition 1.1 for these elements. The statement that λ_{I} ∈ H(𝔪_{I}) follows from [2, Prop. 2.2(2)]. That it is an 𝔪_{I}-unit is [loc. cit., Thm. 2.4] (and this also establishes property 2b of Proposition 1.1). The congruence property with respect to 𝔱 follows from the expression (6.24), and so we have established property 1. Property 2a follows directly from [2, Thm. 2.3].

Finally, we check the L-function evaluation property, which will follow from the Kronecker limit formula. The expression of this formula in [29, (2.4)] is easily seen to imply (via the comparison (6.13) of Oukhaba’s notation to Robert’s)

ζ′_{𝔪_{I}}(0, σ) = \frac{−1}{12w_{I}f_{𝔪_{I}}} ln | (λ̂^{12f_{𝔪_{I}}})^{σ} |^{2},

where σ ∈ Gal(H(𝔪_{I})/k). Let ∞′ be the infinite place of H(𝔪_{I}) ⊆ ℂ induced by the identity on ℂ. Then we have

w_{I}(ζ′_{𝔪_{I}}(0, σ) − N_{𝔱}ζ′_{𝔪_{I}}(0, σσ_{𝔱}^{−1})) = \frac{−1}{12f_{𝔪_{I}}} ln |(λ_{I}^{σ})^{12f_{𝔪_{I}}} |_{∞′},

which gives Proposition 1.1 part (3).

Evidence from special units

We now use the systems of special units defined in Chapter 6 to find evidence for Conjecture 3.5.1. The proof will be combinatorial, based on the distribution relations of Proposition 6.1.1. The local reciprocity symbols appearing in Conjecture 3.5.1 must be related to the global symbols appearing in the Proposition by means of local class field theory. We obtain the statement of Conjecture 3.5.1 multiplied through by an integer b, which may be taken to be w_{k} if O_{k,∞} has class number one or a certain natural sequence of Galois groups is split.

1 Minors of matrices of zero row-sum

This section contains combinatorial preliminaries.

Given an inclusion of nonempty finite sets J ⊆ I, we define an arborescence on I with root J to be a directed graph on the vertices I which contains no loops and is such that every vertex in J has out-degree 0 and every other vertex has out-degree 1. Therefore an arborescence is a finite union of disjoint directed trees (“rooted branchings”), one each with root j for each element j ∈ J, so that each element of I-J is connected to an element of J. We say that there is a unique arborescence T_{∅} on the empty set and no arborescences on non-empty sets with empty root.

Fix a finite ordered set B, and for each pair (i, j) of elements of B with i ≠ j, fix a_{ij} in a given commutative ring with 1. For I ⊆ B, let A^{I} be a square matrix, indexed by I, with (i, j)-th entry a_{ij} for i ≠ j and which has zero row-sums. Therefore the (i, i)-th entry is − ∑_{j∈I-{i}} a_{ij}. For a directed graph T on a subset of B (whose arcs each begin and end at distinct vertices) we denote by A(T) the product ∏_{(i,j)} a_{ij}, where (i, j) runs over all pairs of elements of B such that there is an arc from i to j in T. We define A(T_{∅}) = 1 for an empty product. Note that a function f from a subset of I to a disjoint subset also defines a directed graph by joining i to f(i) for each i in the domain of f. We define A(f) by this graph.

If J ⊆ I then we write A_{J}^{I} for the determinant of the submatrix of A^{I} obtained by deleting the rows and columns indexed by J. If this matrix is 0 × 0 the determinant is taken to be 1. The following proposition shows how to use graphs to calculate these determinants, and generalises the “matrix tree theorem” of Kirchhoff-Tutte.

Proposition 1.1 (S. Chaiken and D. J. Kleitman). Let A^{I} be a square matrix of zero row-sums, indexed by a finite ordered set I. Let J ⊆ I. Then

A_{J}^{I} = (−1)^{#I−#J} ∑_{T} A(T),

where in the sum, T runs through all arborescences on I with root J.

Proof. This is proved in [11]. For illustrative purposes we give a proof here. In the case I = ∅ we obtain A_{∅}^{∅} = A(T_{∅}) = 1. Now assume I ≠ ∅. We proceed by induction on #I. If J = ∅ then both sides

of the desired equality are 0. So we may assume that J is nonempty.

The submatrix of A^{I} obtained by deleting the rows and columns indexed by J is the sum of A^{I-J} and the diagonal matrix with (i, i)-th entry − ∑_{j∈J} a_{ij} for I in I-J. Therefore its determinant is

A_{J}^{I} = ∑_{H⊆I-J} [ ∏_{i∈H} ( − ∑_{j∈J} a_{ij})A_{H}^{I-J}]

Now, since #(I-J) < #I, the induction hypothesis implies that

A_{H}^{I-J} = (−1)^{#I−#J−#H} ∑_{T_{H}} A(T_{H}),

where T_{H} runs through all arborescences on I-J with root H. Also,

∏_{i∈H} ( − ∑_{j∈J} a_{ij}) = (−1)^{#H} ∑_{f_{H}:H→J} A(f_{H}),

where f_{H} runs through all functions from H to J. So

A_{J}^{I} = (−1)^{#I−#J} ∑_{H⊆I-J} ∑_{T_{H}} ∑_{f_{H}} A(T_{H})A(f_{H}).

The product A(T_{H})A(f_{H}) is A(T) where T is the arborescence on I with root J obtained from T_{H} by adjoining the arcs from i to f_{H}(i) for i ∈ H. On the other hand, given any tree T on I with root J, the set of arcs ending in an element of J defines a unique subset H ⊆ I, function f_{H} : H ⟶ J and arborescence T_{H} on I-J such that T is the union of T_{H} and f. Therefore the desired expression for A_{J}^{I} is obtained.

Example 1.2. The minor A_{{3,4}}^{{1,2,3,4}} is the determinant of the matrix

( \begin{matrix}−a_{12} − a_{13} − a_{14} & a_{12} \\ a_{21} & −a_{21} − a_{23} − a_{24} \end{matrix} )

Proposition 1.1 shows that the determinant is a_{13}a_{21} + a_{12}a_{23} + a_{13}a_{23} + a_{13}a_{24} + a_{14}a_{23} + a_{14}a_{24} + a_{12}a_{24} + a_{14}a_{21}, corresponding to the arborescences

\begin{matrix}2 \\ ↓ \\ 1 \\ ↓ \\ 3 & 4 \end{matrix}  \begin{matrix}1 \\ ↓ \\ 2 \\ ↓ \\ 3 & 4 \end{matrix}  \begin{matrix}1 & 2 \\ ↓ & ↙ \\ 3 & 4 \end{matrix}  \begin{matrix}1 & 2 \\ ↓ & ↓ \\ 3 & 4 \end{matrix}  \begin{matrix}2 & 1 \\ ↓ & ↓ \\ 3 & 4 \end{matrix}  \begin{matrix}1 & 2 \\ ↘ & ↓ \\ 3 & 4 \end{matrix}  \begin{aligned}& 1 \\ & ↓ \\ & 2 \\ & ↓ \\ 3 & 4 \end{aligned}  \begin{aligned}& 2 \\ & ↓ \\ & 1 \\ & ↓ \\ 3 & 4 \end{aligned}

We will need to use the following identity which, although it was known in the 18th century, seems to be more often rediscovered than familiar [16]. As in the binomial theorem, we interpret 0^{0} = 1.

Proposition 1.3 (Euler’s formula). Let n be a positive integer and 0 ≤ t < n. Then

∑_{r=0}^{n}(−1)^{r}\binom{n}{r}r^{t} = 0.

Proof. A nice proof is given in [14].

Corollary 1.4. Let n be a positive integer and f be a polynomial of degree less than n. Then

∑_{r=0}^{n}(−1)^{r}\binom{n}{r}f(r) = 0.

We combine the results of this section to obtain an identity involving minors of matrices of zero row sum. Fix a finite ordered set B and fix a_{ij} in a fixed commutative ring with 1 for i ≠ j elements of B. For J ⊆ I ⊆ B define A^{I}, A_{J}^{I} as above, and set I′ = B-I.

Proposition 1.5. Let J ⊆ B and let F be a polynomial of degree less than #J. Then

∑_{I⊆B}(−1)^{#(I′∩J)}F (#I)A_{I∩J}^{I} ∏_{i∈I′-J} ( ∑_{j∈I} a_{ij}) = 0

Proof. By Proposition 1.1 we have

A_{I∩J}^{I} = (−1)^{#(I-J)} ∑_{T_{I}} A(T_{I}),

where T_{I} runs through all arborescences on I with root I ∩ J. Also,

∏_{i∈I′−J} ( ∑_{j∈I} a_{ij}) = ∑_{f_{I}:I′-J→I} A(f_{I})

Combining these shows that the left-hand side of the desired equality is equal to

∑_{I⊆B}(−1)^{#I−#J}F (#I) ∑_{T_{I},f_{I}} A(T_{I})A(f_{I}).

Now A(T_{I})A(f_{I}) = A(T) where T is the arborescence on B with root J obtained from T_{I} by adding the “leaves” f_{I}. On the other hand, we can express any T as such a union for some I, T_{I}, f_{I} by choosing a subset V of the set V^{0}(T) of vertices of T of in-degree 0 and then setting I = B − V, T_{I} to be T with the vertices V and adjoining arcs removed, and f_{I} these removed edges. This gives a bijection (I, T_{I}, f_{I}) ↔ (T, V) so that the sum becomes

∑_{T} A(T) ∑_{V ⊆V^{0}(T)}(−1)^{#B−#J−#V} F (#B − #V).

Let n denote #V^{0}(T). There are \binom{n}{r} sets V of cardinality r for r = 0, 1, …, n. We note that #J ≤ n since each element of J is the root of a subtree of T. So the coefficient of A(T) in the sum is (−1)^{#B−#J} ∑_{r=0}^{n}(−1)^{r}\binom{n}{r}F (#B − r). Since F (#B − r) is a polynomial of degree less than n in r, Corollary 1.4 shows that the coefficient of A(T) is zero.

2 First congruence

Let k be a global field and ∞ a place of k such that all other places are finite. Let O := O_{k,∞} be the ring of ∞-integers. Fix an ideal 𝔪 ≠ 1, 0 of O and set S = Supp(𝔪) ∪ {∞}.

Let 𝔱 be a prime ideal coprime to 3𝔪w_{H(𝔪)}, and write 𝔪 in the form (6.3). Abbreviate B = {1, …, n}. Then we may fix a system λ_{I} ∈ H(𝔪_{I}) as in Proposition 6.1.1 for each I ⊆ B, and let α_{i} = N_{H(𝔪_{i})/H}λ_{i}^{w_{k}/w_{i}} ∈ H, so that α_{i}O_{H} = 𝔭_{i}^{1−N_{𝔱}}O_{H}. Set λ = λ_{B}.

For an ideal 𝔞 of O, we set Γ_{𝔞} := (O/𝔞)^{×} and G_{𝔞} = A_{k,𝔞∞,𝔞}, which we identify with Gal(H(𝔞)/k) by global class field theory. If x ∈ k^{×} is coprime to 𝔞, we denote by [x]_{𝔞} the class of x modulo 𝔞 considered as an element of Γ_{𝔞}. We write [μ_{k}]_{𝔞} for the image in Γ_{𝔞} of the roots of unity μ_{k}. If 𝔞 = 𝔪_{I} for some set I, we abbreviate Γ_{I} := Γ_{𝔪_{I}}, G_{I} := G_{𝔪_{I}} and [x]_{I} = [x]_{𝔪_{I}}. The group G_{1} is the class group of the ring O. The group Γ_{𝔪} is a direct product:

Γ_{𝔪} = Γ_{1} × ⋯ × Γ_{n}.

The sequence (6.1) gives an exact sequence

0 ⟶ [μ_{k}]_{𝔪} ⟶ Γ_{𝔪} ⟶ G_{𝔪} ⟶ G_{1} ⟶ 0  (7.1)

Let s : G_{1} → G_{𝔪} be a section to the projection G_{𝔪} ⟶ G_{1}, and b be an integer such that w_{k} | b and s(x)^{b}s(y)^{b} = s(xy)^{b} for all x and y in G_{𝔱}. Choose s such that b is minimal; the minimum possible b depends upon the structure of the sequence (7.1). If #Γ_{𝔪} and #G_{1} = h_{k,∞} have greatest common divisor which divides w_{k} (which happens, for example, if O has class number 1, i.e. G_{1} is trivial), then we may take b = w_{k}. We write x ∼ y to denote that two elements x and y of a group have equal b-th powers, so that s(x)s(y) ∼ s(xy) for all x, y ∈ G_{1}.

For any abelian group G, we use G^{×b} to denote the subgroup of b-th powers. The function s and the sequence (7.1) define a surjection Γ_{𝔪} × G_{1} → G_{𝔪}, and the composition of this map with raising to the b-th power is a homomorphism.

If I ⊆ B, we have the commutative diagram

Γ_{𝔪} × G_{1} ⟶ G_{𝔪}
↓  ↓
Γ_{I} × G_{1} ⟶ G_{I}.

The natural inclusion Γ_{I} → Γ_{𝔪} and the section s induce a section t_{I} : G_{I} → G_{𝔪} such that t_{I}^{b} is multiplicative.

Now choose a group homomorphism ϕ^{1} : U_{H(𝔪),S,𝔱} → ℤ. For any quotient group G_{𝔪}/G′ of G_{𝔪}, let ϕ^{G_{𝔪}/G′} ∈ Hom_{ℤ[G_{𝔪}/G′]}(U_{H(𝔪)^{G′},S,𝔱}, ℤ[G_{𝔪}/G′]) be the map corresponding to the restriction of ϕ^{1} by Lemma 2.6.5. Let ϕ = ϕ^{G_{𝔪}}.

Lemma 2.1. Let Γ act on H(𝔪) via (7.1). Let I ⊆ C ⊆ {1, …, n}. Using the above notation, we calculate N_{Γ_{C-I}} λ_{C} as follows.

1. If I is not empty, then

N_{Γ_{C-I}} λ_{C} = λ_{I}^{∏_{i∈C-I}(1−σ_{𝔭_{i}}^{−1})}.

2. For any t ∈ C,

N_{Γ_{C}}λ_{C} = α_{t}^{∏_{i≠t}(1−σ_{𝔭_{i}}^{−1})}.

Proof. First assume I ≠ ∅. Taking 𝔞 = 𝔪_{I} and 𝔟 = 𝔪_{C-I} in diagram (6.2) shows that the map Γ_{𝔪_{i}} → Gal(H(𝔪_{I∪{i}})/H(𝔪_{I})) is surjective with kernel of size w_{𝔪_{I}}/w_{𝔪_{C}}. The assertion now follows from repeatedly applying Proposition 6.1.1 (2a).

Next we show (2). Choose t ∈ C. The assertion follows by applying Proposition 1.1 part (2a) for all the i ≠ t and then using part (2b).

Let i ∈ B and i ∉ I ⊆ B. Set σ_{i,I} = (𝔭_{i}, H(𝔪_{I})/k). In G_{𝔪}, we have

s(𝔭_{i})^{−b}t_{I}(σ_{i,I})^{b} = s(𝔭_{i})^{−b#I} ∏_{j∈I} t_{j}(σ_{i,j})^{b}.  (7.2)

Recalling the notation of Proposition 1.5, let a_{ij} = t_{ij}(σ_{i,j}^{−1}) − 1 ∈ ℤ[G]. Then there are defined certain elements A_{J}^{I} ∈ ℤ[G] for J ⊆ I ⊆ B.

For I ⊆ B and an arbitrary choice of t ∈ I, we write

Ω_{I} = ϕ^{1}(N_{G_{1}}α_{t}) ∏_{i∈I−{t}}(1 − s(𝔭_{i})^{−1}) ∈ I_{G_{1}}^{#I−1}.

Note that N_{G_{1}}α_{𝔱} is of the form x^{1−N_{𝔱}} for some x ∈ U_{k,S} and that w_{k} | (1 − N_{𝔱}), so N_{G_{1}}α_{𝔱} is the w_{k}-th power of an element of y of U_{k,S}. We have that y is congruent to a root of unity ζ modulo 𝔱, so ζ^{−1}y is a w_{k}-th root of N_{G_{1}}α_{𝔱} and lies in U_{k,S,𝔱}. This implies that w_{k} | ϕ^{1}(N_{G_{1}}α_{𝔱}) for all ϕ, and hence w_{k} | Ω_{I}. We have \frac{b}{w_{k}} ϕ^{G_{∅}}(N_{Γ_{I}} λ_{I}) ≡ \frac{b}{w_{k}} Ω_{I} (mod I_{G_{∅}}^{#I}), independently of the choice of t, by Lemma 2.1(2).

Proposition 2.2. For all ϕ^{1} we have

ϕ^{G}(λ^{b/w_{𝔪}}) ≡ \frac{b}{w_{k}} ∑_{∅≠J⊆B}(1 − n)^{#J−1}Ω_{J}A_{J}^{B} (mod I_{G}^{n}),  (7.3)

where 0^{0} := 1.

Proof. We prove this by induction on n = #B. Suppose first n = 1. Then

ϕ(λ_{i}^{w_{k}/w_{i}}) ≡ ϕ^{1}(N_{G_{1}}N_{Γ_{1}}λ) ≡ ϕ^{1}(N_{G_{1}}α_{1}) (mod I_{G}).

Hence the statement is true with b = w_{k} in this case.

Now assume that n > 1 and the assertion has been proved for all 𝔪_{I}, ∅ ≠ I ≠ B. Consider the equality in ℤ[G]:

∑_{γ∈Γ_{𝔪}} ∑_{υ∈G_{1}} ϕ^{1}(γ^{−1}s(υ)^{−1}λ)( ∏_{i∈B}([γ]_{𝔪_{i}}^{b} − 1))s(υ)^{b}

= ∑_{γ∈Γ_{𝔪}} ∑_{υ∈G_{1}} ϕ^{1}(γ^{−1}s(υ)^{−1}λ) ∑_{I⊆B}(−1)^{n−#I} ∏_{i∈I}[γ]_{𝔪_{i}}^{b}s(υ)^{b}
= ∑_{I⊆B}(−1)^{n−#I} ∑_{γ∈Γ_{I},υ∈G_{1}} ϕ^{1}(γ^{−1}s(υ)^{−1}N_{Γ_{C-I}} λ)γ^{b}s(υ)^{b}
= ∑_{I⊆B}(−1)^{n−#I}t_{I}^{b}(ϕ^{G_{I}} (N_{Γ_{I′}} λ^{b/w_{I}})).  (7.4)

Now we apply the results of Lemma 2.1. Reducing (7.4) modulo I_{G}^{n} gives

0 ≡ ∑_{I≠∅}(−1)^{n−#I}t_{I}^{b}(ϕ^{G_{I}} (λ_{I}^{b/w_{I}})) ∏_{i∈C-I}(1 − t_{I}(σ_{i,I}^{−1})^{b}) +

(−1)^{#C} ∑_{υ∈G_{1}} t_{∅}^{b}(ϕ^{G_{1}}(α_{t}) ∏_{i≠t})(1 − s(𝔭_{i})^{−b}) (mod I_{G}^{n}).

We now apply the induction hypothesis and Corollary 2.6.3 to obtain

0 ≡ ϕ(λ^{b/w_{𝔪}})
+ \frac{b}{w_{k}} ∑_{I≠∅}(−1)^{n−#I}[ ∑_{∅≠J_{1}⊆I}(1 − #I)^{#J_{1}−1}Ω_{J_{1}}A_{J_{1}}^{I}] ∏_{i∈I′}(1 − t_{I}(σ_{i,I}^{−1}))

+ \frac{b}{w_{k}} (−1)^{n}Ω_{B} (mod I_{G}^{n})

Equation (7.2) shows that b(1 − t_{I}(σ_{i,I}^{−1})) can be replaced by

b ∏_{i∈I′} [ ∑_{j∈I} a_{ij} + b(#I − 1)(1 − s(𝔭_{i}^{−1}))]

= b ∑_{J_{2}⊆I′} [ ∏_{i∈I′-J_{2}} ( ∑_{j∈I} a_{ij})(#I − 1)^{#J_{2}} ∏_{i∈J_{2}}(1 − s(𝔭_{i}^{−1}))].

Therefore we obtain

0 ≡ ϕ(λ^{b/w_{𝔪}})
+ \frac{b}{w_{k}} ∑_{I⊊B} [ ∑_{\begin{matrix}∅≠J_{1}⊆I, \\ J_{2}⊆I′ \end{matrix}}(−1)^{#J_{2}}(1 − #I)^{#J_{1}+#J_{2}−1}Ω_{J_{1}∪J_{2}}A_{J_{1}}^{I} ∏_{i∈C-I-J_{2}} ( ∑_{h∈I} a_{ih})].

If we rearrange the sum over J = J_{1} ∪ J_{2}, note that A_{∅}^{I} = 0 for all I and use Ω_{J_{1}} ∏_{i∈J_{2}}(1 − s(𝔭_{i}^{−1})) = Ω_{J}, we get

0 ≡ ϕ(λ^{b/w_{𝔪}})
+ \frac{b}{w_{k}} ∑_{∅≠J⊆B} Ω_{J}[ ∑_{I⊊B}(−1)^{#(I′∩J)}(1 − #I)^{#J−1}A_{I∩J}^{I} ∏_{i∈I′-J} ( ∑_{h∈I} a_{ih})].

Finally, we apply Proposition 1.5 for F (X) = (1 − X)^{#J−1}. We obtain

ϕ(λ^{b/w_{𝔪}}) ≡ \frac{b}{w_{k}} ∑_{∅≠J⊆B}(1 − n)^{#J−1}Ω_{J}A_{J}^{B} (mod I_{G}^{n}),

as required.

Now the case C = B proves the proposition.

We now give an alternative expression for the right-hand side of (7.3). First we need some linear algebra.

Lemma 2.3. Suppose A is a square n × n matrix and (v_{i})_{i}, (Ω_{i})_{i} are vectors of length n. Suppose Ω_{i}v_{j} = Ω_{j}v_{i} for all i ≠ j. Then the determinant of the n × n matrix with (i, j)-th entry a_{ij} + v_{i} for j ≠ 1 and Ω_{i} for j = 1 is the same as the determinant of the matrix obtained from A by replacing the first column with (Ω_{i})_{i}.

Proof. Expanding by the first column, the coefficients of Ω_{i}v_{j} and Ω_{j}v_{i} are equal and opposite.

If A is a square n × n matrix and i, j ∈ B, let A_{ij} denote the determinant of the matrix obtained from A by deleting row i and column j.

Lemma 2.4. Let A be a square n×n matrix and A′ the sum of A and the diagonal matrix with (j, j)-th entry w_{j}. Then:

A′_{i1} = \begin{cases}(−1)^{i} ∑_{i∈I⊆B} ( ∏_{j∈I-\{i\}} w_{j})A_{I} & if i = 1, \\ (−1)^{i} ∑_{i∈I⊆B-\{1\}} ( ∏_{j∈I-\{i\}} w_{j}) ∑_{σ∈S_{B-I}} sign(σ) ∏_{j∈B-I} a_{j,(1 i)σ(j)} & if i ≠ 1, \end{cases}

where (1 i) ∈ S_{n} is the permutation swapping 1 and i.

Proof. Denote the (j, k)-th entry of A′ by a′_{jk} = a_{jk} + w_{j}δ_{jk}. Then

A′_{i1} = ∑_{σ∈S_{B-{i}}} sign(σ) ∏_{j∈B-{i}} a′_{j,(1 2 ··· i−1 i)σ(j)}
= (−1)^{i} ∑_{σ∈S_{B-{i}}} sign(σ) ∏_{j∈B-{i}} a′_{j,(1 i)σ(j)}.

Now

∏_{j∈B-i} a′_{j,(1 i)σ(j)} = ∑_{J⊆B-{i}} ( ∏_{j∈J} w_{j}δ_{j,(1 i)σ(j)})( ∏_{j∈B-{i}-J} a_{j,(1 i)σ(j)}).

The term for J is nonzero only if j = (1 i)σ(j) for j ∈ J. If i ≠ 1 then this is impossible if 1 ∈ J. So assuming 1 ∉ J if i ≠ 1, the condition implies that σ is any permutation of B-J-{i}. Therefore

A′_{i1} = (−1)^{i} ∑_{J⊆B-{1,i}} ( ∏_{j∈J-{i}} w_{j}) ∑_{σ∈S_{B-J-{i}}} sign(σ) ∏_{j∈B-J-{i}} a_{j,(1 i)σ(j)}.

Now set I = J ∪ {i} to complete the proof.

We now define a matrix to which to apply this theory. We set a_{ih} = t_{h}(σ_{i,h}^{−1}) − 1 ∈ ℤ[G] and υ_{i} = s(𝔭_{i})^{−1} − 1 for i, h ∈ B. Set now

m_{ih} = a_{ih} + (n − 1)υ_{i}δ_{ih} − υ_{i},  (7.5)

defining a matrix M. We consider the expression

∑_{i=1}^{n}(−1)^{i+1}Ω_{i}M_{i1}.

Proposition 2.5.

∑_{i=1}^{n}(−1)^{i}Ω_{i}M_{i1} = ∑_{∅≠J⊆B}(1 − n)^{#J−1}Ω_{J}A_{J}^{B}.

Proof. Noting that Ω_{i}υ_{j} = Ω_{j}υ_{i} = −Ω_{{i,j}} for i ≠ j by Lemma 2.1(2), we see that Lemma 2.3 shows that M_{i1} can be replaced by A′_{i1} in the left-hand side of the equation, where A′ is the n × n matrix with (i, j)-th entry a′_{ih} = a_{ih} + (n − 1)υ_{i}δ_{ih}.

Consider first those J containing 1. In the left-hand-side, such Ω_{J} are produced only by the term for i = 1. Now applying Lemma 2.4 for I = J gives the correct coefficients for these Ω_{J}.

Now suppose 1 ∉ J. We have a contribution to the coefficient of Ω_{J} from terms on the left-hand side corresponding to i for each i ∈ J. Applying Lemma 2.4 with I = J, the coefficient of Ω_{J} on the left-hand side is

−(1 − n)^{#J−1} ∑_{i∈J} ∑_{σ∈S_{B-J}} sign(σ) ∏_{j∈B-J} a_{j,(1 i)σ(j)}

= (1 − n)^{#J−1} ∑_{σ∈S_{B-J}} sign(σ) ∏_{j∈B-J-{σ^{−1}(1)}} a_{j,σ(j)}( − ∑_{i∈J} a_{σ^{−1}(1),i}). (7.6)

Now we use the fact that A has zero row-sums. This gives − ∑_{i∈J} a_{σ^{−1}(1),i} = ∑_{i∈B-J} a_{σ^{−1}(1),i}. Substituting this into the expression (7.6) gives

(1 − n)^{#J−1} ∑_{σ∈S_{B-J}} sign(σ) ∏_{j∈B-J} a_{j,σ(j)}

+ ∑_{i∈B-J-{1}}(1 − n)^{#J−1} ∑_{σ∈S_{B-J}} sign(σ)( ∏_{j∈B-J-{σ^{−1}(1)}} a_{j,σ(j)})a_{σ^{−1}(1),i}.

In the second line, indices i, σ and σ^{−1}(i), (1 i)σ match up with opposite signs. Therefore the second term vanishes and the first term, which is (1 − n)^{#J−1}A_{J}, remains.

Corollary 2.6. For all ϕ^{1} we have

ϕ(λ^{b/w_{𝔪}}) ≡ \frac{b}{w_{k}} ∑_{i=1}^{n}(−1)^{i}ϕ^{1}(N_{G_{1}}α_{i})M_{i1}.

Proof. This combines Propositions 2.2 and 2.5.

3 Reciprocity symbol calculations

We continue with the hypotheses and notation of §2.

Lemma 3.1. If σ ∈ Gal(H(𝔪)/k) restricts to the identity on Gal(H(𝔪_{i})/k) for all i, then σ ∼ id.

Proof. We may represent σ by a principal ideal (a) ∈ A_{k,𝔪∞,𝔪}. The element a ∈ O is then congruent to a root of unity modulo each 𝔪_{i}.

Proposition 3.2. The decomposition group of any prime of H(𝔪) above 𝔭_{i} is an extension of the subgroup of Gal(H(𝔪_{B-{i}})/k) generated by the Frobenius of 𝔭_{i} on H(𝔪_{B-{i}}) by Gal(H(𝔪)/H(𝔪_{B-{i}})). The inertia group of the same prime is Gal(H(𝔪)/H(𝔪_{B-{i}})).

Proof. Choose a prime of H(𝔪) above 𝔭_{i}, and write D for the its decomposition group in G. The groups Gal(H(𝔪)/k) and Gal(H(𝔪_{B-{i}})/k) are connected by the exact sequence on the right-hand side of (6.2) for 𝔞 = 𝔪_{B-{i}} and 𝔟 = 𝔪_{i}.

The projection of D to Gal(H(𝔪_{B-{i}})/k) is the decomposition group of a prime above 𝔭_{i} in the unramified extension H(𝔪_{B-{i}})/k, and so is generated by the Frobenius σ_{𝔭_{i}} on this extension.

Let us now calculate the ramification number of 𝔭_{i} in the extension H(𝔪)/k. Useful formulas for this can be found in [10, Thm. 3.5.3]. These show that the ramification degree of any prime of H(𝔪) over 𝔭_{i} is #(I_{k,S} ∩ P_{k,S-{𝔭_{i}},𝔪_{B-{i}}})/P_{k,S}, which is the size of Gal(H(𝔪)/H(𝔪_{B-{i}})). On the other hand, the ramification number for the subextension H(𝔪_{B-{i}})/k is 1. This completes the proof.

For each i, choose a prime 𝔮_{i} coprime to 𝔪 such that

(𝔮_{i}, H(𝔪)/k) = s(𝔭_{i}),  (7.7)

and hence there is ϵ_{i} ∈ k^{×} such that

𝔭_{i} = 𝔮_{i}(ϵ_{i}).  (7.8)

We abbreviate the local symbol (−, H(𝔪)_{𝔓_{i}}/k_{𝔭_{i}}), where 𝔓_{i} is a prime of H(𝔪) above 𝔭_{i}, to (−, 𝔭_{i}).

Proposition 3.3. Let i ∈ B. Let x ∈ k^{×} be coprime to 𝔪, i.e. v_{𝔭_{j}}(x) = 0 for all j ∈ B. Then

(x, 𝔭_{i}) ∼ [x^{−1}]_{𝔪_{i}}.

Proof. By the product rule of global class field theory, we have

1 = (x, H(𝔪)/k) = ((x), H(𝔪)/k) ∏_{i=1}^{n}(x, 𝔭_{i}) = [x]_{𝔪} ∏_{i=1}^{n}(x, 𝔭_{i}).

We have that (x, 𝔭_{i}) is in the inertia group of 𝔭_{i} because x is a unit at 𝔭_{i}. Projecting to Gal(H(𝔪_{i})/k) and noting that the inertia groups of the other primes (Proposition 3.2) become thereby trivial, we obtain (x, 𝔭_{i}) ∼ [x^{−1}]_{𝔪_{i}} (using Lemma 3.1) as required.

Proposition 3.4. The norm residue symbols of ϵ_{i} in the completions of H(𝔪)/k at finite primes satisfy:

(ϵ_{i}, 𝔭_{i}) ∼ (𝔮_{i}, H(𝔪)/k)[ϵ_{i}]_{𝔪_{B-{i}}}

and

(ϵ_{i}, 𝔭_{j}) ∼ [ϵ_{i}^{−1}]_{𝔪_{j}}

for j ≠ i.

Proof. The product formula for H(𝔪_{B-{i}})/k shows that

[ϵ_{i}^{−1}]_{𝔪_{B-{i}}} ∼ ∏_{j≠i}(ϵ_{i}, 𝔭_{j}).  (7.9)

Projecting this to Gal(H(𝔪_{j})/k) gives the second assertion.

Now the product formula for H(𝔪)/k gives

(𝔮_{i}^{−1}, H(𝔪)/k) ∏_{j=1}^{n}(ϵ_{i}, 𝔭_{j}) = 1.

Using this and (7.9) gives the first assertion.

We are now ready to calculate the local symbols for arbitrary x ∈ k^{×}. Proposition 3.5. If x ∈ k^{×}, we have

(x, 𝔭_{i}) ∼ ((𝔮_{i}, H(𝔪)/k)[ϵ_{i}]_{𝔪_{B-{i}}})^{v_{𝔭_{i}}(x)}[x^{−1}ϵ_{i}^{v_{𝔭_{i}}(x)}]_{i}.

Proof. Let x̃ = ∏_{i=1}^{n} ϵ_{i}^{v_{𝔭_{i}}(x)}. Proposition 3.3 shows that (xx̃^{−1}, 𝔭_{i}) ∼ [x̃x^{−1}]_{i}. On the other hand, Proposition 3.4 calculates (x̃, 𝔭_{i}), and the product is the stated formula for (x, 𝔭_{i}).

We now connect this to the elements σ_{ij}. Proposition 3.6. We have σ_{ij} ∼ s(𝔭_{i})[ϵ_{i}]_{j} for i ≠ j. Proof. Since the elements to compare lie in the subgroup G_{I}, it is sufficient by Lemma 3.1 to compare the corresponding automorphisms of H(𝔪_{I})/k. The result then follows from (7.8) and (7.7).

For any x ∈ U_{k,S}, we define x̃ := ∏_{j=1}^{n} ϵ_{j}^{v_{𝔭_{j}} (x)}. Lemma 3.7. For x ∈ U_{k,S}, we have

[x^{−1}x̃]_{𝔪} ∼ 1.

Proof. The relation (x) = ∏_{j=1}^{n} 𝔭_{j}^{v_{𝔭_{j}} (x)} implies that (x)^{b} = ∏_{j=1}^{n} 𝔮_{j}^{bv_{𝔭_{j}} (x)}(x̃)^{b}, and by multiplicativity of s^{b} we have that the product of 𝔮_{j} lies in P_{k,S,𝔪}. Corollary 3.8. For any I ⊆ B,

[x^{−1}x̃]_{𝔪_{I}} ∼ 1.

We obtain:

Proposition 3.9. Let x ∈ U_{k,S}. Then

(x, H(𝔪)_{𝔓_{j}}/k_{𝔭_{j}}) ∼ (s(𝔭_{j})^{−(n−2)} ∏_{i≠j} σ_{ji})^{v_{𝔭_{j}} (x)} ∏_{i≠j}(σ_{ij}s(𝔭_{i})^{−1})^{v_{𝔭_{i}}(x)}.

Proof. Using Lemma 3.7 to replace x with x̃ when applying Proposition 3.5, we obtain

(x, H(𝔪)_{𝔓_{j}}/k_{𝔭_{j}}) ∼ (s(𝔭_{j})[ϵ_{j}]_{𝔪_{B-{j}}})^{v_{𝔭_{j}} (x)}[x̃^{−1}ϵ_{j}^{v_{𝔭_{j}} (x)}]_{𝔪_{j}}.

Now [ϵ_{j}]_{𝔪_{B-{j}}} = ∏_{i≠j}[ϵ_{j}]_{𝔪_{i}} = ∏_{i≠j}(s(𝔭_{j})^{−1}σ_{ji}) by Proposition 3.6, and also

[x̃^{−1}ϵ_{j}^{v_{𝔭_{j}} (x)}]_{𝔪_{j}} = ∏_{i≠j}[ϵ_{i}^{−v_{𝔭_{i}}(x)}]_{𝔪_{j}} = ∏_{i≠j}(σ_{ij}^{−1}s(𝔭_{i}))^{v_{𝔭_{i}}(x)}, by the same result.

4 Relation to Burns’s conjecture

We continue with all the notation of §2 and §3. The rank of the units of O is zero. Therefore an element x ∈ k^{×} is determined up to a root of unity in O by its valuations, or equivalently a principal fractional ideal has a unique generator up to multiplication by a root of unity. We use these facts repeatedly in this section.

We need to relate our results in terms of global Artin symbols to a basis of U_{k,S,𝔱}. We recall the notation for ideal groups developed in Chapter 2, §1. In addition, we use the notation I_{k,∞}⟨S⟩ to denote the free subgroup of I_{k,∞} generated by the finite primes in S. We have the following exact commutative diagram.

0  0  0
↓  ↓  ↓
0 ⟶ U_{k,S,𝔱} ⟶ I_{k,∞}⟨S⟩ ⟶ I_{k,∞}⟨S⟩/U_{k,S,𝔱} ⟶ 0
↓  ↓  ↓
0 ⟶ P_{k,∞,𝔱} ⟶ I_{k,{𝔱,∞}} ⟶  A_{k,∞,𝔱}  ⟶ 0
↓  ↓  ↓
0 ⟶ P_{k,S,𝔱} ⟶ I_{k,S∪{𝔱}} ⟶  A_{k,S,𝔱}  ⟶ 0
↓  ↓  ↓
0  0  0.

We obtain from the diagram an exact sequence

0 ⟶ I_{k,∞}⟨S⟩/U_{k,S,𝔱} ⟶ A_{k,∞,𝔱} ⟶ A_{k,S,𝔱} ⟶ 0.  (7.10)

The group A_{k,∞,𝔱} fits into the exact sequence

0 ⟶ U_{k,∞} ⟶ (O/𝔱)^{×} ⟶ A_{k,∞,𝔱} ⟶ A_{k,∞} ⟶ 0

The class group A_{k,∞} is the ideal class group of O, with size h_{k,∞} = h_{O}. Therefore we have #A_{k,∞,𝔱} = h_{O}(N_{𝔱} − 1)/w_{k}. If u_{1}, …, u_{n} is an integral basis for U_{k,S,𝔱}, we write V for the matrix with (i, j)-th entry v_{ij} = v_{𝔭_{i}}(u_{j}), a nonsingular n × n matrix. Now in the notation of Chapter 3, §2, we have R = (−1)^{n}V^{t} · diag(ln N_{v_{1}}, ln N_{v_{2}}, … ln N_{v_{n}}). Since also #T = 1, this implies that the sign ξ defined there is the sign of (−1)^{n} det V. The absolute value of det V is the size of I_{k,∞}⟨S⟩/U_{k,S,𝔱}. We conclude from (7.10) that

det V = (−1)^{n}ξ (\frac{N_{𝔱} − 1}{w_{k}}) \frac{h_{O}}{h_{k,S,𝔱}}.

We now use the results of §3 to relate the local Artin symbols for the u_{i} to the global Artin symbols appearing in §2. As in the latter section, we define elements of ℤ[G]

a_{ij} = σ_{i,j}^{−1} − 1, υ_{i} = s(𝔭_{i})^{−1} − 1,

This defines a matrix M = (m_{ij}) as in (7.5). Applying the natural isomorphism G → I_{G}/I_{G}^{2} of Lemma 2.6.2 to the result of Proposition 3.9 and using Corollary 2.6.4 gives the following

Proposition 4.1. Let R be the n × n matrix with (i, j)-th entry f_{𝔭_{j}}(u_{i}). We have

bf_{𝔭_{j}}(u_{i}) ≡ (−1)^{n}b · V M (mod I_{G}^{2}).

We now relate this to Proposition 2.5. For the following calculations, “vector” means a length n column vector over k, ℤ or ℤ[G]. Let V^{adj} represent the adjugate matrix of V, that is, the transpose of its cofactor matrix. Then

V^{adj}V = (det V)1 = (−1)^{n}ξ (\frac{N_{𝔱} − 1}{w_{k}}) \frac{h_{O}}{h_{k,S,𝔱}}1,

where 1 denotes the identity matrix. On the other hand, (1 − N_{𝔱})[H : k]1 = (1 − N_{𝔱})h_{O}1 is the valuation matrix of the vector (N_{Υ}α_{i})_{i}. Therefore if Ω denotes the vector (ϕ^{1}(N_{Υ}α_{i}))_{i}, then Ω = (−1)^{n+1}ξ(w_{k}h_{k,S,𝔱})V^{adj}ω, where ω = (ϕ^{1}(u_{i}))_{i}. Note that the vector ((−1)^{i+1}M_{i1})_{i} is the first column of (M^{adj})^{t}, where t denotes transpose, so we write it as (M^{adj})_{∗1}^{t}. Then Corollary 2.6 states that

ϕ(λ_{𝔪}^{b/w_{𝔪}}) ≡ −\frac{b}{w_{k}} Ω^{t}(M^{adj})_{∗1}^{t} (mod I_{G}^{n}).

The right-hand side of this congruence is \frac{b}{w_{k}} (−1)^{n}ξ(w_{k}h_{k,S,𝔱})ω^{t}(V^{adj})^{t}(M^{adj})_{∗1}^{t}. By standard matrix rules,

b(V^{adj})^{t}(M^{adj})_{∗1}^{t} = (−1)^{n}b((V M)^{adj})_{∗1}^{t} ≡ b(R^{adj})_{∗1}^{t} (mod I_{G}^{2}).

We conclude Theorem 4.2.

ϕ(λ_{𝔪}^{b/w_{𝔪}}) ≡ bξh_{k,S,𝔱} ∑_{i=1}^{n}(−1)^{i+1}ϕ^{1}(u_{i})R_{i1} (mod I_{G}^{n}).

By Proposition 6.1.2, we have η_{H(𝔪)/k,S,{𝔱}}^{w_{k}} = λ_{𝔪}^{w_{k}/w_{𝔪}}. Therefore, we obtain

bϕ(η) ≡ bξh_{k,S,𝔱}(Reg_{G} ∘ ϕ̃)(u_{1} ∧ ⋯ ∧ u_{n}) (mod I_{G}^{n}),

that is, we verify Conjecture 3.5.1 after weakening by b on both sides.

Corollary 4.3. If h_{k,∞} = 1 (i.e. O is a principal ideal domain), or the sequence (7.1) splits, or n = 1, then

w_{k}ϕ(η) ≡ w_{k}ξh_{k,S,𝔱}(Reg_{G} ∘ ϕ̃)(u_{1} ∧ ⋯ ∧ u_{n}) (mod I_{G}^{n}),

Proof. In these cases we may take b = w_{k}.

If K/k is an abelian extension in which all places other than a particular one ∞ are finite and not totally split, h_{k,∞} = 1 and w_{k} is coprime to [K : k], then this result combined with Proposition 4.2.1 proves the whole of Conjecture 3.5.1. To see this, we note that w_{k} is coprime to G and hence multiplication by w_{k} is an automorphism of the quotient I_{G}^{n−1}/I_{G}^{n}, which is a finite abelian group supported above the support of #G. In particular, this proves the conjecture for an odd-degree extension of ℚ, with the infinite place the only place that splits.

Minus-unit conjectures

Suppose K/k is an abelian extension, and k′ is a quadratic Galois extension of k such that K ∩ k′ = k. Set K′ = k′K. Then we have the following diagram of fields

[figure: K; K′; k; k′]

The Galois group Gal(K′/k) is the direct product

Gal(K′/k) = Gal(K/k) × Gal(k′/k).

Set G = Gal(K/k) ≅ Gal(K′/k′) (the isomorphism being given by the natural restriction map). The group Gal(k′/k) is cyclic of order two, and we denote its nontrivial element by τ; by the direct product decomposition, τ may also be considered as an element of Gal(K′/k). Thus we naturally have an involution of K′/k in this situation. We let ω denote the nontrivial character of Gal(k′/k), so ω(τ) = −1.

The kernel of the norm map 1 + τ on a ℤ[Gal(K′/k)]-module M consists of the elements x such that x^{τ} = −x, and this subgroup will be referred to as the “minus-part” of M, written M^{−}. If M is 2-divisible (i.e. multiplication by 2 is an automorphism of M) then M^{−} is a direct summand of M. Another way to put this is that the two idempotents in ℤ[\frac{1}{2}][⟨τ⟩]

e_{+} = \frac{1 + τ}{2} e_{−} = e_{ω} = \frac{1 − τ}{2}

split up any ℤ[\frac{1}{2}][Gal(K′/k)]-module M into a direct sum M^{+}⊕M^{−}, where M^{+} = e_{+}M = M^{⟨τ⟩} is the submodule fixed by τ and the “minus-part” M^{−} is equal to e_{−}M.

In this chapter we use the arrangement of field extensions above and apply the decompositions into plus- and minus-parts to L-functions and unit groups. This provides an interpretation of two conjectures by Darmon and Gross which roughly resemble Gross’s conjecture (Conjecture 3.3.2), but formulated for minus-parts of equivariant L-functions and minus-units. The conjectures are interpreted as providing “base-change” statements for Conjecture 3.5.1 above, or more precisely giving a logical implication from the validity of the conjecture for the extension K/k to its validity for K′/k′. In fact, this only works under a certain simplifying hypothesis (Hypotheses 4.1 and 8.1 below) and the resulting statements are weakened by a power-of-two factor (mainly coming from the problem of inverting 2 for the decompositions described above). Nevertheless, these results provide interpretation for conjectures whose position had not been well understood, and were often regarded as “refinements” or “analogues” of Gross’s conjecture when in fact they appear to relate to behaviour of a natural generalisation of it.

1 Quadratic extension of the base field for L-functions

In this section we assume S and T are disjoint finite sets of places of k such that S is nonempty and contains all infinite places, S contains all places ramifying in K/k and S_{k′} contains all places ramifying in K′/k′. We will use stronger hypotheses in following sections. We compare the definitions of the S-truncated, T -modified L-functions defined in Chapter 2, §3 for the extensions K/k and K′/k′.

The character ω extends to a function on the set of finite places of k by setting ω(v) = ω(σ_{v}) if v is unramified in k′/k and 0 otherwise. Then we have

Lemma 1.1. If v ∉ S, then

∏_{\begin{matrix}v′ place of k′ \\ v′|v \end{matrix}}(1−N_{v′}^{s} ·(v′, K′/k′)^{−1}) = (1−N_{v}^{s} ·(v, K/k)^{−1})(1−ω(v)N_{v}^{s} ·(v, K/k)^{−1}).

Proof. By the fundamental equality for extensions of primes in Galois extensions, we have efg = [k′ : k] = 2, where e is the ramification index of any v′/v, f is its residue extension degree and g is the number of primes in k′ lying above v. Note that N_{w} = N_{v}^{f} and (v′, K′/k′)|_{K} = (v, K/k)^{f}, the second equality holding under the restriction isomorphism Gal(K′/k′) →^{∼} Gal(K/k). In the case where ω(v) = 0, v is ramified in k′/k and so f = g = 1. Each side in the equality of the lemma then has exactly one factor, and these are clearly the same.

In the case where v is split in k′, i.e. g = 2, we have ω(v) = 1 and the equality clearly has two matching factors on each side. In the case where v is inert, i.e. f = 2, we have (v, k′/k) = τ and so ω(v) = −1 and the right-hand side of the equality is (1 − N_{v}^{2s}(v, K/k)^{2}), which is equal to the left-hand side.

Let S′ be the union of S with the set of places of k which ramify in k′/k. Lemma 1.1 shows that the L-functions satisfy the following base-change factorisation when passing from K′/k′ to K′/k:

Θ_{K′/k′,S,T} (s) = Θ_{K′/k′/k,S′,T} (s, ω)Θ_{K/k,S,T} (s),  (8.1)

where

Θ_{K′/k′/k,S′,T} (s, ω) := ( ∏_{v∈T}(1 − ω(v)N_{v}^{1−s}σ_{v}^{−1})) ∑_{χ∈Ĝ} L_{K′/k,S′}(s, ωχ^{−1})e_{χ}(K/k)

= ( ∏_{v∈T}(1 − ω(v)N_{v}^{1−s}σ_{v}^{−1})) ·
( ∑_{σ∈G}(ζ_{K′/k,S′}(s, σ) − ζ_{K′/k,S′}(s, στ))σ^{−1})  (8.2)

is a twisted Stickelberger function (with trivial Euler factors at ramified primes). The validity of equation (8.1) follows from Lemma 1.1 in the region of convergence Re s > 1 and then everywhere by meromorphic continuation.

We also write

δ_{T}^{ω} = ∏_{v∈T}(1 − ω(v)N_{v}σ_{v}^{−1}) ∈ ℤ[G]

for the relative T -modification factor at s = 0.

2 Unit groups in quadratic extensions

Let k′/k be a quadratic Galois extension of global fields with Galois group generated by τ. For this section we only need to assume that S is a finite, nonempty set of places of k containing all infinite places, and that T is any finite disjoint set of places of k. Let #S = n + 1 and let r denote the number of places in S which split in k′/k.

We will be interested in the unit group U_{k′,S,T} and its subgroups U_{k,S,T}, U_{k′,S,T}^{−} = {x ∈ U_{k′,S,T} : x^{τ} = x^{−1}} and U_{k′,S,T}^{1−τ} = {x^{1−τ} : x ∈ U_{k′,S,T} }. Obviously U_{k′,S,T}^{1−τ} ⊆ U_{k′,S,T}^{−}. The quotient is a Tate cohomology group in the sense of [41, Ch. VIII]:

U_{k′,S,T}^{−} /U_{k′,S,T}^{1−τ} = Ĥ^{−1}(⟨τ⟩, U_{k′,S,T}) ≅ H^{1}(⟨τ⟩, U_{k′,S,T}).

We first prove some lemmas calculating indices related to these unit groups.

Lemma 2.1. (U_{k′,S} : U_{k′,S,T}) = (U_{k,S} : U_{k,S,T})(U_{k′,S}^{1−τ} : U_{k′,S,T}^{1−τ})

Proof. Consider the following commutative diagram, in which the rows are exact.

0 ⟶ U_{k,S,T} ⟶ U_{k′,S,T} ⟶^{1−τ} U_{k′,S,T}^{1−τ} ⟶ 0
↓ ↓ ↓
0 ⟶ U_{k,S} ⟶ U_{k′,S} ⟶^{1−τ} U_{k′,S}^{1−τ} ⟶ 0,

The vertical arrows are inclusions. Applying the snake lemma, we obtain the exact sequence

0 ⟶ U_{k,S}/U_{k,S,T} ⟶ U_{k′,S}/U_{k′,S,T} ⟶^{1−τ} U_{k′,S}^{1−τ} /U_{k′,S,T}^{1−τ} ⟶ 0.

This shows the result.

Lemma 2.2. (U_{k′,S}^{1−τ} : U_{k′,S,T}^{1−τ}) = (U_{k′,S}^{−} : U_{k′,S,T}^{−})\frac{#H^{1}(⟨τ⟩,U_{k′,S,T})}{#H^{1}(⟨τ⟩,U_{k′,S})}.

Proof. We have (U_{k′,S}^{−} : U_{k′,S,T}^{1−τ}) = (U_{k′,S}^{−} : U_{k′,S}^{1−τ})(U_{k′,S}^{1−τ} : U_{k′,S,T}^{1−τ}) = (U_{k′,S}^{−} : U_{k′,S,T}^{−})(U_{k′,S,T}^{−} : U_{k′,S,T}^{1−τ}).

We next we adapt the method of [46, §II.2 and Thm. IV.5.4] to show the following.

Lemma 2.3.

(U_{k′,S,T} : U_{k,S,T} U_{k′,S,T}^{−}) = \frac{2^{r}#(U_{k′,S,T}^{−} ∩ \{±1\})}{#H^{1}(⟨τ⟩, U_{k′,S,T})}

Proof. The sequence

0 ⟶ U_{k,S,T} U_{k′,S,T}^{−} ⟶ U_{k′,S,T} ⟶^{1−τ} \frac{U_{k′,S,T}^{1−τ}}{(U_{k′,S,T}^{−})^{2}} ⟶ 0

is exact. This shows that (U_{k′,S,T} : U_{k,S,T} U_{k′,S,T}^{−}) = (U_{k′,S,T}^{1−τ} : (U_{k′,S,T}^{−})^{2}). We also have:

(U_{k′,S,T}^{−} : (U_{k′,S,T}^{−})^{2}) = (U_{k′,S,T}^{−} : U_{k′,S,T}^{1−τ})(U_{k′,S,T}^{1−τ} : (U_{k′,S,T}^{−})^{2}).

The first factor on the right is #H^{1}(⟨τ⟩, U_{k′,S,T}). The factor on the left can be calculated from the standard decomposition of the finitely generated abelian group U_{k′,S,T}^{−}. The free rank of this group is r, and the cokernel of squaring on the torsion part has the same size as the kernel, which is U_{k′,S,T}^{−} ∩ {±1}. The second factor on the right is what we want to calculate. Hence

2^{r}#(U_{k′,S,T}^{−} ∩ {±1}) = #H^{1}(⟨τ⟩, U_{k′,S,T})(U_{k′,S,T} : U_{k,S,T} U_{k′,S,T}^{−}),

as required.

We now compare the regulators for the fields k and k′, as defined in Chapter 3, §2. We let n = #S − 1 and r be the number of places in S which split in k′. Thus #S_{k} = n + 1 + r = n′ + 1. We assume that at least one place in S does not split in k′, i.e. n ≥ r.

We order the places of S as

S = {v_{1}, …, v_{n+1}},  (8.3)

where for 1 ≤ i ≤ r the place v_{i} splits into distinct places v′_{i} and \overline{v′}_{i} of k′ (conjugate by τ) and for i > r there is a unique place v′_{i} of k′ above v_{i}. The set S_{k′} is ordered as follows:

S_{k′} = \{v′_{1}, …, v′_{r}, \overline{v′}_{1}, …, \overline{v′}_{r}, v′_{r+1}, …, v′_{n+1}\}.  (8.4)

Let the r-tuple of S-units ε_{−} = (ε_{1}, …, ε_{r}) be a basis for a maximal free ℤsubmodule ⟨ε_{−}⟩ of U_{k′,S,T}^{−}, and ε_{+} = (ε_{r+1}, …, ε_{r+n}) the same thing for U_{k,S,T}. Let ε = (ε_{1}, …, ε_{r+n}). We wish to use ε instead of a basis for a maximal free ℤ-submodule of U_{k′,S,T} to calculate the regulator R =: R_{k′} as defined in Chapter 3, §2 for the field k′. Define R_{k′}(ε) as follows (using the notation of Chapter 3, §2)

(∧^{n′}λ_{S_{k′}})(ε_{1} ∧ ⋯ ∧ ε_{n′}) = R_{k′}(ε) · x,

and let u = (u_{1}, …, u_{n+r}) be a basis for a maximal free ℤ-submodule of U_{k′,S,T}, so that (U_{k′,S,T} : ⟨u⟩) = w_{k′,T}. By inverting a unit if necessary, we may assume that

(U_{k′,S,T} : ⟨ε⟩) · u_{1} ∧ ⋯ ∧ u_{n+r} = w_{k′,T} · ε_{1} ∧ ⋯ ∧ ε_{n+r}  (8.5)

in ℚ ∧_{ℤ}^{r+n} U_{k′,S,T}. Then

(U_{k′,S,T} : ⟨ε⟩)R_{k′} = w_{k′,T} R_{k′}(ε).  (8.6)

The sign ξ_{k′} is defined to be the ξ defined in Chapter 3, §2 for the field k′, so ξ_{k′} = (−1)^{#T_{k′}+1} · sign(R_{k′}).

Now consider the exact commutative diagram

0 ⟶  0  ⟶ ⟨ε_{−}⟩ × ⟨ε_{+}⟩ ⟶ ⟨ε_{−}⟩ × ⟨ε_{+}⟩ ⟶ 0
↓  ↓  ↓
0 ⟶ U_{k′,S,T}^{−} ∩ U_{k,S,T} ⟶^{Δ} U_{k′,S,T}^{−} × U_{k,S,T} ⟶ U_{k′,S,T}^{−} U_{k,S,T} ⟶ 0,

where Δ(x) = (x^{−1}, x). Applying the snake lemma gives an exact sequence

0 ⟶ U_{k′,S,T}^{−} ∩ U_{k,S,T} ⟶ U_{k′,S,T}^{−} /⟨ε_{−}⟩ × U_{k,S,T} /⟨ε_{+}⟩ ⟶ U_{k′,S,T}^{−} U_{k,S,T} /⟨ε⟩ ⟶ 0,

and hence #(U_{k′,S,T}^{−} ∩ U_{k,S,T})(U_{k′,S,T}^{−} U_{k,S,T} : ⟨ε⟩) = (U_{k′,S,T}^{−} : ⟨ε_{−}⟩)(U_{k,S,T} : ⟨ε_{+}⟩). Now if x ∈ U_{k′,S,T}^{−} ∩U_{k,S,T} then x^{τ} = x = x^{−1} and so x = ±1; therefore the intersection is either {±1} or {1}. This shows that #(U_{k′,S,T}^{−} ∩U_{k,S,T}) = #(U_{k′,S,T}^{−} ∩{±1}). Let w_{k′,T}^{−} denote the size of the torsion subgroup of U_{k′,S,T}^{−}. Then combining these results with Lemma 2.3 gives

(U_{k′,S,T} : ⟨ε⟩) = \frac{2^{r}w_{k′,T}^{−} w_{k,T}}{#H^{1}(⟨τ⟩, U_{k′,S,T})}.  (8.7)

The regulator R_{k′}(ε) is equal to the determinant of the following block matrix:

v′_{1}, …, v′_{r} \overline{v′}_{1}, …, \overline{v′}_{r} v′_{r+1}, …, v′_{n}
ε_{1} ; ⋮ ; ε_{r}  − ln |ε|_{v′} − ln |ε|\overline{v′}  − ln |ε|_{v′}

ε_{r+1} ; ⋮ ; ε_{n′} − ln |ε|_{v′} − ln |ε|\overline{v′}  − ln |ε|_{v′}.

Since ε_{r+1}, …, ε_{n′} are in k, the n × r blocks in the bottom-left and bottommiddle of the matrix are identical, and so subtracting columns shows that R_{k′}(ε) is also the determinant of the matrix

v′_{1}, …, v′_{r}  \overline{v′}_{1}, …, \overline{v′}_{r} v′_{r+1}, …, v′_{n}
ε_{1} ; ⋮ ; ε_{r}  − ln |ε^{1−τ} |_{v′} − ln |ε|\overline{v′}  − ln |ε|_{v′}

ε_{r+1} ; ⋮ ; ε_{n′}  0  − ln |ε|\overline{v′}  − ln |ε|_{v′}

(8.8)

We denote the real numbers R and ξ calculated for the field k as in Chapter 3, §2 by R_{k} and ξ_{k}. We also define a “minus-unit regulator” R^{−} analogously to the R of Chapter 3, §2 but for the units ε_{1}, …, ε_{r} and the places v′_{1}, …, v′_{r}, i.e.

R^{−} = det(− ln |ε_{i}|_{v′_{j}})_{i,j=1}^{r}.  (8.9)

Then the top-left r × r block of (8.8) has determinant 2^{r}R^{−}. We let ξ^{−} = (−1)^{#T_{k′}−#T} sign(R^{−}).

If u ∈ k and v′ is a place of k′ over a place v of k, then we have |u|_{v′} = |u|_{v} if v′/v is split, and |u|_{v′} = |u|_{v}^{2} otherwise. Hence the bottom-right n × n block of (8.8) has determinant 2^{n−r}R_{k}. Putting this together gives

R_{k′}(ε) = 2^{n}R^{−}R_{k}  and  ξ_{k′} = ξ^{−}ξ_{k}.

Therefore, using (8.6) and (8.7), we have

R_{k′} = \frac{w_{k′,T}}{w_{k,T} w_{k′,T}^{−}} 2^{n−r}#H^{1}(⟨τ⟩, U_{k′,S,T})R^{−}R_{k}.

Now by (2.8) we have

ζ_{k′,S,T} (s) = L_{k′/k,S,T} (s, ω)ζ_{k,S,T} (s).

Hence, by (3.3), the coefficient of the leading term of the nontrivial L-function for k′/k is given by

L_{k′/k,S,T}^{r} (0, ω) = \frac{ζ_{k′,S,T}^{n+r} (0)}{ζ_{k,S,T}^{n} (0)} = (−1)^{#T_{k′}−#T} \frac{w_{k,T}}{w_{k′,T}} \frac{h_{k′,S,T}}{h_{k,S,T}} \frac{| R_{k′} |}{| R_{k} |}

= (−1)^{#T_{k′}−#T} \frac{1}{w_{k′,T}^{−}} \frac{h_{k′,S,T}}{h_{k,S,T}} 2^{n−r}#H^{1}(⟨τ⟩, U_{k′,S,T}) | R^{−} | . (8.10)

3 Darmon’s circular unit

We now relate the circular unit defined by Darmon in [12] to a special value of the base-change factor Θ_{K′/k′/k,S′,T} (s, ω) appearing in (8.1). For comparison with [12], we assume the following hypothesis for §§3–7.

Hypothesis 3.1 (Darmon’s setup). Let N and S be coprime integers with N > 2 and S > 1. Let ω be a primitive, quadratic, even Dirichlet character defined modulo N. Set k′ = ℚ(ζ_{N})^{ker ω}, a real quadratic field, and call its nontrivial automorphism τ. Let K be a real subfield of ℚ(ζ_{S}), normal over ℚ and of conductor S. Write K′ = k′K.

Note that our notation N and S agrees with [12], but the fields denoted L and K in loc. cit. are respectively called K′ and k′ here.

We are in the situation of §1 with the further assumptions that k = ℚ and that K and k′ are totally real and have coprime conductors. We define the set S = S_{ℚ} from the integer S in the obvious way: S_{ℚ} = Supp(S) ∪ {∞}. We assume all primes dividing S ramify in K/ℚ, so that there is a unique place ∞ ∈ S_{ℚ} splitting completely in K/ℚ, and exactly two places of k′ splitting completely in K′, namely the two places above ∞ (which are mutually conjugate under τ).

All the characters of these extensions come from even Dirichlet characters because the fields are totally real. Since L-functions of even characters vanish at s = 0 by Proposition 2.3.1, differentiating (8.1) twice shows that we have the following equality in ℂ[G]

Θ_{K′/k′,S,T}^{2} (0) = \frac{1}{2}Θ′′_{K′/k′,S,T} (0) = Θ′_{K′/k′/ℚ,S′,T}(0, ω)Θ′_{K/ℚ,S,T}(0).  (8.11)

We now relate the base-change factor Θ′_{K′/k′/ℚ,S′,T}(0, ω) to the circular unit defined in [12, §4]. The circular unit is the following element of k′(ζ_{S}).

α_{S} := ∏_{σ∈Gal(ℚ(ζ_{NS})/ℚ(ζ_{S}))}σ(ζ_{NS} − 1)^{ω(σ)} = (N_{ℚ(ζ_{NS})/k′(ζ_{S})}(ζ_{NS} − 1))^{1−τ} .

Write ∞′ for the place of K′ corresponding to the embedding of K′ into ℝ induced by ζ_{NS} ↦ e^{2πi/NS}, and \overline{∞}′ its conjugate by τ.

Lemma 3.2. Θ′_{K′/k′/ℚ,S′}(0, ω) · (∞′ − \overline{∞}′) = \frac{1}{2}λ_{S_{K′}} (N_{k′(ζ_{S})/K′}α_{S}).

Proof. Let S′ denote the union of S_{ℚ} and the set of primes dividing N. By (6.5) we have

ζ′_{ℚ(ζ_{NS})^{+}/ℚ,S′}(0, σ) = − ln | 1 − ζ_{NS}^{a} |,

where a is a representative of the class in (ℤ/NSℤ)^{×}/{±1} corresponding to σ. Hence by Proposition 2.3.2 we have

ζ′_{K′/k,S′}(0, σ) = −\frac{1}{2} ln |N_{ℚ(ζ_{NS})/K′}(1 − ζ_{NS})^{σ}|_{∞′}.

Therefore, for σ ∈ G,

ζ′_{K′/k,S′}(0, σ) − ζ′_{K′/k,S′}(0, στ)
= −\frac{1}{2} ln |N_{k′(ζ_{S})/K′}N_{ℚ(ζ_{NS})/k′(ζ_{S})}(1 − ζ_{NS})^{(1−τ)σ}|_{∞′}
= −\frac{1}{2} ln |N_{k′(ζ_{S})/K′}α_{S}^{σ}|_{∞′}.

Write for brevity β = N_{k′(ζ_{S})/K′}α_{S}. Then

λ_{S}(β) = − ∑_{σ∈G} ln |β^{σ}|_{∞′}σ^{−1}∞′ − ∑_{σ∈G} ln |β^{τσ}|_{∞′}σ^{−1}\overline{∞}′,

using the fact that (1 − ζ_{NS}) is a global unit to see that the absolute values at finite places are trivial. But since α_{S}^{τ} = α_{S}^{−1}, i.e. α_{S} is a minus-unit, this is the same as

λ_{S}(β) = ( − ∑_{σ∈G} ln |β^{σ}|_{∞′}σ^{−1})(∞ − \overline{∞}′).

Putting all this together and using the definition (8.2) gives the claimed result.

Definition 3.3. We set η_{ω} := δ_{T}^{ω}N_{k′(ζ_{S})/K′}α_{S}.

Then

Θ′_{K′/k′/ℚ,S′,T}(0, ω) · (∞ − \overline{∞}′) = \frac{1}{2}λ_{S_{K′}} (η_{ω}).  (8.12)

4 Calculation of η

We wish to compare the Stark units of Chapter 2, §1 corresponding to the two sets of data K/k, S, T and K′/k′, S_{k′}, T_{k′}, where T is some set of primes such that U_{K′,S,T} is torsion free (Hypothesis 4.1). The connection will be given by Darmon’s circular unit as defined in the preceding section. We first review our notation and the definition of the Stark units involved.

Consider the extension K/ℚ. The set S = S_{ℚ} consists of the infinite place of ℚ and the finite places corresponding to the primes dividing the integer S. These finite primes may be divided into two kinds: those which split in k′/ℚ and those which are inert (they are all unramified by the assumption of coprimality of conductors in Hypothesis 3.1). We write n_{s} for the number of split primes, n_{i} for the number of inert ones, and n for the total. Since the infinite place is the only place in S splitting in K, we have r_{S} = 1 and n = n_{s} + n_{i} = 1 + d, using the notation r_{S} (Chapter 2, §3) and d (Chapter 3) for the extension K/k. We denote the n_{s} primes in S splitting in k′/ℚ by p_{1}, …, p_{n_{s}}, and the inert primes by q_{1}, …, q_{n_{i}}. Thus we have

S = S_{ℚ} = {∞, p_{1}, …, p_{n_{s}}, q_{1}, …, q_{n_{i}}}.  (8.13)

The Stark unit η_{K/k} ∈ ℝU_{K,S,T} e_{1} is the unique element satisfying

Θ_{K/k,S,T}^{1} (0)(∞′ − w_{0}) = λ_{S}(η_{K/k}),

where ∞′ is the infinite place of K induced by the place ∞′ defined in §3, and v_{0} is any finite place of K.

We turn now to the definition of the Stark unit for the data K′/k′, S_{k′}, T_{k′}. Here we have r_{S_{k′}} = 2, corresponding to the two places ∞′ and \overline{∞}′ of k′ lying above the infinite place of ℚ. Each prime p_{i} ∈ S splits into two primes of k′, which we write as 𝔭_{i} and 𝔭̄_{i}. These are mutually conjugate under the action of τ. The primes q_{i} remain prime in k′, and we denote 𝔮_{i} = q_{i}O_{k′}. Hence we have

S_{k′} = \{∞′, \overline{∞}′, 𝔭_{1}, …, 𝔭_{n_{s}}, 𝔭̄_{1}, … 𝔭̄_{n_{s}}, 𝔮_{1}, …, 𝔮_{n_{i}}\}.  (8.14)

We add dashes to distinguish the integers n and d (as defined at the beginning of Chapter 3) for the set S_{k′}, so that #S_{k′} = r_{S_{k′}} + d′ + 1 = n′ + 1. Then, since #S_{k′} = 2 + 2n_{s} + n_{i}, we have n′ = 1 + 2n_{s} + n_{i} and d′ = 2n_{s} + n_{i} − 1. The Stark unit η_{K′/k′} ∈ ℝ ∧_{ℤ[G]}^{2} U_{K′,S,T} e_{2} is defined by

Θ_{K′/k′,S,T}^{2} (0)(∞′ − w′_{0}) ∧ (\overline{∞}′ − w′_{0}) = (∧^{2}λ_{S_{K′}})(η_{K′/k′})

for a finite place w′_{0} of K′.

The results of §3 allow us to express η_{K′/k′} in terms of η_{ω} and η_{K/k}. We have

2Θ_{K′/k,S,T}^{2} (0)(∞′ − w′_{0}) ∧ (\overline{∞}′ − w′_{0}) =

Θ′_{K′/k′/ℚ,S′,T}(0, ω)(∞′ − \overline{∞}′) ∧ Θ′_{K/ℚ,S,T}(0)(∞′ + \overline{∞}′ − 2w′_{0}). (8.15)

Recall that λ_{S_{K′}} (η_{ω}) was calculated in (8.12) above. We will now calculate λ_{S_{K′}} (η_{K/ℚ}). We note the commutative diagram (2.5) for the extension K′/K, so that λ_{S_{K′}} (x) = i_{K′/K}(λ_{S_{K}}(x)) for x ∈ U_{k,S,T}, where i_{K′/K}(w) = w′ + \overline{w′} for any choice of place w′ of K′ above a place w of K. Hence λ_{S_{K′}} (η_{K/ℚ}) = Θ′_{K/ℚ,S,T}(0)(∞′ + \overline{∞}′ − w′_{0} − \overline{w′_{0}}), where w′_{0} is a place of K′ chosen above w_{0}. If we assume n_{i} ≠ 0, we can choose w_{0} such that w′_{0} = \overline{w′_{0}}. Using this choice of w′_{0} in (8.15) shows that

(∧^{2}λ_{S_{K′}})(η_{ω} ∧ η_{K/ℚ}) = 4Θ_{K′/k′,S,T}^{2} (0)(∞′ − w′_{0}) ∧ (\overline{∞}′ − w′_{0}),

and hence

4η_{K′/k′} = η_{ω} ∧ η_{K/ℚ}.  (8.16)

In [12], Darmon makes a congruence conjecture for his circular unit. We propose to interpret this as a base-change statement for Conjecture 3.5.1, under the following assumptions:

Hypothesis 4.1. 1. n_{i} > 0

2. For every place v in T_{ℚ}, we have ω(v) = 1 (i.e. v splits in k′/ℚ).

Proposition 4.2. Assuming Hypothesis 4.1, δ_{T}^{ω}(U_{K′,S}^{−}) ⊆ U_{K′,S,T}^{−}. Therefore η_{ω} ∈ U_{K′,S,T}^{−}.

Proof. Let x ∈ U_{K′,S}^{−} and y = x^{δ_{T}^{ω}}. Let v ∈ T split into v′, \overline{v′} in k′, let w′ be any place of K′ above v′, and \overline{w′} its conjugate by τ. There is a factor of (1 − N_{v′}σ_{v′}^{−1}) in δ_{T}^{ω} by Lemma 1.1. Hence w′(y − 1) > 0. But we also have \overline{w′}(y − 1) = w′(y^{−1} − 1) = w′((1 − y)/y) = w′(1 − y) > 0. Therefore y ≡ 1 (mod w′) for all w′ ∈ T_{K′} as required. Setting x = N_{k′(ζ_{S})/K′}α_{S} proves the second assertion.

5 Darmon’s conjecture

We continue with the notation of §3, and the assumption of Hypotheses 3.1 and 4.1. We will state Darmon’s conjecture and deduce a modified version involving T.

The free rank of the ℤ-module U_{k′,S}^{−} is n_{s} + 1 (the number of primes of S splitting in k′), as may be seen for example by interpreting it as the kernel of the norm map to U_{k,S,T}. However U_{k′,S}^{−} is not free as a ℤ-module since its torsion subgroup is {±1}. The reader is warned that this appears to have been overlooked in [12, §4]. Accordingly, we assume that either T is empty or U_{k′,S,T}^{−} is torsion free, and we choose a basis ε_{1}, …, ε_{n_{s}+1} for a maximal free ℤ-submodule of U_{k′,S,T}^{−}, which will have index 2 if T = ∅. Let Γ := Gal(k′(ζ_{S})/k′) ≅ (ℤ/Sℤ)^{×}. Following Darmon, we define a regulator

R_{S,T} := ∑_{i=1}^{n_{s}+1}(−1)^{i+1}ε_{i} ⊗ det(f_{𝔭_{k}}(ε_{j}))_{k,j} ∈ U_{k′,S,T} ⊗ \frac{I_{Γ}^{n_{s}}}{I_{Γ}^{n_{s}+1}} ,  (8.17)

where, in the matrix, k runs from 1 to n_{s} and j runs from 1 to n_{s} + 1, omitting i. Note that 2R_{S,∅} is independent of the choice of maximal free subgroup of U_{k′,S,T}^{−}.

Relative to the basis ε_{1}, …, ε_{n_{s}+1} and the ordered list of places ∞′, 𝔭_{1}, …, 𝔭_{n_{s}}, we may define a minus-unit regulator R^{−} ∈ ℝ as in (8.9). Note that (−1)^{#T} ξ^{−} is the sign of R^{−} by Hypothesis 4.1 (2). Due to the fact that the logarithmic regulators Darmon uses are the negatives of ours, his assumption [12, (20)] that a regulator be positive corresponds in our notation to the statement that for T = ∅, we have ξ^{−} = (−1)^{n_{s}+1}. Instead of assuming this, we will introduce an appropriate sign depending on ξ^{−} into statements in the style of Chapter 3, §3.

We define a number m_{T} by L_{k′/k,S,T}^{n_{s}+1} (0, ω) = ξ^{−}m_{T} R^{−}. The calculation (8.10) shows that in the current setup

m_{T} = \frac{1}{w_{k′,T}^{−}} \frac{h_{k′,S,T}}{h_{ℚ,S,T}} 2^{n_{i}−1}#H^{1}(⟨τ⟩, U_{k′,S,T}).  (8.18)

Therefore for T = ∅ we have

m_{∅} = \frac{1}{w_{k′,T}^{−}} h_{k′,S}2^{n_{i}−1}#H^{1}(⟨τ⟩, U_{k′,S,T}).  (8.19)

We study the cohomological factor in this expression.

Lemma 5.1. Suppose k′ is a real quadratic field, and S a set of places of ℚ not ramifying in k′/ℚ. Then #H^{1}(⟨τ⟩, U_{k′,S}) = 2^{a+b}, where

• a = 2 if the fundamental unit of k′ has norm 1 and a = 1 if it has norm −1;

• If C denotes the subgroup of A_{k′} generated by the classes of the primes which ramify in k′/ℚ, then the kernel of the quotient map C → A_{k′,S} is isomorphic to (ℤ/2ℤ)^{b}.

Proof. We abbreviate H_{S} = #H^{1}(⟨τ⟩, U_{k′,S}). Firstly we consider the case where S only consists of the archimedean place. Let u denote the fundamental unit of the real quadratic field k, so that U_{k′} = {±1} × u^{ℤ}. If the norm of u is 1, then U_{k′}^{−} = U_{k′} and U_{k′}^{1−τ} = u^{2ℤ}, and so H^{1}(⟨τ⟩, U_{k′}) = U_{k′}^{−}/U_{k′}^{1−τ} ≅ (ℤ/2ℤ)^{2} and H_{∅} = 4. On the other hand, if the norm of u is −1 then U_{k′}^{−} = {±1} × u^{2ℤ} and U_{k′}^{1−τ} = (−u^{2})^{ℤ}, so H_{∅} = 2.

We now proceed to consider the integer H_{S} when S is not empty. There is a natural map i : I_{ℚ} → I_{k′} given by v_{p} → e_{p} ∑_{𝔭|p} v_{𝔭}, with e_{p} the ramification index

of the prime p in k′/ℚ. Considering the quotient I_{k′,S}^{⟨τ⟩}/I_{ℚ,S}, we see that it is a direct product of cyclic groups of order 2, with each cyclic component generated by the class of a prime which ramifies in k′/ℚ but does not lie in S. Let C_{S} denote the image of these ramified 𝔭 in the S-class group A_{k′,S}, so that C = C_{∅}. Note that C_{S} is always an elementary abelian 2-group. [36, Thm.] gives a commutative diagram with exact rows

0 ⟶ H^{1}(⟨τ⟩, U_{k′}) ⟶ I_{k′}^{⟨τ⟩}/I_{ℚ} ⟶ C ⟶ 0
↓  ↓  ↓
0 ⟶ H^{1}(⟨τ⟩, U_{k′,S}) ⟶ I_{k′,S}^{⟨τ⟩}/I_{ℚ,S} ⟶ C_{S} ⟶ 0

From this we see firstly that if k′/ℚ were unramified outside S then we would have H_{S} = 1 (note that this is stated in [22, Lem. 6.7(i)] erroneously because the assumption on S is missing). Since S contains none of the places which ramify in k′/ℚ, the central vertical arrow of the above diagram is an isomorphism. Therefore, the snake lemma gives that H^{1}(⟨τ⟩, U_{k′,S})/H^{1}(⟨τ⟩, U_{k′}) ≅ ker(C → C_{S}), which is the intersection of C with the subgroup of A_{k′} generated by classes of primes in S_{k′}. Therefore H_{S} = H_{∅}2^{b}. We note that if h_{k′} = 1 then H_{S} = H_{∅} = 2^{a}.

This calculation seems to contradict Darmon’s assertion [12, (22)], which states (in our notation) that m_{∅} = 2^{n_{i}+1}h_{k′,S}. Noting that [12, §4] wrongly states that U_{k′,S}^{−} is torsion free (i.e. w_{k′,S}^{−} = 1), we observe that replacing w_{k′,S}^{−} by 1 in (8.19) gives the same value as [12, (22)] in the case where k′ has class number one and the norm of its fundamental unit is 1, or half that value if the norm of the fundamental unit is −1.

Noting the relation [12, Thm. 4.1] of Darmon’s element α_{S} to double the values of derivatives of L-functions, it seems reasonable that the integer factor in Darmon’s conjecture should be 2m_{∅}, and so that is the value which we use in our statement of the conjecture. This may differ by a power of 2 from the statement in [12]. By the above discussion, our formulation agrees with [12] when k′ has class number one and fundamental unit of norm 1. We note that 2m_{∅} is an integer (and m_{∅} is as well, since we assume Hypothesis 4.1).

Define Γ = Gal(k′(ζ_{S})/k′) ≅ (ℤ/Sℤ)^{×}. We state Darmon’s conjecture under the ring automorphism involution of ℤ[Γ] induced by inversion on Γ. Note that the effect of this involution on an element of I_{Γ}^{n_{s}}/I_{Γ}^{n_{s}+1} is the same as multiplication by (−1)^{n_{s}}, by repeated application of Corollary 2.6.3. Then Darmon’s conjecture [12, Conj. 4.3] states that

Conjecture 5.2 (Darmon). Let T = ∅. We have the following equality in U_{k′(ζ_{S}),S} ⊗_{ℤ} I_{Γ}^{n_{s}}/I_{Γ}^{n_{s}+1}:

∑_{σ∈Γ} σ^{−1}α_{S} ⊗ σ = ξ^{−}2m_{∅}R_{S,∅}.

We consider a T -modified version. This will fit with our general framework, and avoids the problem of torsion in the unit group.

Lemma 5.3. Under Hypothesis 4.1, m_{T} is an integer.

Proof. By Hypothesis 4.1 (2), each v in T splits into w and \overline{w} in k′, with N_{w} = N_{v}. Then by (2.3), we have the following:

h_{ℚ,S,T} = h_{ℚ,S} · \frac{∏_{v∈T} (N_{v} − 1)}{(U_{ℚ,S} : U_{ℚ,S,T})}, h_{k′,S,T} = h_{k′,S} · \frac{∏_{v∈T} (N_{v} − 1)^{2}}{(U_{k′,S} : U_{k′,S,T})} ,

where we note h_{ℚ,S} = 1. The quotient is

\frac{h_{k′,S,T}}{h_{ℚ,S,T}} = h_{k′,S} · \frac{∏_{v∈T} (N_{v} − 1)}{(U_{k′,S}^{1−τ} : U_{k′,S,T}^{1−τ})} = h_{k′,S} · \frac{∏_{v∈T} (N_{v} − 1)}{(U_{k′,S}^{−} : U_{k′,S,T}^{−})} \frac{#H^{1}(⟨τ⟩, U_{k′,S})}{#H^{1}(⟨τ⟩, U_{k′,S,T})},

using Lemmas 2.1 and 2.2. Therefore, by (8.18) we have

m_{T} = \frac{1}{w_{k′,T}^{−}} h_{k′,S} · \frac{∏_{v∈T} (N_{v} − 1)}{(U_{k′,S}^{−} : U_{k′,S,T}^{−})}2^{n_{i}−1}#H^{1}(⟨τ⟩, U_{k′,S}).

We have w_{k′,T}^{−} = 1 or 2, n_{i} −1 ≥ 0 (by Hypothesis 4.1 (1)) and 2 | #H^{1}(⟨τ⟩, U_{k′,S}) (by Lemma 5.1). Therefore it is sufficient to show that (U_{k′,S}^{−} : U_{k′,S,T}^{−}) divides ∏_{v∈T} (N_{v} − 1).

The natural sequence

0 ⟶ U_{k′,S,T}^{−} ⟶ U_{k′,S}^{−} ⟶ ⨁_{v∈T} 𝔽_{v′}^{×}

is exact, where for each v, we choose a place v′ of k′ arbitrarily above v. To see the exactness, we note that if u ∈ U_{k′,S}^{−} reduces to 1 modulo each v′, then \overline{v′}(u − 1) = v′(ū − 1) = v′((1 − u)/u) = v′(1 − u) > 0, as in Proposition 4.2. Hence u ∈ U_{k′,S,T} ∩ U_{k′,S}^{−} = U_{k′,S,T}^{−}. This shows that (U_{k′,S}^{−} : U_{k′,S,T}^{−}) | ∏_{v∈T} (N_{v} − 1), as required.

We propose the following generalisation of Darmon’s conjecture:

Conjecture 5.4. We have the following equality in U_{k′(ζ_{S}),S,T} ⊗ I_{Γ}^{n_{s}}/I_{Γ}^{n_{s}+1}:

∑_{σ∈Γ} σ^{−1}α_{S}^{δ_{T}^{ω}} ⊗ σ = ξ^{−}2m_{T} R_{S,T} .  (8.20)

If we put T = ∅ in this statement, then we recover Conjecture 5.2. On the other hand, let us look at how Conjecture 5.4 varies when we replace T by T ∪{v}. If T is empty, then for the comparison statement we will have to assume that the regulator in Conjecture 5.2 is calculated with respect to a maximal free subgroup of U_{k′,S} which contains U_{k′,S,{v}}.

Passing from T to T ∪ {v} has the effect of multiplying the left-hand side of (8.20) by 1 ⊗ (1 − N_{v}σ_{v}^{−1}), or equivalently its augmentation 1 − N_{v}. On the other side, the index of a maximal free ℤ-submodule of U_{k′,S,T ∪{v}} in a maximal free

ℤ-submodule of U_{k′,S,T} is y := \frac{(U_{k′,S,T}^{−}:U_{k′,S,T ∪\{v\}}^{−})}{w_{k′,T}}. We have (1 − N_{v})ξ_{T}^{−}m_{T} R_{T}^{−} =

ξ_{T ∪{v}}^{−}m_{T ∪{v}}yR_{T}^{−}, and R_{S,T ∪{v}} = yR_{S,T}. This shows that equality (8.20) is preserved by enlarging T.

The consequence of Darmon’s conjecture that we wish to use is the following.

Proposition 5.5. Assume the setup of Hypothesis 3.1 and let T satisfy Hypothesis 4.1 (2) with U_{K′,S,T} torsion-free. Then Conjecture 5.4 implies that for each ϕ ∈ Hom_{ℤ[G]}(U_{K′,S,T} , ℤ[G]), we have

ϕ(η_{ω}) ≡ ξ^{−}2m_{T} ϕ^{1}(R_{S,T}) (mod I_{G}^{n_{s}+1}).

Proof. We apply the natural projection U_{k′(ζ_{S}),S} ⊗ℤ[Γ] ⟶ U_{k′(ζ_{S}),S} ⊗ℤ[G], which maps the left-hand side of (8.20) to ∑_{σ∈G} σ^{−1}(δ_{T}^{ω}N_{k′(ζ_{S})/K′}α_{S})⊗σ in U_{K′,S} ⊗ℤ[G]. Then Conjecture 5.4 implies the following equality in U_{K′,S} ⊗ I_{G}^{n_{s}}/I_{G}^{n_{s}+1}:

∑_{σ∈G} σ^{−1}η_{ω} ⊗ σ = ξ^{−}2m_{T} R_{S,T} ,

with η_{ω} from Definition 3.3.

We define a homomorphism ϕ^{1} corresponding to ϕ as in Lemma 2.6.5. Applying the homomorphism

ϕ^{1} ⊗ id : U_{K′,S,T} ⊗ I_{G}^{n_{s}}/I_{G}^{n_{s}+1} ⟶ I_{G}^{n_{s}}/I_{G}^{n_{s}+1}

gives the stated result.

6 Factorisation of the regulator

We assume T is such that U_{K′,S,T} is torsion free, and that Hypothesis 4.1 is satisfied. We let ε_{1}, …, ε_{n_{s}+1} be a basis for U_{k′,S,T}^{−} and ε_{n_{s}+2}, …, ε_{n′} be a basis for U_{ℚ,S,T}. Then these ε_{i} form a basis for U_{ℚ,S,T}U_{k′,S,T}^{−}. The index of this group in U_{k′,S,T} was calculated in Lemma 2.3. We calculate the regulator from Conjecture 3.5.1 for these ε_{i}. Let Φ = ϕ_{1} ∧ ϕ_{2} ∈ ∧_{ℤ[G]}^{2}Hom_{ℤ[G]}(U_{K′,S,T} , ℤ[G]). Let R_{S,T} be the regulator defined in §5 in terms of the ε_{i}. Write ε_{ℚ} = ε_{n_{s}+2} ∧ ⋯ ∧ ε_{n′}. We refer to §4 above for the definitions and orderings of the sets S and S_{k′}, and the definition of the integers n_{s}, n_{i}, n, d, n′ and d′.

Proposition 6.1. We have the following equality in ℤ[G]/I_{G}^{d′+1}:

Reg_{K′/k′}(Φ̃(ε_{1} ∧ ⋯ ∧ ε_{n′})) = (−2)^{n_{s}} | \begin{matrix}2^{n_{i}−1}ϕ_{1}^{1}(R_{S,T}) & 2^{n_{i}−1}ϕ_{2}^{1}(R_{S,T}) \\ Reg_{K/ℚ}(ϕ̃_{1}(ε_{ℚ})) & Reg_{K/ℚ}(ϕ̃_{2}(ε_{ℚ})) \end{matrix} | .

Proof. We saw in Lemma 2.5.3 that

Φ̃(ε_{1} ∧ ⋯ ∧ ε_{n′}) = ∑_{σ∈\left[{n′ \atop 2}\right]} sign(σ) | \begin{matrix}ϕ_{1}^{1}(ε_{σ(1)}) & ϕ_{2}^{1}(ε_{σ(1)}) \\ ϕ_{1}^{1}(ε_{σ(2)}) & ϕ_{2}^{1}(ε_{σ(2)}) \end{matrix} | ε_{σ(3)} ∧ ⋯ ∧ ε_{σ(n′)}. (8.21)

The terms ε_{σ(3)} ∧ ⋯ ∧ ε_{σ(n′)} are made by choosing two of the ε_{i} for the integer determinant. So each σ excludes 0, 1 or 2 units of the U_{k′,S,T}^{−} basis from the wedge of units. Let m_{σ} = #(σ({3, 4, …, n′}) ∩ {1, …, n_{s} + 1}) be the number of minus-units included in ε_{σ(3)} ∧ ⋯ ∧ ε_{σ(n′)} in the term corresponding to σ, so m_{σ} = n_{s} − 1, n_{s} or n_{s} + 1.

We calculate the matrix determining the map Reg_{K′/k′} (see Chapter 3, §3) with respect to the following places of k′, using Hypothesis 4.1 to exclude 𝔮_{n_{i}}:

𝔭_{1}, …, 𝔭_{n_{s}}, 𝔭̄_{1}, …, 𝔭̄_{n_{s}}, 𝔮_{1}, …, 𝔮_{n_{i}−1}.

This means we have the determinant of the following d′ ×d′ matrix to calculate for Reg_{K′/k′}(ε_{σ(3)} ∧ ⋯ ∧ ε_{σ(n′)}):

\begin{aligned}& 𝔭_{1}, …, 𝔭_{n_{s}} & 𝔭̄_{1}, …, 𝔭̄_{n_{s}} & 𝔮_{1}, …, 𝔮_{n_{i}−1} \\ ε_{σ(3)} ; ⋮ ; ε_{σ(m_{σ}+2)} & f_{𝔭_{j}}(ε_{i}) & f𝔭̄_{j}(ε_{i}) & f_{𝔮_{j}}(ε_{i}) \\ ε_{σ(m_{σ}+3)} ; ⋮ ; ε_{σ(n′)} & f_{𝔭_{j}}(ε_{i}) & f𝔭̄_{j}(ε_{i}) & f_{𝔮_{j}}(ε_{i}) \end{aligned}  (8.22)

where the units at the top are in U_{k′,S,T}^{−} and the units at the bottom are in U_{ℚ,S,T}. We will distinguish between the cases where m_{σ} takes the different values.

First consider the case m_{σ} = n_{s} + 1. We may add the column for 𝔭_{j} to the column for 𝔭̄_{j} for j = 1, …, n_{s} without altering the value of the determinant. The (i, j)-th entry in the top-centre (n_{s} + 1) × (n_{s}) block is then congruent mod I_{G}^{2} to f_{𝔭_{j}}(ε̄_{i}ε_{i}) = 0. On the other hand, f_{𝔮_{j}}(ε_{i}) = f_{q_{j}}(ε_{i}^{1+τ}) under the natural restriction map corresponding to Gal(K′/k′) → Gal(K/ℚ), and so f_{𝔮_{j}}(ε_{i}) = 0 for these ε_{i}. Therefore the entire top-right (n_{s} + 1) × (n − 1) block is zero. Hence there are at most n_{s} columns which are nonzero in their first n_{s} rows. Therefore the determinant is zero.

Now in (8.22) we subtract the 𝔭̄_{j} column from the 𝔭_{j} column for j = 1, …, n_{s} to show the determinant is the same as the determinant of the following matrix.

\begin{aligned}& 𝔭_{1}, …, 𝔭_{n_{s}} & 𝔭̄_{1}, …, 𝔭̄_{n_{s}} & 𝔮_{1}, …, 𝔮_{n_{i}−1} \\ ε_{σ(3)} ; ⋮ ; ε_{σ(m_{σ}+2)} & f_{𝔭_{j}}(ε_{i}) − f𝔭̄_{j}(ε_{i}) & f𝔭̄_{j}(ε_{i}) & f_{𝔮_{j}}(ε_{i}) \\ ε_{σ(m_{σ}+3)} ; ⋮ ; ε_{σ(n′)} & 0 & f𝔭̄_{j}(ε_{i}) & f_{𝔮_{j}}(ε_{i}) \end{aligned}  (8.23)

If m_{σ} = n_{s} − 1, the first n_{s} columns have all zeros except perhaps in the first n_{s} − 1 rows. Therefore the determinant is again zero.

We are left with the case m_{σ} = n_{s}. In this case the matrix (8.23) is blockupper-triangular. Let us consider the top-left n_{s} × n_{s} block first. We note that f_{𝔭_{j}}(ε_{σ(i)})−f𝔭̄_{j}(ε_{σ(i)}) ≡ 2f_{𝔭_{j}}(ε_{σ(i)}) (mod I_{G}^{2}). So the top-left block has determinant 2^{n_{s}} det(f_{𝔭_{j}}(ε_{σ(i)}))_{i,j}. Note the relation to the regulator R_{S,T} of (8.17).

Now we calculate the determinant of the bottom-right block. We have f𝔭̄_{j}(ε) = f_{p_{j}}(ε), f_{𝔮_{j}}(ε) = f_{q_{j}}(ε^{2}) for each j and each ε appearing. So the bottom-right block is 2^{n_{i}−1}Reg_{K/ℚ}(ε_{σ(m_{σ}+3)} ∧ ⋯ ∧ ε_{σ(n′)}).

Referring back to (8.21), the only terms which appear in the sum after applying Reg_{K′/k′} are those for σ such that σ(1) ≤ n_{s} + 1 and σ(2) > n_{s} + 1. We put this in correspondence with a pair (i, j), 1 ≤ i ≤ n_{s} + 1, 1 ≤ j ≤ d + 1 such that σ(1) = i, σ(2) = n_{s} + 1 + j. Then one may check that sign(σ) = (−1)^{n_{s}}(−1)^{i+1}(−1)^{j+1}.

Putting all this together with (8.21) gives the stated result.

7 Base change for the congruence

We are now ready to show the base change statement for Conjecture 3.5.1.

We choose a basis ε_{1}, …, ε_{n′} as in §6. We also choose a basis u_{1}, …, u_{n′} for U_{k′,S,T} such that u_{i} = ε_{i} for i ≥ n_{s} + 2 (which is possible by Lemma 2.4.3) and, if we write u = u_{1} ∧ ⋯ ∧ u_{n′} and ε = ε_{1} ∧ ⋯ ∧ ε_{n′}, that these are similarly oriented in the sense that (8.5) is satisfied. Set u_{ℚ} = u_{n_{s}+2} ∧ ⋯ ∧ u_{n′}.

Theorem 7.1. We use the setup of Hypothesis 3.1 and assume Hypothesis 4.1. Assume Conjecture 3.5.1 holds for the extension K/ℚ, ie. that

ϕ(η_{K/ℚ}) ≡ ξ_{ℚ}h_{ℚ,S,T}Reg_{K/ℚ}(ϕ̃(u_{ℚ})) (mod I_{G}^{d+1}),

for all ϕ ∈ Hom_{G}(U_{K,S,T} , ℤ[G]). Assume also that the modified Darmon conjecture 5.4 holds. Then Conjecture 3.5.1 holds for the extension K′/k′ up to a power of 2. Explicitly, for all Φ ∈ ∧_{ℤ[G]}^{2}Hom_{ℤ[G]}(U_{K′,S,T} , ℤ[G]), we have

4 · 2^{n_{s}}Φ(η_{K′/k′}) ≡ ξ_{k′}4 · 2^{n_{s}}h_{k′,S,T} Reg_{K′/k′}(Φ̃(u)) (mod I_{G}^{d′+1}).

Proof. Recall from §4 that 4η_{K′/k′} = η_{ω} ∧ η_{K/ℚ}. Set (as in Chapter 5) H := #H^{1}(⟨τ⟩, U_{k′,S,T}). The conjectures tell us, using Proposition 5.5,

4Φ(η_{K′/k′}) = | \begin{matrix}ϕ_{1}(η_{ω}) & ϕ_{2}(η_{ω}) \\ ϕ_{1}(η_{K/ℚ}) & ϕ_{2}(η_{K/ℚ}) \end{matrix} |

≡ | \begin{matrix}ξ^{−}2m_{T} ϕ_{1}^{1}(R_{S,T}) & ξ^{−}2m_{T} ϕ_{2}^{1}(R_{S,T}) \\ ξ_{ℚ}h_{ℚ,S,T}Reg_{K/ℚ}(ϕ̃_{1}(u_{ℚ})) & ξ_{ℚ}h_{ℚ,S,T}Reg_{K/ℚ}(ϕ̃_{2}(u_{ℚ})) \end{matrix} |

modulo I_{G}^{d+n_{s}+1}, where R_{S,T} is calculated with respect to the ℤ-basis ε_{1}, …, ε_{n_{s}+1} of U_{k′,S,T}^{−}. Noting that d′ = d + n_{s}, this is

≡ ξ^{−}ξ_{ℚ}2m_{T} h_{ℚ,S,T} | \begin{matrix}ϕ_{1}^{1}(R_{S,T}) & ϕ_{2}^{1}(R_{S,T}) \\ Reg_{K/ℚ}(ϕ̃_{1}(u_{ℚ})) & Reg_{K/ℚ}(ϕ̃_{2}(u_{ℚ})) \end{matrix} | (mod I_{G}^{d′+1}).

By (8.18), 2m_{T} h_{ℚ,S,T} = 2^{n_{i}}h_{k′,S,T} H. Hence by Proposition 6.1, we have

4 · 2^{n_{s}}Φ(η_{K′/k′}) ≡ ξ^{−}ξ_{ℚ}(−1)^{n_{s}}h_{k′,S,T} · 2HReg(Φ̃(ε)) (mod I_{G}^{d′+1}).

Now we know from Lemma 2.3 that (U_{k′,S,T} : U_{ℚ,S,T}U_{k′,S,T}^{−}) = 2^{n_{s}+1}/H, so ε = (2^{n_{s}+1}/H)u. Therefore

4 · 2^{n_{s}}Φ(η_{K′/k′}) ≡ ξ^{−}ξ_{ℚ}(−1)^{n_{s}}h_{k′,S,T} · 2^{n_{s}+2}Reg_{K′/k′}(Φ̃(u_{ℚ})) (mod I_{G}^{d′+1}).

Concerning the signs, we note that we may use the results of §2 to compare ξ_{ℚ}, ξ^{−} and ξ_{k′} if we permute the ordering (8.14) of S_{k′} so that it relates to the ordering (8.13) of S in the same way that (8.4) relates to (8.3). This may be achieved by applying a cycle of length n_{s} + 1 to transform the sequence \overline{∞}′, 𝔭_{1}, 𝔭_{2}, …, 𝔭_{n} into 𝔭_{1}, 𝔭_{2}, …, 𝔭_{n}, \overline{∞}′, which is a permutation which has sign (−1)^{n_{s}}, corresponding to multiplication by this sign of the regulator R_{k′}. Thus, using the results of §2 and noting Hypothesis 4.1, we have ξ_{k′} = ξ^{−}ξ_{ℚ}(−1)^{n_{s}}.

Putting all this together gives the result.

Note that if #G is odd, this last congruence is the full statement of Conjecture 3.5.1.

Remark 7.2. Using the method of [12, Lem. 8.1], it is possible to prove that ϕ(N_{k′(ζ_{S})/K′}α_{S}) ∈ I_{G}^{n_{s}} for all ϕ ∈ Hom_{G}(U_{k′,S}^{−}, ℤ[G]), without assuming the validity of Darmon’s conjecture. It then follows that ϕ′(η_{ω}) ∈ I_{G}^{n_{s}} for all ϕ′ ∈ Hom_{G}(U_{K,S,T} , ℤ[G]). Thus if Conjecture 3.5.1 holds for K/ℚ, then for K′/k′ we have 4Φ(η_{K′/k′}) ∈ I_{G}^{d′}, for all Φ ∈ ∧_{ℤ[G]}^{2}Hom_{ℤ[G]}(U_{K′,S,T} , ℤ[G]), consistent with the “order of vanishing” implied by Conjecture 3.5.1.

8 Gross’s conjecture on the L-functions of tori

In [18, §8], Gross makes a conjecture motivated by considering algebraic tori. Similarly to Darmon’s later conjecture, this involves a quadratic extension of the base field and consideration of the “minus-units” in this extension. It also involves a “Θ” element which is twisted by the nontrivial character of the extension. In the previous section we saw that Darmon’s conjecture, which was related to the first derivative of the relative factor in (8.11), gave us a base change property for Conjecture 3.5.1 where the order of vanishing, r_{S}, increased by 1. Gross’s conjecture, by contrast, concerns the value (zero-th derivative) of the relative factor and correspondingly it gives us a base change property where r_{S} doesn’t change.

Our setup is as follows. Let k be a global field and K/k and k′/k linearly disjoint abelian extensions, with [k′ : k] = 2. Setting K′ = k′K, we are in the situation of §1, and we use the notation G, τ, ω from that section. Let S = S_{k} be a set of places of k containing all infinite places and all places ramifying in K′/k. That is, both K/k and k′/k are unramified outside S (and so, in the notation of §1, we have S′ = S). Take T = T_{k} such that U_{K′,S,T} is torsion free. We assume that r_{S} = r_{S′}, that is, the places splitting completely in K/k do not split in k′/k. We define n_{s} to be the number of places in S splitting in the quadratic extension k′/k, and refer to the other #S − n_{s} inert or ramified places as nonsplit. This is the setup of [18, §8], with slightly different notation in that the objects denoted L, σ, χ and n there are respectively called k′, τ, ω and n_{s} here.

We may order S as follows:

S = \{ \overbrace{v_{1}, …, v_{r_{S}},}^{r_{S} nonsplit in k′} \overbrace{v_{r_{S}+1}, …, v_{r_{S}+n_{s}}}^{n_{s} split in k′}, \overbrace{v_{r_{S}+n_{s}+1}, …, v_{n+1}}^{d−n_{s}+1 nonsplit in k′} \},  (8.24)

and then, with v′_{i} denoting a place of k′ above v_{i} and \overline{v′}_{i} := τv′_{i}, we order

S_{k′} = \{v′_{1}, …, v′_{r_{S}}, v′_{r_{S}+1}, …, v′_{r_{S}+n_{s}}, \overline{v′}_{r_{S}+1}, …, \overline{v′}_{r_{S}+n_{s}}, v′_{r_{S}+n_{s}+1}, …, v′_{n+1}\}.

(8.25)

We impose the following assumption, which is the same as Hypothesis 4.1 (i).

Hypothesis 8.1. There is a place in S which is nonsplit in k′/k and does not split completely in K/k. That is, d ≥ n_{s}.

We fix a basis u_{1}, …, u_{n′} of U_{k′,S,T} such that u_{n_{s}+1}, …, u_{n′} is a basis of U_{k,S,T}, which is possible by Lemma 2.4.3. Then this basis together with the orderings (8.24) and (8.25) define regulators R_{k} and R_{k′} as in (3.2). Also, choosing a basis ε_{1}, …, ε_{n_{s}} for U_{k′,S,T}^{−} such that (8.5) is satisfied defines a minus-unit regulator R^{−} as in §2.

We wish to apply the results of §2 to compare R_{k′}, R_{k} and R^{−}. However, we note that the orderings of sets (8.24) and (8.25) are those required for use in Chapter 3, §5 and are not of the form required by §2. Such a form may be achieved by rotating the first r_{S} + n_{s} terms of sequence (8.24) left r_{S} times, and rotating the first r_{S} + 2n_{s} terms of sequence (8.25) left r_{S} times, permutations which have signs (−1)^{(r_{S}+n_{s}+1)r_{S}} and (−1)^{(r_{S}+2n_{s}+1)r_{S}} respectively. Then the results of §2 show that, for R_{k′} and R_{k} calculated with respect to (8.24) and (8.25), we have (−1)^{(r_{S}+2n_{s}+1)r_{S}}R_{k′} = (−1)^{(r_{S}+n_{s}+1)r_{S}}2^{n−n_{s}}(U_{k′,S,T} : U_{k,S,T})R_{k}R^{−}, where R^{−} is the minus-unit regulator calculated relative to {v′_{r_{S}+1}, …, v′_{r_{S}+n_{s}}}. Hence

ξ_{k′} = (−1)^{n_{s}r_{S}}ξ_{k}ξ^{−}.  (8.26)

By (8.10), the leading term at s = 0 of the L-function of the nontrivial character ω of k′/k is

L_{k′/k,S,T}^{n_{s}} (0, ω) = ξ^{−} \frac{h_{k′,S,T}}{h_{k,S,T}} 2^{n−n_{s}}R^{−}#H^{1}(⟨τ⟩, U_{k′,S,T}).

The reader is warned that the factor 2^{n−n_{s}} is missing in [18, (8.5)]. Writing the leading term as L_{k′/k,S,T}^{n_{s}} (0, ω) = ξ^{−}m_{ω}R^{−}, so that (−1)^{n_{s}}ξ^{−}m_{ω} is the number denoted m_{τ} in [18, (8.4)] (the sign (−1)^{n_{s}} coming from the differing signs in logarithmic regulators), we have

m_{ω} = \frac{h_{k′,S,T}}{h_{k,S,T}} 2^{n−n_{s}}#H^{1}(⟨τ⟩, U_{k′,S,T}).

Lemma 8.2. Assuming Hypothesis 8.1, m_{ω} is an integer.

Proof. The hypothesis shows that n − n_{S} ≥ 0. Also k′/k is a quadratic extension unramified outside S such that at least one place in S is inert. Therefore Lemma 1.4 (i) shows that h_{k,S,T} divides h_{k′,S,T}. This gives the result.

We define an algebraic minus-unit regulator R_{G}^{−} ∈ ℤ[G]/I_{G}^{n_{s}+1} by the determinant of the n_{s}×n_{s} matrix with (i, j)-th entry f_{v′_{r_{S} +j}}(ε_{i}) for 1 ≤ i, j ≤ n_{s}, where the ε_{i} correspond to the u_{i} as in (8.5). This R_{G}^{−} corresponds to (−1)^{n_{s}} det_{G}(λ_{τ}) in the notation of [18] (with sign change for the same reason as noted in the discussion following the statement of Conjecture 3.3.2 above).

We can now state Gross’s tori conjecture [18, Conj. 8.8], which in our setup, assuming Hypothesis 8.1 in order to have the conclusion of Lemma 8.2, states the following

Conjecture 8.3 (Gross). We have

Θ_{K′/k′/k,S,T} (0, ω) ≡ ξ^{−}m_{ω}R_{G}^{−} (mod I_{G}^{n_{s}+1}).

We now show how the term Θ_{K′/k′/k,S,T} (0, ω) appearing in Gross’s conjecture relates the Stark units η_{K′/k′} and η_{K/k}. Differentiating the base-change factorisation of the L-functions (8.1) r_{S} times and evaluating at s = 0 gives

Θ_{K′/k′,S,T}^{r_{S}} (0) = Θ_{K′/k′/k,S,T} (0, ω)Θ_{K/k,S,T}^{r_{S}} (0).  (8.27)

The base-change factor Θ_{K′/k′/k,S,T} (0, ω) lies in ℤ[G] by the argument following [18, (8.7)], where the corresponding element is denoted θ_{G}(χ). The argument is essentially that, since no places in S split completely in K′ by our assumptions, the integrality follows from the integrality of Θ_{K′/k,S,T} (0) given by Theorem 3.3.1. Gross’s tori conjecture concerns Θ_{K′/k′/k,S,T} (0, ω), and we will show that its validity would imply that Conjecture 3.5.1 for K′/k′, S_{k′}, T_{k′} (weakened by powers of 2, similarly to the case of Darmon’s conjecture) follows from the conjecture for K/k, S, T.

The place v_{n+1} is inert in k′ by Hypothesis 8.1. Choose w′_{i} a place of K′ above v_{i} for i = 1, …, n+1. Set x′ = (w′_{1}−w′_{n+1})∧⋯∧(w′_{r_{S}} −w′_{n+1}). Write w_{i} for the place of K induced by w′_{i} for i = 0, …, r_{S}, and set x = (w_{1} −w_{n+1})∧⋯∧(w_{r_{S}} −w_{n+1}). Then according to Chapter 2, §1 we have that η_{K/k} ∈ ℂ ∧_{ℤ[G]}^{r_{S}} U_{K,S,T} is defined by the equation

λ_{S_{K}}(η_{K/k}) = Θ_{K/k,S,T}^{r_{S}} (0)x.

The commutative diagram (2.5) holds here for the extension K′/K. Since the places v_{1}, …, v_{r_{S}} and v_{n+1} are all nonsplit in k′/k, there is a unique w′_{i} over w_{i} for i = 1, …, r_{S} and i = n + 1. Hence

λ_{K′}(η_{K/k}) = Θ_{K/k,S,T}^{r_{S}} (0)((2w′_{1}−2w′_{n+1})∧⋯∧(2w′_{r_{S}} −2w′_{n+1})) = 2^{r_{S}}Θ_{K/k,S,T}^{r_{S}} (0)x′.

Therefore we have, using (8.27), λ_{K′}(η_{K/k}^{Θ_{K′/k′/k,S,T} (0,ω)}) = 2^{r_{S}}Θ_{K′/k′,S,T}^{r_{S}} (0)x′. It follows that

η_{K′/k′} = \frac{1}{2^{r_{S}}} η_{K/k}^{Θ_{K′/k′/k,S,T} (0,ω)}.  (8.28)

9 Base change via the tori conjecture

Keeping the assumptions and notation of the previous section, we now go on to study the regulators involved in the various conjectures.

The analogue of Proposition 6.1 in this situation is the following.

Proposition 9.1. We have the following in ℤ[G]/I_{G}^{d+n_{s}+1}:

Reg_{K′/k′}(Φ̃(u_{1} ∧ ⋯ ∧ u_{n′})) =

(−1)^{r_{S}n_{s}}2^{d−n_{s}}R_{G}^{−}Reg_{K/k}(Φ̃(u_{1+n_{s}} ∧ ⋯ ∧ u_{n′}))#H^{1}(⟨τ⟩, U_{k′,S,T}).

Proof. The regulator on the left is

∑_{σ∈\left[{n′ \atop r_{S}}\right]} sign(σ) det (ϕ_{j}^{1}(u_{σ(i)}))_{1≤i,j≤r_{S}}Reg_{K′/k′}(u_{σ(r_{S}+1)} ∧ ⋯ ∧ u_{σ(n′)}), (8.29)

where, after manipulations as in the proof of Proposition 6.1, Reg_{K′/k′}(u_{σ(r_{S}+1)} ∧ ⋯ ∧ u_{σ(n′)}) is seen to be the determinant in ℤ[G]/I_{G}^{d+n_{s}+1} of the matrix

v′_{r_{S}+1}, …, v′_{r_{S}+n} \overline{v′}_{r_{S}+1}, …, \overline{v′}_{r_{S}+n} v′_{r_{S}+n+1}, …, v′_{r_{S}+n+d}
u_{σ(r_{S}+1)} ; ⋮ ; u_{σ(r_{S}+m_{σ})}  f_{v′_{j}}(u_{i}^{1−τ})  f\overline{v′_{j}}(u_{i})  f_{v′_{j}}(u_{i})

u_{σ(r_{S}+m_{σ}+1)} ; ⋮ ; u_{σ(r_{S}+d+n)}  0  f_{v_{j}}(u_{i})  2(f_{v_{j}}(u_{i}))

in which m_{σ} = #[σ({r_{S} + 1, …, n′}) ∩ {1, …, n_{s}}]. Now if m_{σ} < n_{s} then this determinant is clearly 0. So for nonzero terms in the sum (8.29) we must have m_{σ} = n_{s}, ie. σ(r_{S} + 1) = 1, …, σ(r_{S} + n_{s}) = n_{s}. The determinant of the top left n_{s} ×n_{s} block is (U_{k′,S,T}^{−} : U_{k′,S,T}^{1−τ})R_{G}^{−} (where the sign has been fixed by (8.5)). The determinant of the bottom-right d × d block is 2^{d−n_{s}}Reg_{K/k}(u_{σ(r_{S}+n_{s}+1)} ∧ ⋯ ∧ u_{σ(n′)}).

Note that for such σ, the map σ ∘ (1 2 … r_{S} + n_{s})^{r_{S}} is a permutation of \{n_{s} + 1, …, n′\} of the form n_{s} + k ↦ n_{s} + σ′(k) for σ′ ∈ \left[{n \atop r_{S}}\right]. Thus sign(σ) = (−1)^{r_{S}(r_{S}+n_{s}+1)}sign(σ′) = (−1)^{r_{S}n_{s}}sign(σ′), since (−1)^{r_{S}(r_{S}+1)} = 1. We have

Reg_{K′/k′}(Φ̃(u_{1} ∧ ⋯ ∧ u_{n′})) = 2^{d−n_{s}}(U_{k′,S,T}^{−} : U_{k′,S,T}^{1−τ})R_{G}^{−} ∑_{σ′}((−1)^{r_{S}n_{s}}sign(σ′)

det (ϕ_{j}^{1}(u_{n_{s}+σ′(i)}))_{1≤i,j≤r_{S}}Reg_{K/k}(u_{n_{s}+σ′(r_{S}+1)} ∧ ⋯ ∧ u_{n_{s}+σ′(r_{S}+d)})),

which gives the result.

We may now give the base-change theorem.

Theorem 9.2. Let K/k, k′/k be finite linearly disjoint abelian extensions of a global field k, with [k′ : k] = 2. Set K′ = k′K. Assume S = S_{k}, T = T_{k} are such that K′/k is unramified outside S and U_{K′,S,T} is torsion-free. Assume that r_{S_{k′}} = r_{S}, and that Hypothesis 8.1 holds for these data.

Assume that Conjecture 3.5.1 holds for K/k, S, T and that Conjecture 8.3 holds. Then for all Φ ∈ ∧_{ℤ[G]}^{r_{S}}Hom_{ℤ[G]}(U_{K′,S,T} , ℤ[G]) we have

2^{r_{S}}Φ(η_{K′/k′}) ≡ ξ_{k′}2^{r_{S}}h_{k′,S,T} (Reg_{G} ∘ Φ̃) (mod I_{G}^{d′+1}).

That is, the conclusion of Conjecture 3.5.1 for K′/k′, S_{k′}, T_{k′} holds with a factor of 2^{r_{S}} on each side.

Proof. By (8.28) we have 2^{r_{S}}Φ(η_{K′/k′}) = Θ_{K′/k′/k,S,T} (0, ω)Φ(η_{K/k}). Multiplying the congruences of Conjecture 8.3 and Conjecture 3.5.1 gives

2^{r_{S}}Φ(η_{K′/k′}) ≡ ξ^{−} \frac{h_{k′,S,T}}{h_{k,S,T}} 2^{n−n_{s}}(U_{k′,S,T}^{−} : U_{k′,S,T}^{1−τ})R_{G}^{−} ·
ξ_{k}h_{k,S,T} Reg_{K/k}(Φ̃(u_{1+n_{s}} ∧ ⋯ ∧ u_{n′})) (mod I_{G}^{d+n_{s}+1})
≡ ξ_{k′}2^{r_{S}}h_{k′,S,T} Reg_{K′/k′}(Φ̃(u_{1} ∧ ⋯ ∧ u_{n′})) (mod I_{G}^{d′+1}),

by the regulator calculation in Proposition 9.1. Note that for the last congruence we have used (8.26).

Note that if r_{S} = 0 then this shows that, under Hypothesis 8.1, Gross’s conjecture on tori actually gives a base-change property with no weakening factor for [18, Conj. 4.1].

The augmentation filtration

Let G be a finite abelian group, and ℤ[G] its integral group ring. Recall from Chapter 2, §6 that the augmentation ideal I_{G} ⊆ ℤ[G] is the kernel of the augmentation map aug : ℤ[G] → ℤ. The powers I_{G}^{d} form a filtration of ℤ[G] by ideals. We note that Conjecture 3.3.2 and its generalisation, Conjecture 3.5.1, predict that a certain element of ℤ[G] should belong to some I_{G}^{d} and satisfy a congruence modulo the ideal I_{G}^{d+1}. The latter is equivalent to giving a value for its projection to the quotient I_{G}^{d}/I_{G}^{d+1}. In Chapter 5, §3, we make use of the fact that for G a group with two elements, I_{G}^{d}/I_{G}^{d+1} is a group with two elements; this is one example where it is useful to know about the structure of the quotients I_{G}^{d}/I_{G}^{d+1}. In this chapter we summarise some facts about this structure and give the results of a computer investigation which determined this structure for some small groups G.

By [31, Prop. 3.25 (ii)], the quotients I_{G}^{d}/I_{G}^{d+1} decompose as

I_{G}^{d}/I_{G}^{d+1} ≅ ⨁_{p} I_{G_{p}}^{d}/I_{G_{p}}^{d+1}

where G_{p} is the Sylow p-subgroup of G, so to address questions of structure it is sufficient to assume that G is a p-group. We assume this from now on. For pgroups, useful structural results are known when G is cyclic (e.g. [31, Thm. 3.1]), where we have I_{G}^{d}/I_{G}^{d+1} ≅ G for all d > 0, and if G is elementary (i.e. exponent p) there is a useful and fairly explicit description in terms of quotients of polynomial rings given by [33, §3]. Therefore we will consider the case where G is not cyclic and has exponent bigger than p.

If H is a subgroup of G, there is a natural surjective homomorphism ℤ[G] ⟶ ℤ[G/H]. One might hope that it is sufficient in our case to study the image of I_{G}^{d} in those quotient rings for which G/H is cyclic, where the structure is well understood. David Burns and Joongul Lee have asked (in private correspondence) whether

I_{G}^{d}/I_{G}^{d+1} ⟶ ∏_{H} I_{G/H}^{d}/I_{G/H}^{d+1}  (9.1)

is an injection, which is the same as saying that if x ∈ ℤ[G] is in I_{G/H}^{d+1} after projection to each ℤ[G/H], then it is in I^{d+1}. This is known to be true in the elementary case [25, Cor. 5] and obviously in the cyclic case. In fact, it turns out not to be true in general. We will study an example where it fails. This means that, in studying Conjectures 3.3.2 and 3.5.1, it is not sufficient to consider only cyclic subextensions L/k of K/k.

The calculations were carried out on a computer, using the matrix manipulation provided by the PARI system [30].

1 Method

Let f be the number of cyclic factors of G and fix a generator a_{i} ∈ G for the i-th factor.

Fix an ordering of the elements of G, g_{1}, …, g_{#G}. Such an ordering gives us an obvious isomorphism of ℤ-modules

\begin{matrix}ℤ[G] & ⟶ & ℤ^{#G} \\ ∑_{i} n_{i}g_{i} & ⟼ & (n_{i})_{i}. \end{matrix}

Via this isomorphism we can use linear algebra to study the structure of the augmentation filtrations. Multiplication by an element x ∈ ℤ[G] induces a ℤendomorphism of ℤ[G] by x ∑_{i} n_{i}g_{i} = ∑_{i}(∑_{j} m_{ij}n_{j})g_{i}, that is, xy = A_{x}y where (A_{x})_{ij} = m_{ij} and y is considered as a vector in ℤ^{#G}. On the other hand, for an integer d ≥ 1, the d-th power I_{G}^{d} of the augmentation ideal is generated over ℤ[G] by the finite set

S = {(a_{1} − 1)^{r_{1}}(a_{2} − 1)^{r_{2}} ⋯ (a_{f} − 1)^{r_{f}} : r_{i} ≥ 0, r_{1} + … + r_{f} = d}.

Therefore the ℤ-submodule I_{G}^{d} ⊆ ℤ[G] is equal to the ℤ-span of the columns of the matrix A′ = (A_{x_{1}}| ⋯ |A_{x_{#S}}) where x_{i} runs through the elements of S. Given A′, there exists a unique matrix M_{I_{G}^{d}} in Hermite Normal Form [10, §2.4.2] whose column span over ℤ is the same as that of A′. The columns of M_{I_{G}^{d}} form a basis for the free ℤ-module I_{G}^{d}. We observe that the rank of M_{I_{G}^{d}} is always #G − 1, since this may easily be seen to be the rank of I_{G}^{1}, and the quotients I_{G}^{d}/I_{G}^{d+1} are finite.

For any d ≥ 1, the ℤ-module I_{G}^{d+1} is a submodule of I_{G}^{d} with finite index. Thus it is possible to express the basis M_{I_{G}^{d+1}} of I_{G}^{d+1} in terms of the basis M_{I_{G}^{d}} of I_{G}^{d} as M_{I_{G}^{d+1}} = M_{I_{G}^{d}}B, where B is some (#G − 1)-square matrix with integer entries. B may now be reduced to Smith normal form by elementary row and column operations. If U, V are square invertible matrices over ℤ such that UBV = diag(d_{1}, …, d_{#G−1}) with d_{1} | d_{2} | … | d_{#G−1}, d_{i} positive integers, then I_{G}^{d}/I_{G}^{d+1} ≅ ℤ/d_{1}ℤ × ⋯ × ℤ/d_{#G−1}ℤ and the columns of M_{I_{G}^{d}}U^{−1} are generators of each cyclic factor. In this way the structure of the I_{G}^{d}/I_{G}^{d+1} and a minimal set of representatives in ℤ[G] were determined.

We now do some simple manipulations useful for studying the augmentation filtrations, relating to cyclic groups or subgroups. Let g ∈ G have order n and write N_{g} = ∑_{i=0}^{n−1} g^{i}. Then the following equalities are easy to check:

(g − 1)N_{g} = 0  (9.2)
(g − 1)(−g^{n−2} − 2g^{n−3} − 3g^{n−4} − … − (n − 2)g − (n − 1)) = n − N_{g} (9.3)
(g − 1)(n − N_{g}) = n(g − 1)  (9.4)

This shows that n(g − 1) ∈ I_{G}^{2}, and in particular it proves that I_{G}^{d}/I_{G}^{d+1} is annihilated by the exponent of G for d ≥ 1.

It is easy to “divide” by the augmentation element (g − 1) in I_{⟨g⟩}. If y = ∑_{i=0}^{n−1} y_{i}g^{i} is an element with augmentation 0, then y = (g − 1)x where x = −(y_{0}) − (y_{0} + y_{1})g − … − (y_{0} + … + y_{n−2})g^{n−2}. Also, by (9.2) we may add multiples of N_{g} to x until its augmentation is in the range 0 ≤ aug x < n. This is then uniquely determined by y, and it tells us whether y is in I_{G}^{2} by the next result.

Proposition 1.1. Let Γ be a cyclic group of order p^{m} generated by γ. Then every x ∈ ℤ[Γ] has a unique expression

x = ∑_{i=0}^{∞} x_{i}(γ − 1)^{i}

with 0 ≤ x_{i} < p^{m} for i > 0.

Proof. Such an expression exists, with x_{0} = aug(x), by the “division” process described above. Suppose ∑ x′_{i}(γ − 1)^{i} is a different expression for x. Let r be the first i where x′_{i} ≠ x_{i} and assume (without loss of generality) that 0 < t := x′_{i} − x_{i}. Then t(γ − 1)^{r} ∈ I_{Γ}^{r+1}. But multiplication by γ − 1 induces an isomorphism I_{Γ}^{d} ⟶ I_{Γ}^{d+1} for all d ≥ 1 (see eg. [26, Lem. 3]). Hence t(γ − 1) ∈ I_{Γ}^{2}. But this means γ^{t} = 1 in G, which is impossible since 0 < t < p^{m}. Hence the expression for x is unique.

Thus we see that by iterating the above process we can find the highest power of the augmentation ideal containing a given element of ℤ[⟨g⟩]. This method is used to determine the kernel of the map (9.1).

2 Results

We first show that the map (9.1) is not always injective.

Let G = ℤ/3ℤ×ℤ/9ℤ be the smallest odd, non-elementary, non-cyclic p-group. Write a = (1, 0) and b = (0, 1) for the generators of the first and second cyclic factors respectively. We will first determine all the nontrivial cyclic quotients of G. The subgroups H such that G/H is cyclic are easily seen to be the kernels of the characters χ : G ⟶ ℂ^{×}. Since G has exponent 9, the image of any χ is contained in the group of ninth roots of unity, which is isomorphic to ℤ/9ℤ. So we consider all homomorphisms ω : G ⟶ ℤ/9ℤ.

We have either ω(a) = 0, 3 or 6. Firstly assume ω(a) = 0, so b ∈ H := ker ω. The order of ω(b) is either 1, 3 or 9. If ω(b) = 0 then H = G gives the trivial quotient, which we denote by Γ_{0}. If ω(b) has order 3 then H = ⟨a, b^{3}⟩ giving quotient Γ_{1}. Otherwise H = ⟨a⟩, giving quotient Γ_{2}.

Note that the Galois group 𝔊 = (ℤ/9ℤ)^{×} acts on ℤ/9ℤ and permutes the ω, the kernel H being an invariant of the orbit. Every character with ω(a) = 6 is in the same orbit as a character with ω(a) = 3 by the action of −1 ∈ 𝔊. So we may assume ω(a) = 3 to find the rest of the kernels. Now the possible 𝔊-orbits of ω(b) are {0}, {1, 4, 7}, {2, 5, 8},{3} and {6}, giving subgroups as follows:

ω(b)  0  1  2  3  6
H  ⟨b⟩ ⟨a^{2}b^{3}⟩ ⟨ab^{3}⟩ ⟨a^{2}b⟩ ⟨ab⟩
G/H name Γ_{3} Γ_{4}  Γ_{5}  Γ_{6}  Γ_{7}

Thus there are 7 nontrivial cyclic quotients Γ_{1}, …, Γ_{7}.

We now present an element which shows that the map (9.1) is not injective for d = 5. Let

x := 2(a − 1)^{2}(b − 1)^{3} + (a − 1)^{4}(b − 1).

Then x ∈ I_{G}^{5} is clear. It is also clear that x projects to 0 in ℤ[Γ_{1}], ℤ[Γ_{2}] and ℤ[Γ_{3}] since the corresponding H contains a or b. So we wish to show it projects to an element of I_{Γ_{i}}^{6} in ℤ[Γ_{i}] for i = 4, 5, 6, 7. In the following table, x̄ denotes the projection of x in ℤ[Γ_{i}] in terms of a chosen generator σ of Γ_{i}, the choice of which is also indicated in the table.

Quotient σ #Γ_{i}  x̄
Γ_{4}  b̄ 9 (σ − 1)^{6}(12 + 15σ + 10σ^{2} + σ^{3} − 9σ^{4}
−14σ^{5} − 13σ^{6} − 6σ^{7} + 4σ^{8})
Γ_{5} b̄ 9 (σ − 1)^{6}(5 − 3σ − 8σ^{2} − 9σ^{3} − 6σ^{4}
−2σ^{5} + 4σ^{6} + 9σ^{7} + 10σ^{8})
Γ_{6}  ā 3  (σ − 1)^{6}(−1 + σ^{2})
Γ_{7}  ā^{2} 3  (σ − 1)^{6}

Therefore x is in the kernel of the map (9.1). On the other hand, x is not in I_{G}^{6}. This is proved as follows: the matrix M_{I_{G}^{6}} whose columns form a Hermite normal form basis for the ℤ-module I^{6} ⊆ ℤ[G] has top-right entry −51. But reducing the matrix (M_{I_{G}^{6}}|x) to Hermite normal form produces a matrix with top-right entry −53. Because the Hermite normal form is an invariant of the ℤ-span of the columns of the matrix, this shows that x ∉ I_{G}^{6}. Therefore x is a nontrivial element of the kernel of the map (9.1).

We remark furthermore that all the proper quotients of G are cyclic or elementary. Since the map (9.1) is known to be injective for G elementary, this shows that x ∈ I_{G/H}^{6} for all proper subgroups H.

The following table shows the structure of the first few augmentation quotients for some small p-groups. The notation C_{a}^{b}C_{c}^{d} is short for (ℤ/aℤ)^{b} × (ℤ/cℤ)^{d}. K denotes the kernel of the map (9.1). Where the size of K is not recorded it is equal to 1.

We observe that it is proved in [1] that the structure of the quotients I_{G}^{d}/I_{G}^{d+1} is eventually constant. This can be seen in the following table for the smaller groups.

G  #G I_{G}^{2}/I_{G}^{3} I_{G}^{3}/I_{G}^{4} I_{G}^{4}/I_{G}^{5}  I_{G}^{5}/I_{G}^{6}  I_{G}^{6}/I_{G}^{7}  I_{G}^{7}/I_{G}^{8}
C_{2}C_{4}  8 C_{2}^{2}C_{4} C_{2}^{3}C_{4} C_{2}^{3}C_{4}  C_{2}^{3}C_{4}  C_{2}^{3}C_{4}  C_{2}^{3}C_{4}
C_{4}C_{4}  16  C_{4}^{3}  C_{2}C_{4}^{3} C_{2}^{2}C_{4}^{3}  C_{2}^{3}C_{4}^{3}  C_{2}^{3}C_{4}^{3}  C_{2}^{3}C_{4}^{3}
C_{2}C_{8}  16 C_{2}^{2}C_{8} C_{2}^{3}C_{8} C_{2}^{4}C_{8}  C_{2}^{4}C_{8}  C_{2}^{4}C_{8}  C_{2}^{4}C_{8}
#K = 2 #K = 2 #K = 2 #K = 2
C_{2}C_{2}C_{4} 16 C_{2}^{5}C_{4} C_{2}^{8}C_{4} C_{2}^{9}C_{4} C_{2}^{9}C_{4} C_{2}^{9}C_{4} C_{2}^{9}C_{4}
#K = 2 #K = 2 #K = 2 #K = 2
C_{4}C_{8}  32 C_{4}^{2}C_{8} C_{4}^{3}C_{8} C_{2}C_{4}^{3}C_{8} C_{2}^{2}C_{4}^{3}C_{8} C_{2}^{3}C_{4}^{3}C_{8} C_{2}^{4}C_{4}^{3}C_{8}
#K = 2 #K = 4
C_{2}C_{4}C_{4} 32 C_{2}^{3}C_{4}^{3} C_{2}^{7}C_{4}^{3} C_{2}^{10}C_{4}^{3} C_{2}^{12}C_{4}^{3} C_{2}^{13}C_{4}^{3} C_{2}^{13}C_{4}^{3}
#K = 2 #K = 4 #K = 8 #K = 8
C_{8}C_{8} 64 C_{8}^{3} C_{4}C_{8}^{3} C_{4}^{2}C_{8}^{3} C_{4}^{3}C_{8}^{3} C_{2}C_{4}^{3}C_{8}^{3} C_{2}^{2}C_{4}^{3}C_{8}^{3}
#K = 2 #K = 4
C_{3}C_{9} 27 C_{3}^{2}C_{9} C_{3}^{3}C_{9} C_{3}^{4}C_{9} C_{3}^{5}C_{9} C_{3}^{5}C_{9} C_{3}^{5}C_{9}
#K = 3 #K = 3 #K = 3
C_{9}C_{9}  81  C_{9}^{3}  C_{9}^{4}  C_{3}C_{9}^{4}  C_{3}^{2}C_{9}^{4}  C_{3}^{3}C_{9}^{4}  C_{3}^{4}C_{9}^{4}
C_{3}C_{27} 81 C_{3}^{2}C_{27} C_{3}^{3}C_{27} C_{3}^{4}C_{27} C_{3}^{5}C_{27}  C_{3}^{6}C_{27}  C_{3}^{7}C_{27}
#K = 3 #K = 9 #K = 27
C_{9}C_{27} 243 C_{9}^{2}C_{27} C_{9}^{3}C_{27} C_{9}^{4}C_{27} C_{9}^{5}C_{27} C_{3}C_{9}^{5}C_{27} C_{3}^{2}C_{9}^{5}C_{27}
#K = 3 #K = 3 #K = 3
C_{5}C_{25} 125 C_{5}^{2}C_{25} C_{5}^{3}C_{25} C_{5}^{4}C_{25} C_{5}^{5}C_{25} C_{5}^{6}C_{25} C_{5}^{7}C_{25}
#K = 5

We see that K may in general be nontrivial but tends to be very small compared to the size of the group.

Bibliography

[1] F. Bachmann and L. Grünenfelder, The periodicity in the graded ring associated with an integral group ring, J. Pure and Applied Algebra 5 (1974), 253–264.

[2] W. Bley, Wild Euler systems of elliptic units and the equivariant Tamagawa number conjecture, to appear in J. Reine Angew. Math. (2004).

[3] W. Bley and D. Burns, Explicit units and the equivariant Tamagawa number conjecture, Amer. J. Math. 123 (2001), 931–949.

[4] D. Burns and M. Flach, On Galois structure invariants associated to Tate motives, Amer. J. Math. 120 (1998), no. 6, 1343–1397.

[5] D. Burns and M. Flach, Tamagawa numbers for motives with (noncommutative) coefficients, Doc. Math. 6 (2001), 501–570.

[6] D. Burns, Equivariant Tamagawa numbers and refined abelian Stark conjectures, J. Math. Soc. Univ. Tokyo 10 (2003), 225–259.

[7] D. Burns, Congruences between derivatives of abelian L-functions at s = 0, Preprint (2004).

[8] D. Burns, C. Popescu, J. Sands, D. Solomon (eds.), Proceedings of the conference on Stark’s conjecture and related topics, Contemporary Math., American Mathematical Society (2004).

[9] H. Cartan and S. Eilenberg, Homological Algebra, Princeton Math. Ser. 19, Princeton (1956).

[10] H. Cohen, Advanced topics in computational number theory, Graduate Texts in Mathematics 193, Springer Verlag (2000).

[11] S. Chaiken and D. J. Kleitman, Matrix tree theorems, J. Combinatorial Theory (Ser. A) 24 (1978), 377–381.

[12] H. Darmon, Thaine’s method for circular units and a conjecture of Gross, Canadian J. Math. 47 (1995) 302–317.

[13] P. Deligne and K. Ribet, Values of abelian L-functions at negative integers over totally real fields, Invent. Math. 59 (1980), 227–286.

[14] F. Gerrish, Surjections from an m-set to an n-set, Mathl. Gaz. 63 (1979), No. 426, 259–260.

[15] D. Goss, Basic structures of function field arithmetic, Ergebnisse der Mathematik und ihre Grenzgebiete, 3. Folge, Bd 35, Springer Verlag (1996).

[16] H. W. Gould, Euler’s formula for nth differences of powers, Amer. Mathl. Monthly 85 (1978), 450–467.

[17] C. Greither and R. Kučera, The lifted root number conjecture for fields of prime degree over the rationals: an approach via trees and Euler systems, Ann. Inst. Fourier 53 (2002), no. 3, 735–777.

[18] B. H. Gross, On the values of abelian L-functions at s=0, J. Fac. Sci. Univ. Tokyo, Sect 1A, Math. 35 (1988) 177–197.

[19] B. H. Gross and M. Rosen, Fourier series and the special values of Lfunctions, Adv. Math. 69 (1988), 1–31.

[20] D. Hayes, Explicit class field theory in global function fields, Adv. in Math. Suppl. Stud. 6, Academic Press (1979), 173–217.

[21] D. Hayes, Stickelberger elements in function fields, Compos. Math. 55 (1985), 209–239.

[22] A. Hayward, A class number formula for higher derivatives of abelian Lfunctions, Compos. Math. 140 (2004) 99–129.

[23] S. Lang, Algebraic Number Theory, Second Edition, Graduate Texts in Mathematics 110, Springer Verlag (1994).

[24] S. Lang, Algebra (revised 3rd ed.), Graduate Texts in Mathematics 211, Springer Verlag (2002).

[25] J. Lee, On Gross’s refined class number formula for elementary abelian extensions, J. Math. Sci. Univ. Tokyo 4 (1997), 373–383.

[26] J. Lee, Stickelberger element for cyclic extensions and the order of vanishing of abelian L-functions at s = 0, Compos. Math. 138 (2003), no. 2, 157–163.

[27] J. Martinet, Character theory and Artin L-functions, in Algebraic Number Fields, ed. A. Fröhlich, Academic Press (1977).

[28] J. Neukirch, Algebraic Number Theory, Grundlehren der mathematischen Wissenschaften 322, Springer Verlag (1999).

[29] H. Oukhaba, Index formulas for ramified elliptic units, Compos. Math. 137 (2003), 1–22.

[30] The PARI Group, PARI, Version 2.1.4, Bordeaux (2002), (available from

    http://pari.math.u-bordeaux.fr/).

[31] I. B. S. Passi, Polynomial maps on groups, J. Algebra 9 (1968), 121–151.

[32] I. B. S. Passi, Polynomial functors, Proc. Cambridge Philos. Soc. 66 (1972), 25–32.

[33] I. B. S. Passi and L. R. Vermani, The associated graded ring of an integral group ring, Math. Proc. Camb. Phil. Soc. 82 (1977), 25–33.

[34] C. D. Popescu, Base change for Stark-type conjectures “Over ℤ”, J. Reine und Angew. Math. 542 (2002) 85–111.

[35] M. Reid, Gross’ conjecture for extensions ramified over three points of ℙ^{1}, J. Math. Sci. Univ. Tokyo 10 (2003), no. 1, 119–138.

[36] D. S. Rim, An exact sequence in Galois cohomology, Proc. Amer. Math. Soc. 16 (1965), 837–840.

[37] G. Robert, Concernant la relation de distribution satisfaite par la fonction φ associée à un réseau complexe, Invent. Math. 100 (1990) 231–357.

[38] G. Robert, Concernant la relation de distribution satisfaite par la fonction φ associée à un réseau complexe, Invent. Math. 100 (1990) 231–357.

[39] K. Rubin, A Stark conjecture “over ℤ” for abelian L-functions with multiple zeros, Ann. Inst. Fourier 46 (1996) 33–62.

[40] R. Schertz, Galoismodulstruktur und elliptische Funktionen, J. Number Theory 39 (1991), no. 3, 285–326.

[41] J.-P. Serre, Local Fields, Graduate Texts in Mathematics 67, Springer Verlag (1979).

[42] H. M. Stark, Values of L-functions at s = 1. I. L-functions for quadratic forms, Adv. Math. 7 (1971), 301–343.

[43] H. M. Stark, L-functions at s = 1. II. Artin L-functions with rational characters, Adv. Math. 17 (1975), no. 1, 60–92.

[44] H. M. Stark, L-functions at s = 1. III. Totally real fields and Hilbert’s twelfth problem, Adv. Math. 22 (1976), no. 1, 64–84.

[45] H. M. Stark, L-functions at s = 1. IV. First derivatives at s = 0, Adv. Math. 35 (1980), 197–235.

[46] J. T. Tate, Les conjectures de Stark sur les fonctions L d’Artin en s=0, Progress in Mathematics 47, Birkhäuser (1984).

[47] L. C. Washington, Introduction to cyclotomic fields, Second Edition, Graduate Texts in Mathematics 83, Springer Verlag (1997).

[48] H. Weber, Lehrbuch der Algebra, 2. Auflage, 3. Band, Vieweg und Sohn (1908).

[49] A. Weil, Basic Number Theory, Grundlehren der mathematischen Wissenschaften 144, Springer Verlag (1967).