Congruences satisfied by Stark units Anthony Hayward King’s College, London

http://www.mth.kcl.ac.uk/∼hayward I. What are Stark units?

\begin{matrix}L \\ |G \\ K \end{matrix} abelian extension of global fields. S, T disjoint

finite sets of places of K such that S ≠ ∅ contains all infinite and ramified primes and U_{L,S,T} torsion-free. U_{L,S,T} = {x ∈ L^{×} : w(x) = 0 for w|_{K} ∉ S,

w(x − 1) > 0 for w|_{K} ∈ T }.

(S, T)-modified Artin (Dirichlet) L-function

Θ_{S,T}(s) = ∑_{χ∈Ĝ} L_{L/K,S,T}(s, χ^{−1})e_{χ} ∈ ℂ[G] (s ∈ ℂ)

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

L_{L/K,S,T}(s, χ) = ∏_{𝔭∈T}(1−N_{𝔭}^{1−s}χ(σ_{𝔭})) ∏_{𝔭∉S}(1−N_{𝔭}^{−s}χ(σ_{𝔭}))^{−1}

“Refined abelian Stark conjectures” are special value conjectures for Θ_{S,T} at 0. r splitting places in S force vanishing to order r in all L-functions.

Θ_{S,T}^{(r)}(0) = lim_{s→0} s^{−r}Θ_{S,T}(s)

r-th term in Taylor series.

• Stark (r = 1): Θ′_{S}(0) ↔ R(η)

η ∈ U_{L,S} with special properties, R Dirichlet logarithmic regulator over splitting place.

• Rubin[-Stark]: Θ_{S,T}^{(r)}(0) = R(η)

η ∈ ℂ ⊗_{ℤ} ∧_{ℤ[G]}^{r}U_{L,S,T} such that Φ(η) ∈ ℤ[G] for all Φ ∈ ∧_{ℤ[G]}^{r}Hom_{ℤ[G]}(U_{L,S,T}, ℤ[G]). R = Dirichlet logarithmic regulator for r splitting places = det(− ∑_{σ∈G} ln | u^{σ^{−1}} |_{w} σ).

• Gross (r = 0): Θ_{S,T}(0) ≡ h_{K,S,T}Reg_{K,S,T} (mod I_{G}^{#S})

h_{K,S,T} = a class number, Reg_{K,S,T} ∈ ℤ[G] algebraic regulator = det((u, L_{w}/K_{v}) − 1), u ∈ U_{K,S,T}, v ∈ S.

• ETNC Θ_{S}^{∗}(0)ℤ[G] = ϑ_{R}(Det_{ℤ[G]}(C^{•}))

Θ_{S}^{∗} = characterwise leading term ϑ_{R} = trivialisation by logarithmic regulator C^{•} = an étale cohomology complex Kato, Kodai Math. J. (1993). Generalised by Burns and Flach.

By interpreting ETNC as existence of certain systems of units, Burns was led to combine Rubin’s and Gross’s conjectures. S, T, r as above, so Θ_{S,T}^{(r)}(0) = R(η). Rubin:

Φ(η) ∈ ℤ[G] for all Φ ∈ ∧_{ℤ[G]}^{r}Hom_{ℤ[G]}(U_{L,S,T}, ℤ[G])

Postulate a congruence for Φ(η) in the spirit of Gross’s conjecture. Φ = ϕ_{1} ∧ ⋯ ∧ ϕ_{r}, and ϕ_{i} ∈ Hom_{ℤ[G]}(U_{L,S,T}, ℤ[G]) ≅ Hom_{ℤ}(U_{L,S,T}, ℤ) ∋ ϕ_{i}^{1} Let f_{v} = (−, L_{w}/K_{v}) local reciprocity map, #S = r + d + 1.

Reg^{Φ} = | \begin{matrix}ϕ_{1}^{1}(u_{1}) & ⋯ & ϕ_{1}^{1}(u_{r+d}) \\ ⋮ & ... & ⋮ \\ ϕ_{r}^{1}(u_{1}) & ⋯ & ϕ_{r}^{1}(u_{r+d}) \\ f_{v_{1}}(u_{1}) − 1 & ⋯ & f_{v_{1}}(u_{r+d}) − 1 \\ ⋮ & ... & ⋮ \\ f_{v_{d}}(u_{1}) − 1 & ⋯ & f_{v_{d}}(u_{r+d}) − 1 \end{matrix} |

where u_{i} are a ℤ-basis for U_{K,S,T}. There is a natural interpretation of Reg^{Φ} in terms of

maps “∧^{r+d}U_{K,S,T} ⟶^{Φ^{1}} ∧^{d}U_{K,S,T} ⟶^{f} I_{G}^{d}/I_{G}^{d+1}”

Reg^{Φ} = ∑(−1)^{?} \underbrace{det(ϕ_{i}^{1}(u_{j}))}_{r×r det /ℤ} \underbrace{det(f_{i}(u_{j}) − 1)}_{d×d det /ℤ[G]}.

So it lies in I_{G}^{d}. Conjecture (Burns) Φ(η) ≡ h_{K,S,T}Reg^{Φ} (mod I_{G}^{d+1})

• This is a strengthening of Rubin’s conjecture.

• It recovers Gross’s conjecture as the special case r = 0.

• It is actually a consequence of ETNC (very recent work of Burns). The “E” for “equivariant” is important here. Difficult to recognise I_{G}^{d} working one character at a time. ETNC is known for absolutely abelian fields and an analogous conjecture is known for function fields with ([L : K], char(K)) = 1.

• A natural refinement exists, introducing an intermediate field E, which compares η_{E/K} to η_{L/K}. Conjecture above is then the special case E = K. This is also a consequence of ETNC. It generalises Gross’s “refined abelian p-adic Stark conjecture”.

The conjecture in quadratic extensions If [L : K] = 2 we can express L(s, χ) as ratio of zetafunctions, and then the analytic class number formula gives η explicitly. Can use genus theory to verify Burns’s conjecture when h_{L,S,T} = 1. The conjecture allows us to increase S to kill the class group - we also use I_{G}/I_{G}^{2} ⟶_{∼}^{×(σ−1)^{n−1}} I_{G}^{n}/I_{G}^{n+1}.

• Get new neater proof of Gross for quadratic extensions.

• Explain observed high power of 2 factor for η.

Comparison to analytic class number formula

Suppose all places in S split completely.

ACNF ⟺ η = \frac{h_{K,S,T}}{#G^{r}} u_{1} ∧ ⋯ ∧ u_{r}, where u_{i} are a

ℤ-basis for U_{K,S,T}. ⇒ Φ(η) = \frac{h_{K,S,T}}{#G^{r}} N_{G}^{r} det(ϕ_{i}^{1}(u_{j})) = \frac{h_{K,S,T}}{#G} N_{G} det(ϕ_{i}^{1}(u_{j}))

We have #G | h_{K,S,T} because L/K is unramified.

Burns: Φ(η) ≡ h_{K,S,T} det(ϕ_{i}^{1}(u_{j})) (mod I_{G}). Image of ACNF under (arbitrary) Φ.

Relation to certain “minus-unit” conjectures If F/K is a disjoint quadratic extension, can form minus-units with respect to Gal(F/K).

1) Darmon’s conjecture: for K = ℚ, F real, L maximal real subfield of cyclotomic field, this is a “first order analogue” of Gross’s conjecture for circular minusunits.

2) Gross’s conjecture on L-functions of tori.

These can be interpreted as “base-change” type statements for Burns’s conjecture

L/K ⇝ LF/F
Darmon: r = 1 ⇝ r = 2
Gross: r ⇝ r

II. Explicit Stark units Now focus on case r = 1. Let v denote the splitting place in S. Let #S = n + 1. The conjecture above says there exists

η ∈ U_{L,S,T}

such that, for a certain w | v,

Θ′_{L/K,S,T}(0) = − ∑_{σ∈G} ln | η^{σ} |_{w} σ (= R(η))

and, for all ϕ ∈ Hom_{ℤ[G]}(U_{L,S,T}, ℤ[G]),

ϕ(η) ≡ ∑_{i=1}^{n}(−1)^{i+1}ϕ^{1}(u_{i}) \underbrace{| \begin{matrix}f_{v_{1}}(u_{1}) − 1 & ⋯ & f_{v_{1}}(u_{n}) − 1 \\ ⋮ & ... & ⋮ \\ f_{v_{n−1}}(u_{1}) − 1 & ⋯ & f_{v_{n−1}}(u_{n}) − 1 \end{matrix} |}_{excluding u_{i}} (mod I_{G}^{n})

I consider case where every other place but v =: ∞ is finite. That is, either

1) K = ℚ,

2) K = ℚ(\sqrt{−d}), or

3) K is a global function field.

A = ring of ∞-integers of K, ie. (1) ℤ, (2) O_{ℚ(\sqrt{−d})}, or (3) ring of functions regular away from ∞.

Ideal 𝔪 = 𝔭_{1}^{e_{1}} ⋯ p_{n}^{e_{n}}. For I ⊆ {1, …, n}, write 𝔪_{I} = ∏_{i∈I} 𝔭_{i}^{e_{i}}. H(𝔪) = ray class field modulo 𝔪 totally split at ∞, which is the field associated to the “cycle” 𝔪 by global class field theory.

Gal(H(𝔪)/K) = I(𝔪)/P(𝔪)
= \frac{\{I : (I, 𝔪) = 1\}}{\{(x) : x ≡ 1 (mod 𝔪)\}}

0 ⟶ [μ]_{𝔪} ⟶ (A/𝔪)^{×} ⟶ I(𝔪)/P(𝔪) ⟶ I/P ⟶ 0 is exact. Let w(𝔪) to be the number of roots of unity congruent to 1 (mod 𝔪).

Fix T = {𝔱} one prime coprime to W_{H(𝔪)}𝔪.

There is a system of elements λ_{I} satisfying

• λ_{I} ∈ U_{H(𝔪_{I}),Supp(𝔪_{I}),T}.

• Distribution relations:

▶ 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}} ▶ N_{H(𝔪_{i})/H}(λ_{i}^{w/w(i)}) = α_{i} ∈ O_{H},

where α_{i}O_{H} = 𝔭_{i}^{N_{𝔱}−1}O_{H}.

• L-function evaluation:

w(𝔪_{I})Θ′_{H(𝔪_{I})/K,𝔪_{I},𝔱}(0) = R(λ_{I})

These units are as follows:

1) K = ℚ : λ_{I} = (1 − e^{2πi/m_{I}})^{N_{𝔱}−σ_{𝔱}}.

2) Elliptic units, constructed from special values of Weierstrass σ-function.

3) If λ̂_{I} is an 𝔪_{I}-division point on a rank 1 sign-normalised Drinfeld module, then λ_{I} = λ̂_{I}^{N_{𝔱}−σ_{𝔱}}.

We let L = H(𝔪)^{Δ}, where Δ is such that

\xymatrix{Gal(H(𝔪)/K)/Δ \ar[r] & Cl \ar[l]^{s} \ar[r] & 0}

splits. Here s is a multiplicative section. The archetype for the argument is the case

[figure: ℚ(ζ_{pq}); ℚ(ζ_{p}); Γ_{q}; ℚ(ζ_{q}); Γ_{p}; ℚ; Γ_{p}; Γ_{q}]

Here, (p, ℚ(ζ_{pq})q̃/ℚ_{q}) = ((p), ℚ(ζ_{q})/ℚ)^{−1}. In general, if Γ_{i} := (A/𝔪_{i})^{×}, there is a surjection

∏_{i} Γ_{i} × s(Cl) ↠ G.

Roughly speaking, for i ≠ j, lift σ_{p_{i}} on H(𝔪_{j})^{Δ} to an element σ_{ij} ∈ G. A^{I} = square matrix indexed by I of zero row sum with off-diagonal entries σ_{ij}−1, and A = A^{{1,...,n}}. If J ⊆ I, then A_{J}^{I} is its J-minor determinant.

[figure: H(𝔪); H(𝔪/𝔪_{n}); ⋯; H(𝔪/𝔪_{2}); H(𝔪/𝔪_{1}); H(𝔪/𝔪_{1,2}); ⋮; ⋮; ⋮; H(𝔪_{1,2}); H(𝔪_{1}); H(𝔪_{2}); ⋯; H(𝔪_{n}); H; K]

Consider N_{Γ_{I}}λ_{I}. For K = ℚ this is always 1. In general it is a unit α_{j}^{∏_{i∈I-{j}}(1−s(𝔭_{i})^{−1})} of H. Set

Ω_{I} = ϕ^{1}(N_{Cl}α_{j}) ∏_{i∈I-{j}}(1 − s(𝔭_{i})^{−1})

(independent of choice of j ∈ I!) Proposition

ϕ(N_{Δ}λ_{𝔪}) ≡ ∑_{\begin{matrix}∅≠J \\ ⊆\{1,...,n\} \end{matrix}}(1 − n)^{#J−1}Ω_{J}A_{J} (mod I_{G}^{n})

Sketch proof - forget Δ. From expanding

∑_{γ∈Γ_{𝔪}} ∑_{υ∈Cl} ϕ^{1}(γ^{−1}s(υ)^{−1}λ_{𝔪}) ∏_{j}([γ]_{𝔪_{j}} − 1)s(υ),

get expression for ϕ(λ_{𝔪}) in terms of ϕ^{G_{I}}(N_{Γ_{I′}}λ_{𝔪}) for I ⊊ {1, …, n} [where G_{I} ≈ Gal(H(𝔪_{I})/K)]. We can then apply distribution relations.

The proof proceeds essentially by induction on n. We find we need the relation, for each J ⊆ {1, …, n},

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

This proof is combinatorial, relying only on the fact that A is a matrix with zero row-sums.

After establishing the proposition, we relate the RHS to Reg^{ϕ} using global class field theory and the fact that rank A^{×} = 0, and conclude

Theorem

wϕ(η) ≡ wh_{K,S,T}Reg^{ϕ} (mod I_{G}^{n})

The combinatorics The combinatorial tricks for studying matrices of zero row sum have various applications when considering explicit units (cf. recent work of Greither and Kuc̆era) and such matrices arise naturally when considering regulators. The main tool is

Matrix Tree Theorem (S. Chaiken and D. J. Kleitman) Let A^{I} be a matrix with zero row sums, indexed by I. Let J ⊆ I. Then we have A_{J}^{I} = (−1)^{#I−#J} ∑_{T} A(T), where T runs through all arborescences on I with root J.

Example

A_{\{3,4\}}^{\{1,2,3,4\}} = det ( \begin{matrix}−a_{12} − a_{13} − a_{14} & a_{12} \\ a_{21} & −a_{21} − a_{23} − a_{24} \end{matrix} )

= 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

\overbrace{\begin{matrix}2 \\ ↓ \\ 1 \\ ↓ \\ 3 & 4 \end{matrix}}^{a_{13}a_{21}}  \overbrace{\begin{matrix}1 \\ ↓ \\ 2 \\ ↓ \\ 3 & 4 \end{matrix}}^{a_{12}a_{23}}  \overbrace{\begin{matrix}1 & 2 \\ ↓ & ↙ \\ 3 & 4 \end{matrix}}^{a_{13}a_{23}}  \overbrace{\begin{matrix}1 & 2 \\ ↓ & ↓ \\ 3 & 4 \end{matrix}}^{a_{13}a_{24}}  \overbrace{\begin{matrix}2 & 1 \\ ↓ & ↓ \\ 3 & 4 \end{matrix}}^{a_{14}a_{23}}  \overbrace{\begin{matrix}1 & 2 \\ ↘ & ↓ \\ 3 & 4 \end{matrix}}^{a_{14}a_{24}}  \overbrace{\begin{aligned}& 1 \\ & ↓ \\ & 2 \\ & ↓ \\ 3 & 4 \end{aligned}}^{a_{12}a_{24}}  \overbrace{\begin{aligned}& 2 \\ & ↓ \\ & 1 \\ & ↓ \\ 3 & 4 \end{aligned}}^{a_{14}a_{21}}

Let A be an n × n matrix with zero row sums. We wanted to know

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

Let F (X) = (1−X)^{#J−1}. Apply matrix tree theorem:

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

If m = #V^{0}(T), there are \binom{m}{r} sets V of size r for r = 0, …, n. So

LHS = ± ∑_{T} A(T) ∑_{r=0}^{m}(−1)^{r}\binom{m}{r}F (n − r)

We use Euler’s formula: If 0 ≤ t < m then

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

Cor: If deg f < m then ∑_{r=0}^{m}(−1)^{r}\binom{m}{r}f(r) = 0.

Apply this with f(X) = F (n − X). Note #J ≤ m since for each “root” we have a “leaf”. The matrices are even richer in relations than we needed!