# Hilbert modular forms and the Gross{Stark Hilbert modular forms and the Gross{Stark conjecture Samit

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Hilbert modular forms and the Gross–Stark conjecture

Samit Dasgupta

Henri Darmon Robert Pollack

November 12, 2009

Abstract

Let F be a totally real field and χ an abelian totally odd character of F . In

1988, Gross stated a p-adic analogue of Stark’s conjecture that relates the value of the

derivative of the p-adic L-function associated to χ and the p-adic logarithm of a p-unit

in the extension of F cut out by χ. In this paper we prove Gross’s conjecture when F is

a real quadratic field and χ is a narrow ring class character. The main result also applies

to general totally real fields for which Leopoldt’s conjecture holds, assuming that either

there are at least two primes above p in F , or that a certain condition relating the L -

invariants of χ and χ−1 holds. This condition on L -invariants is always satisfied when

χ is quadratic.

Contents

1 Duality and the L -invariant 12 1.1 Local cohomology groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.2 Global cohomology groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.3 A formula for the L -invariant . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2 Hilbert modular forms 18 2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.2 Eisenstein Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.3 A product of Eisenstein series . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.4 The ordinary projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.5 Construction of a cusp form . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3 Λ-adic forms 31 3.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.2 Λ-adic Eisenstein series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.3 A Λ-adic cusp form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.4 The weight 1 + ε specialization . . . . . . . . . . . . . . . . . . . . . . . . . 34

1

4 Galois representations 37 4.1 Representations attached to ordinary eigenforms . . . . . . . . . . . . . . . . 37 4.2 Construction of a cocycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

Introduction

Let F be a totally real field of degree n, and let

χ : GF := Gal(F̄ /F ) → Q ×

be a character of conductor n. Such a character cuts out a finite cyclic extension H of F , and can be viewed as a function on the ideals of F in the usual way, by setting χ(a) = 0 if a is not prime to n. Let Na = NormF/Q(a) denote the norm of a. Fix a rational prime p and a choice of embeddings Q ⊂ Qp ⊂ C that will remain in effect throughout this article. The character χ may be viewed as having values in Qp or C via these embeddings.

Let S be any finite set of places of F containing all the archimedean places. Associated to χ is the complex L-function

LS(χ, s) := ∑

(a,S)=1

χ(a) Na−s = ∏

p/∈S

(1 − χ(p) Np−s)−1, (1)

which converges for Re(s) > 1 and has a holomorphic continuation to all of C when χ 6= 1. By work of Siegel [13], the value LS(χ, n) is algebraic for each n ≤ 0. (See for instance the discussion in §2 of [8], where LS(χ, n) is denoted aS(χ, n).)

Let E be a finite extension of Qp containing the values of the character χ. Let

ω : Gal(F (µ2p)/F ) → (Z/2p)× → Z×p denote the p-adic Teichmuller character. If S contains all the primes above p, Deligne and Ribet [3] have proved the existence of a continuous E-valued function LS,p(χω, s) of a variable s ∈ Zp characterized by the interpolation property

LS,p(χω, n) = LS(χω n, n) for all integers n ≤ 0. (2)

The function LS,p(χω, s) is meromorphic on Zp, regular outside s = 1, and regular everywhere when χω is non-trivial.

If p ∈ S is any non-archimedean prime, and R := S − {p}, then

LS(χ, 0) = (1 − χ(p))LR(χ, 0).

In particular, LS(χ, s) vanishes at s = 0 when χ(p) = 1, and equation (2) implies that the same is true of the p-adic L-function LS,p(χω, 0). Assume for the remainder of this article that the hypothesis χ(p) = 1 is satisfied.

For x ∈ Z×p , let 〈x〉 = x/ω(x) ∈ 1 + pZp. If p does not divide p, the formula

LS,p(χω, s) = (1 − 〈Np〉−s)LR,p(χω, s)

2

implies that L′S,p(χω, 0) = logp(Np)LR(χ, 0), (3)

where logp : Q × p → Qp denotes the usual Iwasawa p-adic logarithm.

If p divides p, which we assume for the remainder of this article, a formula analogous to (3) comparing L′S,p(χω, 0) and LR(χ, 0) has been conjectured in [8]. This formula involves the group O×H,S of S-integers of H—more precisely, the χ−1-component Uχ of the E-vector space O×H,S ⊗ E:

Uχ := (O×H,S ⊗ E)χ −1

:= {

u ∈ O×H,S ⊗ E such that σu = χ−1(σ)u }

. (4)

Dirichlet’s unit theorem implies that Uχ is a finite-dimensional E-vector space and that

dimE Uχ = #{v ∈ S such that χ(v) = 1} = ords=0 LS(χ, s)

= ords=0 LR(χ, s) + 1.

In particular, the space Uχ is one-dimensional if and only if LR(χ, 0) 6= 0. Assume that this is the case, and let uχ be any non-zero vector in Uχ.

The choice of a prime P of H lying above p determines two Z-module homomorphisms

ordP : O×H,S → Z, LP : O×H,S → Zp, (5)

where the latter is defined by

LP(u) := logp(NormHP/Qp(u)). (6)

Let ordP and LP also denote the homomorphisms from Uχ to E obtained by extending scalars to E. Following Greenberg (cf. equation (4’) of [6] in the case F = Q), the L - invariant attached to χ is defined to be the ratio

L (χ) := − LP(uχ) ordP(uχ)

∈ E. (7)

This L -invariant is independent of the choice of non-zero vector uχ ∈ Uχ, and it is also independent of the choice of the prime P lying above p. When LR(χ, 0) = 0, we (arbitrarily) assign the value of 1 to L (χ).

The following is conjectured in [8] (cf. Proposition 3.8 and Conjecture 3.13 of loc. cit.):

Conjecture 1 (Gross). For all characters χ of F and all S = R ∪ {p}, we have

L′S,p(χω, 0) = L (χ)LR(χ, 0). (8)

When LR(χ, 0) = 0, Conjecture 1 amounts to the statement L ′ S,p(χω, 0) = 0. As explained

in Section 1, this case of the conjecture follows from Wiles’ proof of the Main Conjecture for totally real fields (assuming that χ is of “type S”; see Lemma 1.2). We will therefore

3

assume that LR(χ, 0) 6= 0. In this setting, Gross’s conjecture suggests defining the analytic L -invariant of χ by the formula

Lan(χ) := L′S,p(χω, 0)

LR(χ, 0) =

d

dk Lan(χ, k)k=1, (9)

where

Lan(χ, k) := −LS,p(χω, 1 − k)

LR(χ, 0) . (10)

Conjecture 1 can then be rephrased as the equality L (χ) = Lan(χ) between algebraic and analytic L -invariants. The main result of this paper is:

Theorem 2. Assume that Leopoldt’s conjecture holds for F .

1. If there are at least two primes of F lying above p, then Conjecture 1 holds for all χ.

2. If p is the only prime of F lying above p, assume further that

ordk=1(Lan(χ, k) + Lan(χ −1, k)) = ordk=1 Lan(χ

−1, k). (11)

Then Conjecture 1 holds for both χ and χ−1.

Remark 3. The somewhat mysterious condition formulated in (11) makes no a priori as- sumption on the order of vanishing of LS,p(χω, s) at s = 0. It is automatically satisfied (after possibly interchanging χ and χ−1) when Lan(χ, k) and Lan(χ

−1, k) have different orders of vanishing. When these orders of vanishing agree, it stipulates that the sum of the leading terms at k = 1 of Lan(χ, k) and Lan(χ

−1, k) should be nonzero. In the setting that we are considering, where LR(χ, 0) 6= 0, it is expected that the functions Lan(χ, k) and Lan(χ−1, k) both vanish to order 1 at k = 1. If this is true, condition (11) amounts to the condition

Lan(χ) + Lan(χ −1) 6= 0. (12)

One can show using the methods of section 4 that when p is the unique prime of F lying above p and Leopoldt’s conjecture holds for F , we have

L (χ) 6= 0 and L (χ−1) 6= 0 =⇒ L (χ) + L (χ−1) 6= 0.

Therefore, condition (12) is always expected to hold. The condition on the non-vanishing of the algebraic L -invariant attached to χ may however be quite deep, and condition (11) appears to be a substantial hypothesis in the formulation of Theorem 2.

Remark 3 notwithstanding, Theorem 2 leads to the following two unconditional results.

Corollary 4. Let F be a real quadratic field, and let χ be a narrow ring class character of F . Then Conjecture 1 holds for χ.

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Proof. Corollary 4 is unconditional because Leopoldt’s conjecture is trivial for real quadratic F . Furthermore, since χ is a ring class character, the representations induced from F to Q by χ and χ−1 are equal and therefore the L-functions (both classical and p-adic) attached to these characters agree. It follows that the leading terms of Lan(χ, k) and Lan(χ

−1, k) are equal, and therefore condition (11) is satisfied. Corollary 4 follows.

Corollary 5. Let F be a totally real field satisfying Leopoldt’s conjecture, and let χ be a narrow ray class character of F . Then Conjecture 1 holds for χ in either of the following two cases:

1. There are at least two primes of F above the rational prime p, or

2. The character χ is quadratic.

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