A conjectural class number formula for derivatives of abelian L-functions

Anthony Hayward King’s College, London.

Principal References

1. Karl Rubin, A Stark conjecture “over ℤ” for abelian L-functions with multiple zeros, 1996

2. Benedict H. Gross, On the values of abelian L-functions at s = 0, 1988

3. Henri Darmon, Thaine’s method of circular units and a conjecture of Gross, 1995 1 - Basic Setup - note that this is identical to that of Rubin’s conjecture.

L/K finite abelian extension of global fields with group G.

S, T finite sets of places of K, r a nonnegative integer. S_{L} etc. places in L over these.

Hypothesis

1. S ⊇ infinite places

2. S ⊇ ramified places

3. r places in S split completely in L/K

4. #S ≥ r + 1

5. T ∩ S = ∅ and U_{L,S,T} torsion free.

U_{L,S,T} = {S-units ≡ 1 (mod w) ∀w ∈ T_{L}} Artin L-function

L_{L/K,S}(s, χ) = _{v∉S_{K}}^{∏}(1 − N_{v}^{−s}χ(Frob _{v}))^{−1}
= ∑ ; \begin{matrix}I◁K \\ (I,S)=1 \end{matrix} χ(I)NI^{−s}

For χ ∈ Ĝ, write e_{χ} = \frac{1}{#G} ∑_{g∈G} χ(g)g^{−1} Θ_{L/K,S,T} (s) := δ_{T} (s) _{χ∈Ĝ}^{∑} L_{L/K,S}(s, χ^{−1})e_{χ}

where δ_{T} (s) = ∏_{p∈T_{K}}(1 − N_{p}^{1−s}Frob _{p}^{−1}). So χ(Θ_{L/K,S,T} (s)) = L_{L/K,S,T} (s, χ^{−1})

We study the rth term in the Taylor expansion at 0:

Θ_{L/K,S,T}^{r} (0) := lim_{s→0} s^{−r}Θ_{L/K,S,T} (s).

Idempotent e_{r} := ∑_{r(χ)=r} e_{χ}.

Dirichlet regulator map

\begin{aligned}λ_{S_{L}} : & U_{L,S_{L}} & ⟶ & ℝX_{S_{L}} \\ & u & ⟼ & − ∑_{w∈S_{L}} ln ||u||_{w} · w \end{aligned}

A_{K,S,T} = (S,T) ray-class group of K, size

h_{K,S,T} = h_{K,S} \frac{∏_{p∈T} (N_{p} − 1)}{(U_{K,S} : U_{K,S,T})}

λ : ℝU_{L,S,T} ^{⟶} ℝX_{S_{L}} is an ℝ[G] isom. Analytic class number formula ζ_{K,S,T} = Θ_{K/K,S,T} (s)

≡ ±h_{K,S,T} det (λ)s^{#S−1} (mod s^{#S} )

To generalise, relate Θ_{L/K,S,T}^{r} (0) to units. Choosing W = (w_{1}, …, w_{r}) above r splitting places leads to a ℂ[G]-isomorphism

W^{∗} ∘ λ_{S_{L}} : (ℂ ∧_{ℤ[G]}^{r} U_{L,S,T})e_{r} ⟶ ℂ[G]e_{r}

where W^{∗}(w_{1} ∧ ⋯ ∧ w_{r}) = 1 ∈ ℂ[G]. So there is a unique element

ε = ε_{L/K,S,T,r,W} ∈ (ℂ ∧_{ℤ[G]}^{r} U_{L,S,T})e_{r}

mapping to Θ_{L/K,S,T}^{r} (0).

What are its integrality properties?

Take homomorphisms

Φ = ϕ_{1} ∧ ⋯ ∧ ϕ_{r} ∈ ∧_{ℤ[G]}^{r}Hom _{ℤ[G]}(U_{L,S,T} , ℤ[G])

Look at element Φ(ε) ∈ ℂ[G], where Φ(u_{1} ∧ ⋯ ∧ u_{r}) = det (ϕ_{j}(u_{i}))_{i,j}.

Conjecture (Rubin) Φ(ε) ∈ ℤ[G].

This conjecture, for r = 1 and all T, is equivalent to the abelian Stark conjecture. Note r = 1 ⟹ ϵ ∈ U_{L,S,T}. We have ϵ = λ^{δ_{T}} where λ = u^{1/W_{L}} is a root of the Stark unit u.

We will demand more of Φ(ε) …

Taster: conjecture Φ(ε) ∈ I_{G}^{d}, where d = #S − r − 1 and I_{G} = augmentation ideal.

Furthermore, we’ll conjecture a value for Φ(ε) in I_{G}^{d}/I_{G}^{d+1} .

2 - Formulation of Congruence

If ϕ ∈ Hom _{ℤ[G]}(M, ℤ[G]), define ϕ^{1}(m) to be the coefficient of 1_{G} in ϕ(m), which gives ϕ^{1} ∈ Hom _{ℤ}(M, ℤ). We have

ϕ(m) = _{g∈G}^{∑} ϕ^{1}(g^{−1}m)g,

and ϕ ↦ ϕ^{1} is a group isomorphism.

If 0 < n ∈ ℤ then such a ϕ^{1} induces

ϕ̃ : ∧_{ℤ}^{n}M ⟶ ∧_{ℤ}^{n−1}M

by ϕ̃(m_{1} ∧ ⋯ ∧ m_{n}) = ∑_{i=1}^{n} (−1)^{i+1} ϕ^{1}(m_{i})m′_{i}, where m′_{i} is the wedge of all the m_{j} except m_{i}. We can iterate this so that Φ = ϕ_{1} ∧ ⋯ ∧ ϕ_{r} induces a map Φ̃ which knocks down exterior powers by r.

For v a place of K, local reciprocity map

f_{v} = (−, L_{w}/K_{v}) : K^{×} ⟶ G

Choose ordering v_{1}, …, v_{d+1} for S, excluding the r splitting places from W. Define, following Gross, a homomorphism Reg _{G} from ∧_{ℤ}^{d}U_{K,S,T} to ℤ[G]/I_{G}^{d+1} by

u_{1} ∧ ⋯ ∧ u_{d} ⟼ det (f_{v_{j}}(u_{i}) − 1)_{1≤i,j≤d}.

Depends on ordering of places up to sign.

u_{1}, …, u_{r+d} ℤ-basis for U_{K,S,T}. Set

Reg _{G}^{Φ} = Reg _{G}(Φ̃ (u_{1} ∧ ⋯ ∧ u_{r+d} ))

N.B. independent of basis & places up to sign.

Reg _{G}^{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} |

= ∑_{\begin{matrix}σ:σ(1)<...<σ(r) \\ σ(r+1)<...<σ(r+d) \end{matrix}} sign (σ) \overbrace{| \begin{matrix}ϕ_{1}^{1}(u_{σ(1)} ) & ⋯ & ϕ_{1}^{1}(u_{σ(r)}) \\ ⋮ & ... & ⋮ \\ ϕ_{r}^{1}(u_{σ(1)} ) & ⋯ & ϕ_{r}^{1}(u_{σ(r)}) \end{matrix} |}^{r × r det. /ℤ} ×

\underbrace{| \begin{matrix}f_{v_{1}}(u_{σ(r+1)} ) − 1 & ⋯ & f_{v_{d}}(u_{σ(r+d)} ) − 1 \\ ⋮ & ... & ⋮ \\ f_{v_{d}}(u_{σ(r+1)} ) − 1 & ⋯ & f_{v_{d}}(u_{σ(r+d)} ) − 1 \end{matrix} |}_{d × d det. /ℤ[G]}

where u_{1}, …, u_{r+d} is a ℤ-basis for U_{K,S,T}

Conjecture (Burns) “C_{≡}(L/K, S, T, r)”

Φ(ε) ≡ ±h_{K,S,T} Reg _{G}^{Φ} (mod I_{G}^{d+1} )

3 - Basic properties

1) L/M/K: C_{≡}(L/K, S, T, r) ⇒ C_{≡}(M/K, S, T, r)

2) C_{≡}(L/K, S, T, r) ⇒ C_{≡}(L/K, S, T ∪ {v}, r)

3) C_{≡}(L/K, S ∪{v}, T, r +1) ⇒ C_{≡}(L/K, S, T, r) if v splits completely in L/K.

4) C_{≡}(L/K, S, T, r) ⇒ C_{≡}(L/K, S ∪ {v}, T, r) for all v ∉ S ∪ T.

4 - Special cases If r = 0, ε ∈ ℂ ∧_{ℤ[G]}^{0} U_{L,S,T} e_{r} is Θ_{L/K,S,T} (0) ∈ ℂ[G]e_{r}. C_{≡}(L/K, S, T) says that modulo I_{G}^{d+1} , have

Θ_{L/K,S,T} (0) ≡ ±h_{K,S,T} Reg _{G}(u_{1} ∧ ⋯ ∧ u_{d}), u_{i} a ℤ-basis for U_{K,S,T}. That is,

Θ_{L/K,S,T} (0) ≡ ±h_{K,S,T} | \begin{matrix}f_{v_{1}}(u_{1}) − 1 & ⋯ & f_{v_{1}}(u_{d}) − 1 \\ ⋮ & ... & ⋮ \\ f_{v_{d}}(u_{1}) − 1 & ⋯ & f_{v_{d}}(u_{d}) − 1 \end{matrix} |

(mod I_{G}^{d+1} ), where d = #S − 1. This is ± Gross’s Conjecture for this data.

If more than r places in S split completely, all the L-functions for nontrivial characters vanish to order > r.

d > 0: C_{≡}(L/K, S, T) says 0 ≡ 0, since the zeta-function vanishes too.

d = 0: Only the zeta-function doesn’t vanish. C_{≡}(L/K, S, T) is the homomorphic image of A.C.N.F. for K. Let u_{i} be a ℤ-basis for U_{K,S,T}. A.C.N.F. ⟺ ε = h_{K,S,T} /#G^{r} u_{1} ∧ ⋯ ∧ u_{r} C_{≡}(L/K, S, T, r) ⟺

Φ(ε) ≡ ±h_{K,S,T} det (ϕ_{j}^{1}(u_{i}))_{i,j}

5 - Some evidence

Theorem 1 (H.) C_{≡}(L/K, S, T, r) holds if L/K is quadratic.

Ingredients in the proof:

• A.C.N.F. gives formulae for ε (Rubin). Suffices for d = 0 case (equality in ℤ).

• Can always add a split place to S and increase r.

• G cyclic ⇒ we can add inert places to S too, for d > 0.

• So can increase S to kill A_{L,S,T}.

• Now RHS collapses to a single term. Proof of Gross’s conjecture in quadratic case ⇝ criterion for nonvanishing of this regulator.

Theorem 2 (H.) 2C_{≡}(L/ℚ, S, T, 1) holds with W = ∞_{L} if L is real.

ie. 2ϕ(ε) ≡ ±2h_{ℚ,S,T} ∑_{i}(−1)^{i+1} ϕ^{1}(u_{i})Reg _{G}(u′_{i}).

Ingredients in the proof:

• L ↪ ℚ(ζ_{m})

• Formula for L′_{L,S}(0, χ) ⇝

2ε = N_{ℚ(ζ_{m})/L}(1 − ζ_{m})^{δ_{T}}

• Distribution relations for cyclotomic elements • Careful induction on #{p | m}

• Lower top field from ℚ(ζ_{m}), add T conditions in at the end.

6 - Interpretation of Darmon’s conjecture

N, S coprime integers. K ⊆ ℚ(ζ_{N}) quadratic field. L̃ ⊆ ℚ(ζ_{S}) real.

Henri Darmon conjectured about a “circular unit” - explicit element α_{S} ∈ K(ζ_{S}) made from cyclotomic elements. We can relate α_{S} to the base-change factor from Θ′_{L̃/ℚ,S,T} (0) to Θ_{L̃K/K,S,T}^{2} (0).

Conjecture

_{σ∈Γ_{S}}^{∑} σ^{−1}α_{S} ⊗ σ = ±2^{∗}h_{K,S}R_{S}

in U_{K(ζ_{S}),S,T} ⊗ I_{Γ}^{n}/I_{Γ}^{n+1} , n = number of primes dividing S which split in K/ℚ. R_{S} is a sum of terms of the form u_{i} ⊗Reg made out of minusunits.

_{σ∈Γ_{S}}^{∑} σ^{−1}α_{S}^{δ_{T}} ⊗ σ = ±2^{∗}\frac{h_{K,S,T}}{h_{ℚ,S,T}} R_{S,T} .

Hit this conjecture with ϕ^{1} ⊗ id to get a congruence statement in ℤ[G].

A T -modified Darmon conjecture implies a “base change” statement for C_{≡}.

Theorem 3 (H.) Assume C_{≡}(L̃/ℚ, S, T, 1) and Darmon’s conjecture hold. Then can conclude (4 · 2^{n}) C_{≡}(L̃K/K, S, T, 2); ie.

(4·2^{n})(ϕ_{1} ∧ϕ_{2})(ε) ≡ (4·2^{n})(±h_{K,S,T} ∑_{i<j}(−1)^{i+1} (−1)^{j+1}

| \begin{matrix}ϕ_{1}^{1}(u_{i}) & ϕ_{2}^{1}(u_{i}) \\ ϕ_{1}^{1}(u_{j}) & ϕ_{2}^{1}(u_{j}) \end{matrix} | det (f_{v_{k}}(u_{l}) − 1)_{l≠i,j} ) (mod I_{G}^{d+1} )

7 - Relations between abelian Stark-type conjectures