Hilbert modular forms and the Gross–Stark conjecture^{∗}

Samit Dasgupta Henri Darmon Robert Pollack

August 27, 2012

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

∗appeared in Annals of Mathematics 174 (2011), 439–484

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

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 : G_{F} := Gal(F̄/F) → \overline{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 = Norm_{F/Q}(𝔞) denote the norm of 𝔞. Fix a rational prime p and a choice of embeddings \overline{Q} ⊂ \overline{Q}_{p} ⊂ C that will remain in effect throughout this article. The character χ may be viewed as having values in \overline{Q}_{p} 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

L_{S}(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 ≠ 1. By work of Siegel [14], the value L_{S}(χ, n) is algebraic for each n ≤ 0. (See for instance the discussion in §2 of [9], where L_{S}(χ, n) is denoted a_{S}(χ, n).)

Let E be a finite extension of Q_{p} containing the values of the character χ. Let

ω : Gal(F (μ_{2p})/F) → (Z/2p)^{×} → Z_{p}^{×}

denote the p-adic Teichmuller character. If S contains all the primes above p, Deligne and Ribet [4] have proved the existence of a continuous E-valued function L_{S,p}(χω, s) of a variable s ∈ Z_{p} characterized by the interpolation property

L_{S,p}(χω, n) = L_{S}(χω^{n}, n)  for all integers n ≤ 0.  (2)

The function L_{S,p}(χω, s) is meromorphic on Z_{p}, 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

L_{S}(χ, 0) = (1 − χ(𝔭))L_{R}(χ, 0).

In particular, L_{S}(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 L_{S,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 + pZ_{p}. If 𝔭 does not divide p, the formula

L_{S,p}(χω, s) = (1 − ⟨N𝔭⟩^{−s})L_{R,p}(χω, s)

implies that

L′_{S,p}(χω, 0) = log_{p}(N𝔭)L_{R}(χ, 0),  (3)

where log_{p} : Q_{p}^{×} → Q_{p} 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 L_{R}(χ, 0) has been conjectured in [9]. This formula involves the group O_{H,S}^{×} of S-integers of H—more precisely, the χ^{−1}-component U_{χ} of the E-vector space O_{H,S}^{×} ⊗ E:

U_{χ} := (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 U_{χ} is a finite-dimensional E-vector space and that

dim_{E} U_{χ} = #{v ∈ S such that χ(v) = 1}
= ord_{s=0} L_{S}(χ, s)
= ord_{s=0} L_{R}(χ, s) + 1.

In particular, the space U_{χ} is one-dimensional if and only if L_{R}(χ, 0) ≠ 0. Assume that this is the case, and let u_{χ} be any non-zero vector in U_{χ}.

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

ord_{𝔓} : O_{H,S}^{×} → Z,  L_{𝔓} : O_{H,S}^{×} → Z_{p},  (5)

where the latter is defined by

L_{𝔓}(u) := log_{p}(Norm_{H_{𝔓}/Q_{p}}(u)).  (6)

Let ord_{𝔓} and L_{𝔓} also denote the homomorphisms from U_{χ} to E obtained by extending scalars to E. Following Greenberg (cf. equation (4’) of [7] in the case F = Q), the ℒ - invariant attached to χ is defined to be the ratio

ℒ (χ) := −\frac{L_{𝔓}(u_{χ})}{ord_{𝔓}(u_{χ})} ∈ E.  (7)

This ℒ -invariant is independent of the choice of non-zero vector u_{χ} ∈ U_{χ}, and it is also independent of the choice of the prime 𝔓 lying above 𝔭. When L_{R}(χ, 0) = 0, we (arbitrarily) assign the value of 1 to ℒ (χ).

The following is conjectured in [9] (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)L_{R}(χ, 0).  (8)

When L_{R}(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 assume that L_{R}(χ, 0) ≠ 0. In this setting, Gross’s conjecture suggests defining the analytic ℒ -invariant of χ by the formula

ℒ_{an}(χ) := \frac{L′_{S,p}(χω, 0)}{L_{R}(χ, 0)} = \frac{d}{dk} ℒ_{an}(χ, k)_{k=1},  (9)

where

ℒ_{an}(χ, k) := \frac{−L_{S,p}(χω, 1 − k)}{L_{R}(χ, 0)}.  (10)

Conjecture 1 can then be rephrased as the equality ℒ (χ) = ℒ_{an}(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

ord_{k=1}(ℒ_{an}(χ, k) + ℒ_{an}(χ^{−1}, k)) = ord_{k=1} ℒ_{an}(χ^{−1}, k).  (11)

Then Conjecture 1 holds for both χ and χ^{−1}.

Remark 3. In [3], the leading terms at s = 0 of the p-adic zeta functions of F and H, defined following Katz’s approach, are related to p-adic regulators formed from p-adic logarithms of S-units. If χ is a quadratic character, Conjecture 1 can be deduced from a conjectural factorization formula for the p-adic zeta function of H (cf. [3, Conjecture 7]) combined with this leading term formula.

Remark 4. The somewhat mysterious condition formulated in (11) makes no a priori assumption on the order of vanishing of L_{S,p}(χω, s) at s = 0. It is automatically satisfied (after possibly interchanging χ and χ^{−1}) when ℒ_{an}(χ, k) and ℒ_{an}(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 ℒ_{an}(χ, k) and ℒ_{an}(χ^{−1}, k) should be nonzero. In the setting that we are considering, where L_{R}(χ, 0) ≠ 0, it is expected that the functions ℒ_{an}(χ, k) and ℒ_{an}(χ^{−1}, k) both vanish to order 1 at k = 1. If this is true, condition (11) amounts to the condition

ℒ_{an}(χ) + ℒ_{an}(χ^{−1}) ≠ 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) ≠ 0 and ℒ (χ^{−1}) ≠ 0 ⟹ ℒ (χ) + ℒ (χ^{−1}) ≠ 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 4 notwithstanding, Theorem 2 leads to the following two unconditional results.

Corollary 5. Let F be a real quadratic field, and let χ be a narrow ring class character of F. Then Conjecture 1 holds for χ.

Proof. Corollary 5 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 ℒ_{an}(χ, k) and ℒ_{an}(χ^{−1}, k) are equal, and therefore condition (11) is satisfied. Corollary 5 follows.

Corollary 6. 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.

Proof. The first case is direct consequence of Theorem 2, and the second follows from the fact that ℒ_{an}(χ, k) = ℒ_{an}(χ^{−1}, k) when χ is quadratic.

Remark 7. When F_{𝔭} = Q_{p}, Conjecture 1 leads to a p-adic analytic construction of nontrivial 𝔭-units in abelian extensions of F by exponentiating the first derivatives of the appropriate partial p-adic L-series. In this way, Conjecture 1 supplies a p-adic solution to Hilbert’s twelfth problem for certain abelian extensions of F, just like Stark’s original archimedean conjectures. See [9, Proposition 3.14] for a more detailed discussion of the application of Conjecture 1 to the analytic construction of class fields.

Remark 8. Conjecture 1 has been proved in [9, §4] in the case F = Q using an explicit expression for the Gross-Stark unit u_{χ} in terms of Gauss sums, which are related to values of the p-adic Gamma function by the Gross-Koblitz formula. The p-adic Gamma function is in turn related to the p-adic L-functions over Q by the work of Ferrero and Greenberg [5]. In contrast, the approach we have followed to handle more general totally real F does not construct the Gross-Stark unit u_{χ} directly. Instead, it exploits the two-dimensional padic representations attached to certain families of Hilbert modular forms to construct an annihilator for u_{χ} under the local Tate pairing. The explicit construction of these families allows us to relate these annihilators, and hence u_{χ} itself, to p-adic L-functions and to ℒ_{an}(χ).

We now present a more detailed outline of the strategy used to prove Theorem 2, and give an overview of the contents of this article.

Cohomological interpretation of Conjecture 1

Let

ϵ_{cyc} : G_{F} → Z_{p}^{×}  (13)

denote the cyclotomic character defined by

σ(ζ) =: ζ^{ϵ_{cyc}(σ)}  (14)

for any p-power root of unity ζ ∈ \overline{F}. Write E(χ)(1) for the one-dimensional E-vector space equipped with the continuous action of G_{F} via χϵ_{cyc}, and let E(χ^{−1}) denote its Kummer dual, on which G_{F} acts via χ^{−1}. The first step is to exploit Tate’s local duality and the reciprocity law of global class field theory to give an alternate description of ℒ (χ), following Greenberg [7]. This alternate description involves the subgroup H_{𝔭}^{1}(F, E(χ^{−1})) of the global cohomology group H^{1}(F, E(χ^{−1})) consisting of (continuous) classes whose restrictions to the inertia subgroups I_{𝔮} ⊂ G_{F} are unramified for all primes 𝔮 ≠ 𝔭 of F. Under the assumptions L_{R}(χ, 0) ≠ 0 and χ(𝔭) = 1, we show that

dim_{E} H_{𝔭}^{1}(F, E(χ^{−1})) = [F_{𝔭} : Q_{p}], dim_{E} H^{1}(F_{𝔭}, E(χ^{−1})) = [F_{𝔭} : Q_{p}] + 1,

and that the natural restriction map

H_{𝔭}^{1}(F, E(χ^{−1}))⟶H^{1}(F_{𝔭}, E(χ^{−1}))  (15)

is injective.

Since χ(𝔭) = 1, the group H^{1}(F_{𝔭}, E(χ^{−1})) = H^{1}(F_{𝔭}, E) = Hom_{cts}(G_{F_{𝔭}}, E) contains two distinguished elements: the unique unramified homomorphism

κ_{nr} ∈ Hom(Gal(F_{𝔭}^{nr}/F_{𝔭}), O_{E})

sending the Frobenius element Frob_{𝔭} to 1, and the restriction to G_{F_{𝔭}} of the p-adic logarithm of the cyclotomic character:

κ_{cyc} := log_{p}(ϵ_{cyc}) ∈ Hom(G_{F} , E) = H^{1}(F, E).  (16)

Let H^{1}(F_{𝔭}, E)^{cyc} denote the two-dimensional subspace of H^{1}(F_{𝔭}, E) spanned by κ_{nr} and κ_{cyc}, and let H_{𝔭}^{1}(F, E(χ^{−1}))^{cyc} denote its inverse image in H_{𝔭}^{1}(F, E(χ^{−1})) under the restriction map at 𝔭. It is proved in Section 1 that

dim_{E} H_{𝔭}^{1}(F, E(χ^{−1}))^{cyc} = 1.

If κ ∈ H_{𝔭}^{1}(F, E(χ^{−1}))^{cyc} ⊂ H^{1}(F_{𝔭}, E)^{cyc} is any non-zero class, we may thus write

res_{𝔭}(κ) =: x · κ_{nr} + y · κ_{cyc}, with x, y ∈ E.

The ratio x/y—the “slope” of the global line relative to the natural basis (κ_{nr}, κ_{cyc})—does not depend on the choice of κ. The main result of Section 1 is that y ≠ 0 and that

ℒ (χ) = −x/y.

Thanks to this result, the problem of proving Theorem 2 is transformed into the problem of constructing a global cohomology class κ ∈ H_{𝔭}^{1}(F, E(χ^{−1}))^{cyc} whose “coordinates” x and y can be computed explicitly and related to p-adic L-functions at 0—more precisely, such that

res_{𝔭}(κ) = −ℒ_{an}(χ) · κ_{nr} + κ_{cyc}.  (17)

Construction of a cusp form

The construction of κ borrows heavily from the techniques initiated in [12] and extended and developed in [18] to prove the main conjecture of Iwasawa theory for totally real fields. We briefly outline the main steps in the notationally simpler case where F = Q and the prime p is odd. Let m ≥ 1 denote the conductor of the odd Dirichlet character χ. Let R be the set of primes dividing m∞, and let S = R ∪ {p}.

For an integer k ≥ 1, denote by M_{k}(m, χ) = M_{k}(Γ_{1}(m), χ) the space of classical modular forms of weight k, level m and character χ with Fourier coefficients in E. Let (η, ψ) be a pair of (not necessarily primitive) Dirichlet characters of modulus m_{η} and m_{ψ}, respectively, such that ηψ(−1) = (−1)^{k}. A key role is played in the argument by the weight k Eisenstein series E_{k}(η, ψ). After recalling the definitions of Hilbert modular forms and their q-expansions in Section 2.1, the Eisenstein series for the Hilbert modular group are defined in Section 2.2. A key result in that section is the calculation of their constant terms at certain cusps. When F = Q, the Eisenstein series E_{k}(η, ψ) ∈ M_{k}(m_{η}m_{ψ}, ηψ) for k ≥ 1 and (k, η, ψ) ≠ (2, 1, 1) are given by:

E_{k}(η, ψ) := C_{k}(η, ψ) + ∑_{n=1}^{∞}( ∑_{d|n} η (\frac{n}{d}) ψ(d)d^{k−1})q^{n},  (18)

where

C_{k}(η, ψ) := \frac{1}{2} \{ \begin{matrix}L(ψ, 1 − k) & if η = 1; \\ L(η, 1 − k) & if ψ = 1; \\ 0 & otherwise. \end{matrix}

In particular, the Eisenstein series

E_{1}(1, χ) := \frac{1}{2}L_{R}(χ, 0) + ∑_{n=1}^{∞}( ∑_{d|n} χ(d))q^{n}

E_{k−1}(1, ω^{1−k}) := \frac{1}{2}L(ω^{1−k}, 2 − k) + ∑_{n=1}^{∞}( ∑_{d|n} ω^{1−k}(d)d^{k−2})q^{n}  (19)

= \frac{1}{2}ζ_{p}(2 − k) + ∑_{n=1}^{∞}( ∑_{d|n} ω^{−1}(d)⟨d⟩^{k−2})q^{n}

belong to the spaces M_{1}(m, χ) and M_{k−1}(p, ω^{1−k}) respectively. Here, ζ_{p}(s) = L_{p}(1, s) denotes the p-adic zeta-function of Kubota-Leopoldt. In (19), the character ω^{1−k} is always viewed as having modulus p, even when k ≡ 1 (mod p − 1). Let

G_{k−1}(1, ω^{1−k}) := 2ζ_{p}(2 − k)^{−1}E_{k−1}(1, ω^{1−k})  (20)
= 1 + 2ζ_{p}(2 − k)^{−1} ∑_{n=1}^{∞}( ∑_{d|n} ω^{−1}(d)⟨d⟩^{k−2})q^{n}  (21)

be the associated normalized Eisenstein series of weight k − 1.

In Section 2.3, we consider the product

P_{k} := E_{1}(1, χ)G_{k−1}(1, ω^{1−k}) ∈ M_{k}(mp, χω^{1−k}).  (22)

As explained in Section 2.3 in the setting of Hilbert modular forms, it follows from general principles that the series P_{k} can be expressed uniquely as the sum of a cusp form and a linear combination of the Eisenstein series (18) in M_{k}(mp, χω^{1−k}):

P_{k} = ( \begin{matrix}Cusp \\ form \end{matrix} ) + ∑_{(η,ψ)∈J} a_{k}(η, ψ)E_{k}(η, ψ),  (23)

where (η, ψ) ranges over a set J of pairs of (not necessarily primitive) Dirichlet characters of modulus m_{η} and m_{ψ}, respectively, satisfying

m_{η}m_{ψ} = mp,  ηψ = χω^{1−k}.  (24)

The main result of Section 2.3 is the computation of certain coefficients in (23) for k > 2:

a_{k}(1, χω^{1−k}) = −ℒ_{an}(χ, k)^{−1},  a_{k}(χ, ω^{1−k}) = −ℒ_{an}(χ^{−1}, k)^{−1}⟨m⟩^{k−1},  (25)

where ℒ_{an}(χ, k) is the quantity defined in (10). The derivation of (25) proceeds by comparing the constant terms of both sides of (23) at various cusps.

In Section 2.4, we consider Hida’s idempotent

e : M_{k}(mp, χω^{1−k})⟶M_{k}(mp, χω^{1−k}),

which is defined as

e := lim U_{p}^{n!}

on the submodule of modular forms with Fourier coefficients in O_{E}, and extended to the space M_{k}(mp, χω^{1−k}) by E-linearity. The image of e,

M_{k}^{o}(mp, χω^{1−k}) := eM_{k}(mp, χω^{1−k}),

has dimension that is bounded independently of k, and is called the ordinary subspace. The operator e preserves the space of cusp forms. For all (η, ψ) satisfying (24), and k > 1, it can be checked that

eE_{k}(η, ψ) = \{ \begin{matrix}E_{k}(η, ψ) & if p ∤ m_{η}; \\ 0 & if p | m_{η}. \end{matrix}

Hence the modular form P_{k}^{o} := eP_{k} can be written

P_{k}^{o} = ( \begin{matrix}A cusp \\ form \end{matrix} ) + ∑_{(η,ψ)∈J^{o}} a_{k}(η, ψ)E_{k}(η, ψ),  (26)

where the sum on the right is taken over the set J^{o} of pairs (η, ψ) ∈ J for which p ∤ m_{η}.

Let u_{k}, v_{k} and w_{k} be scalars defined for integer k > 2 by

u_{k} := \frac{ℒ_{an}(χ, k)^{−1}}{c_{k}} ,  v_{k} := \frac{ℒ_{an}(χ^{−1}, k)^{−1}⟨m⟩^{k−1}}{c_{k}} ,  w_{k} := \frac{1}{c_{k}} ,  (27)

where

c_{k} := ℒ_{an}(χ, k)^{−1} + ℒ_{an}(χ^{−1}, k)^{−1}⟨m⟩^{k−1} + 1.

The vector (u_{k}, v_{k}, w_{k}) is proportional to (a_{k}(1, χω^{1−k}), a_{k}(χ, ω^{1−k}), −1). From (26), it follows that the modular form

H_{k} := u_{k}E_{k}(1, χω^{1−k}) + v_{k}E_{k}(χ, ω^{1−k}) + w_{k}P_{k}^{o}  (28)

can be written as a linear combination

H_{k} = ( \begin{matrix}A cusp \\ form \end{matrix} ) + ∑_{η≠1,χ^{(η,ψ)∈J^{o}}} w_{k}a_{k}(η, ψ)E_{k}(η, ψ).  (29)

In Section 2.5, we introduce a Hecke operator t that satisfies

tE_{1}(1, χω^{0}) = E_{1}(1, χω^{0})  (30)

and annihilates all the Eisenstein series contributions in (29), so that

F_{k} := tH_{k}

is a cusp form. Equation (30) explains the necessity of calculating the coefficients in (25) instead of applying a Hecke operator t directly to P_{k}^{o}—any Hecke operator that annihilates E_{k}(1, χω^{1−k}) or E_{k}(χ, ω^{1−k}) necessarily annihilates their common weight 1 specialization E_{1}(1, χω^{0}) = E_{1}(χ, ω^{0}).

p-adic interpolation of modular forms

In Section 3 we describe the p-adic interpolation (in the variable k) of the Eisenstein series E_{k}(η, ψ) as well as of the forms P_{k}, P_{k}^{o}, H_{k} and F_{k}. For this purpose, the Iwasawa algebra Λ is defined in Section 3.1. In our application, it is most useful to view Λ as a complete subring of the ring C(Z_{p}, E) of continuous E-valued functions on Z_{p} equipped with the topology of uniform convergence inherited from the sup norm. The ring Λ is the completion of the O_{E}- subalgebra of C(Z_{p}, E) generated by the functions k ↦ a^{k} as a ranges over 1 + pZ_{p}. For each k ∈ Z_{p}, write ν_{k} : Λ⟶E for the evaluation homomorphism ν_{k}(h) := h(k). The algebra Λ is known to be isomorphic to the power series ring O_{E}[[T]], and in particular, by the Weierstrass preparation theorem, any element of Λ has finitely many zeroes. If h belongs to the fraction field F_{Λ} of Λ, it follows that the evaluation ν_{k}(h) ∈ E is defined for all but finitely many k ∈ Z_{p}. A Λ-adic modular form of tame level m and character χ is, by definition, a formal q-expansion 𝒢 ∈ F_{Λ} ⊗ Λ[[q]] satisfying

ν_{k}(𝒢) belongs to M_{k}(mp, χω^{1−k}) for almost all integers k > 1.

(Here, ν_{k}(𝒢) ∈ E[[q]] is simply the power series obtained from 𝒢 by applying ν_{k} to its coefficients.) Such a 𝒢 is said to be a Λ-adic cusp form if ν_{k}(𝒢) is a cusp form for all but finitely many k > 1. The F_{Λ}-vector spaces of Λ-adic modular forms and cusp forms are denoted M(m, χ) and S(m, χ), respectively. The ordinary projection e acts on M(m, χ) and S(m, χ) in a manner compatible with the projections ν_{k}. The images of e in M(m, χ) and S(m, χ) are denoted M^{o}(m, χ) and S^{o}(m, χ), respectively. These definitions are all recalled in the context of Hilbert modular forms in Section 3.1.

Basic examples of Λ-adic forms are given by the Λ-adic Eisenstein series ℰ (η, ψ) satisfying

ν_{k}(ℰ (η, ψ)) = E_{k}(η, ψω^{1−k}) for k ∈ Z^{≥2}
= \frac{1}{2}δ_{η=1}L_{S,p}(ψω, 1 − k) + ∑_{n=1}^{∞}( ∑_{{d|n \atop p∤d}}η (\frac{n}{d}) ψ(d)⟨d⟩^{k−1})q^{n}.

(By convention, the character ψω^{1−k} is viewed as having modulus divisible by p, even if p does not divide its conductor; in particular ψω^{1−k}(p) = 0 for all k.) The existence and basic properties of the Λ-adic Eisenstein series, which are intimately related to those of the p-adic L-functions attached to abelian characters of G_{F}, are recalled in Section 3.2.

In Section 3.3 we define elements 𝒫 ∈ M(m, χ), ordinary forms 𝒫^{o}, ℋ ∈ M^{o}(m, χ), and a cusp form ℱ ∈ S^{o}(m, χ) satisfying

ν_{k}(𝒫) = P_{k}, ν_{k}(𝒫^{o}) = P_{k}^{o}, ν_{k}(ℋ ) = H_{k}, ν_{k}(ℱ) = F_{k}

for almost all k ∈ Z^{>2}. The proof consists essentially in observing that there are elements u, v, and w ∈ F_{Λ} such that, for almost all k > 2,

ν_{k}(u) = u_{k}, ν_{k}(v) = v_{k}, ν_{k}(w) = w_{k},

where u_{k}, v_{k}, and w_{k} are the constants introduced in (27). In Section 3.3, we describe the main properties of the Λ-adic cusp form ℱ, most notably:

1. Let

Λ_{(1)} := \{\frac{f}{g} , with f, g ∈ Λ and g(1) ≠ 0\}

denote the localization of Λ at ker ν_{1}. Under the assumption (11) when 𝔭 is the unique prime above p, it can be seen that the elements u, v and w belong to Λ_{(1)}, and that u even belongs to Λ_{(1)}^{×}. In particular, the cusp form ℱ belongs to Λ_{(1)} ⊗_{Λ} Λ[[q]], and hence its weight one specialization is defined. Furthermore, assuming Leopoldt’s conjecture for F, we have

ν_{1}(ℱ) = E_{1}(1, χω^{0}).

2. Let

ν_{1+ε} : Λ_{(1)}⟶E[ε]/ε^{2},  ν_{1+ε}(f) := f(1) + f′(1)ε  (31)

be the natural lift of ν_{1} to the ring Ẽ := E[ε]/ε^{2} of dual numbers over E. The expression F_{1+ε} := ν_{1+ε}(ℱ) can be thought of as a “cusp form of weight 1+ε”. Let u_{1} := u(1) and v_{1} = v(1) be the values at k = 1 of the elements u, v. A direct calculation, explained in Section 3.3, shows that for every prime q,

a_{q}(F_{1+ε}) = \{ \begin{matrix}(1 + v_{1}κ_{cyc}(q)ε) + χ(q)(1 + u_{1}κ_{cyc}(q)ε) & if q ∤ mp, \\ 1 + v_{1}κ_{cyc}(q)ε & if q | m, \\ 1 + u_{1}ℒ_{an}(χ)ε & if q = p. \end{matrix}  (32)

An examination of the Fourier coefficients of F_{1+ε} reveals that it is an eigenform relative to the natural action of the Hecke operators on Ẽ[[q]] inherited from the specialization map ν_{1+ε} : M(m, χ)⟶Ẽ[[q]]. In descriptive terms, the Λ-adic cusp form ℱ is an “eigenform in a first order infinitesimal neighborhood of weight one.”

Let T denote the Λ-algebra generated by the Hecke operators acting on the finite-dimensional F_{Λ}-vector space S^{o}(m, χ). The eigenform F_{1+ε} gives rise to surjective Λ_{(1)}-algebra homomorphisms

ϕ_{1} : T ⊗ Λ_{(1)}⟶E, ϕ_{1+ε} : T ⊗ Λ_{(1)}⟶Ẽ

sending t ∈ T ⊗ Λ_{(1)} to the associated eigenvalues of t acting on F_{1} and F_{1+ε}, respectively. Write T_{(1)} for the localization of T ⊗ Λ_{(1)} at the maximal ideal 𝔪 := ker ϕ_{1}. The homomorphisms ϕ_{1} and ϕ_{1+ε} factor through T_{(1)}, and we will often view them as defined on this quotient of T ⊗ Λ_{(1)}. Likewise, we will write 𝔪 for the maximal ideal of T_{(1)}.

Representations associated to Λ-adic forms

In Section 4 we exploit the homomorphism ϕ_{1+ε} to construct the desired cocycle κ. Let F_{(1)} denote the total ring of fractions of T_{(1)}:

F_{(1)} := \{\frac{a}{b}, a, b ∈ T_{(1)}, b is not a zero divisor\} .

A key ingredient in the construction of κ is the two-dimensional Galois representation

ρ : G_{F} ⟶GL_{2}(F_{(1)}); ρ(σ) = ( \begin{matrix}a(σ) & b(σ) \\ c(σ) & d(σ) \end{matrix} )

attached to the space of ordinary Λ-adic cusp forms. The existence and basic properties of this representation, which were established by Wiles in [17], are recalled in Section 4.1. It is shown in Section 4.1 that ρ can be conjugated so that, for all σ ∈ G_{F},

a(σ), d(σ) belong to T_{(1)}^{×}.

Under the assumption that L_{R}(χ, 0) ≠ 0, it is further shown in Section 4.2 that there is a multiple b̃ of the matrix entry b (by some element of F_{(1)}) for which:

1. The function σ ↦ K(σ) := b̃(σ)/d(σ) takes values in the maximal ideal 𝔪 ⊂ T_{(1)}.

2. Up to scaling by E^{×}, the function κ given by the formula

ϕ_{1+ε}(K(σ)) =: κ(σ)ε

has the required properties—namely, it is a 1-cocycle representing a class

[κ] ∈ H_{𝔭}^{1}(F, E(χ^{−1}))^{cyc}

that satisfies

res_{𝔭}([κ]) = −ℒ_{an}(χ)κ_{nr} + κ_{cyc}.

1 Duality and the ℒ -invariant

We begin by recalling some notations and conventions regarding characters of F. Fix an ordering of the n real places of F, and let sgn_{i} : R^{×} → {±1} denote the associated sign function. For a vector r ∈ (Z/2Z)^{n} write sgn(a)^{r} = ∏_{i=1}^{n} sgn_{i}(a)^{r_{i}}.

Let 𝔟 be an integral ideal of F. Let I_{𝔟} denote the group of fractional ideals of F that are relatively prime to 𝔟. A narrow ray class character modulo 𝔟 is a homomorphism

ψ : I_{𝔟} → \overline{Q}^{×}

that is trivial on any principal ideal (α) generated by a totally positive element α ≡ 1 (mod 𝔟). The function ψ may be extended to a function on the set of all integral ideals of F by defining ψ(𝔪) = 0 if 𝔪 is not relatively prime to 𝔟. The character ψ may be viewed as a character modulo 𝔟𝔞 for any integral ideal 𝔞. If there exists a narrow ray class character ψ_{0} modulo 𝔠 for some proper divisor 𝔠 of 𝔟, with ψ_{0}(𝔪) = ψ(𝔪) for all 𝔪 relatively prime to 𝔟, then ψ is called an imprimitive character. The minimal such divisor of 𝔟 is called the conductor of ψ, and ψ_{0} is called the associated primitive character.

Given a narrow ray class character ψ modulo 𝔟, there exists an r ∈ (Z/2Z)^{n} such that

ψ((α)) = sgn(α)^{r} for α ≡ 1 (mod 𝔟).

The vector r is called the sign of ψ. The character ψ is said to be totally even if r = (0, 0, …, 0) and totally odd if r = (1, 1, …, 1).

We place ourselves in the situation of the introduction regarding the character χ and the finite sets R and S of places of the totally real field F. More precisely, we assume that χ is a primitive character of conductor 𝔫 and that S consists exactly of the set of places dividing 𝔫p∞, while R = S − {𝔭}. Recall that E is a finite extension of Q_{p} containing the values of the character χ.

Lemma 1.1. The following are equivalent:

1. The special value L_{R}(χ, 0) is non-zero.

2. For all places v ∈ R, we have χ(v) ≠ 1. (In particular, the character χ is totally odd.)

3. The vector space U_{χ} is one-dimensional over E.

Proof. Since χ is non-trivial, the L-series L_{R}(χ^{−1}, s) (viewing χ as a complex character) is holomorphic and non-vanishing at s = 1. A consideration of the local factors in the functional equation relating L_{R}(χ^{−1}, s) and L_{R}(χ, 1 − s) shows that

ord_{s=0} L_{R}(χ, s) = #{v ∈ R such that χ(v) = 1}.  (33)

This implies the equivalence of (1) and (2). (See, for example, Prop. 3.4 in Ch. I of [15].) Dirichlet’s S-unit theorem implies the equivalence of (2) and (3). (See also Ch. I.4 of [15].)

Recall that the character χ is said to be of “type S” in the terminology of Greenberg (cf. the first paragraph in [18]) if H ∩ F_{∞} = F, where F_{∞} is the cyclotomic Z_{p}-extension of F. This condition is satisfied, for example, if χ has order prime to p. The following lemma (whose proof is deep, relying on the full force of the main conjecture for totally real fields) essentially disposes of Conjecture 1 in the case where L_{R}(χ, 0) = 0.

Lemma 1.2. Suppose that H ∩ F_{∞} = F. If L_{R}(χ, 0) = 0, then L′_{S,p}(χ, 0) = 0, and therefore Conjecture 1 holds for χ and R.

Proof. Theorem 1.2 of [18] implies that

ord_{s=0} L_{S,p}(χω, s) = ord_{s=0} f_{χ},  (34)

where f_{χ} corresponds to the characteristic power series of the Λ = Z_{p}[[Gal(H_{∞}/H)]]-module X_{χ} ⊂ Gal(L_{∞}/H_{∞}) defined on p. 409 of [16]. (Here H_{∞} denotes the cyclotomic Z_{p}-extension of H and L_{∞} is the maximal unramified abelian pro-p-extension of H_{∞}.) It is known by an explicit construction (cf. the statement at the top of p. 410 of [16], or [1], [6]) that

#{v ∈ S such that χ(v) = 1} ≤ ord_{s=0} f_{χ}.  (35)

Since χ(𝔭) = 1 and 𝔭 ∈ S, equations (33) and (35) combine with (34) to give

ord_{s=0} L_{S,p}(χω, s) ≥ ord_{s=0} L_{R}(χ, s) + 1.

In particular, L′_{S,p}(χω, 0) = 0 when L_{R}(χ, 0) = 0.

Thanks to Lemma 1.2, we assume for the rest of this article χ satisfies the equivalent assumptions of Lemma 1.1.

Let E(1)(χ) and E(χ^{−1}) be the continuous one-dimensional representations of G_{F} defined in the introduction, following equation (14). We will now study certain Galois cohomology groups (both local and global) associated to these p-adic representations.

If K is a field and M is a finite G_{K}-module, we denote by H^{i}(K, M) the Galois cohomology group of continuous i-cocycles on G_{K} with values in M, modulo i-coboundaries. If O_{E} denotes the rings of integers of E and π is a uniformizing element of O_{E}, then recall the definitions:

H^{1}(K, O_{E}(χ^{−1})) := lim_{←,n} H^{1}(K, O_{E}/π^{n}(χ^{−1})),
H^{1}(K, E(χ^{−1})) := H^{1}(K, O_{E}(χ^{−1})) ⊗_{O_{E}} E.

1.1 Local cohomology groups

Let v be any place of F, and let G_{v} and I_{v} ⊂ G_{v} denote a choice of decomposition and inertia group at v in G_{F}. Given any finite G_{F} -module M, the inflation-restriction sequence attached to I_{v} yields the exact sequence

0⟶H^{1}(G_{v}/I_{v}, M^{I_{v}})⟶H^{1}(F_{v}, M) ⟶^{res_{Iv}} H^{1}(I_{v}, M)^{G_{v}/I_{v}}.  (36)

A class in H^{1}(F, M) is said to be unramified at v if its restriction to H^{1}(F_{v}, M) lies in the kernel of res_{I_{v}}.

Recall that local Tate duality gives rise to a perfect pairing

H^{1}(F_{v}, Z/p^{n}Z) × H^{1}(F_{v}, μ_{p^{n}})⟶H^{2}(F_{v}, μ_{p^{n}}) = Z/p^{n}Z.

After tensoring this with O_{E}/π^{n}, twisting by χ^{−1}, and passing to the limit as n → ∞, one obtains perfect O_{E}-linear and E-linear pairings

⟨ , ⟩_{v} : H^{1}(F_{v}, O_{E}/π^{n}(χ^{−1})) × H^{1}(F_{v}, O_{E}/π^{n}(1)(χ))⟶O_{E}/π^{n},  (37)
⟨ , ⟩_{v} : H^{1}(F_{v}, E(χ^{−1})) × H^{1}(F_{v}, E(1)(χ))⟶E.  (38)

If χ(v) ≠ 1, then H^{1}(G_{v}/I_{v}, O_{E}/π^{n}(χ^{−1})^{I_{v}}) ≅ (O_{E}/π^{n}(χ^{−1})^{I_{v}})/(χ^{−1}(v) − 1) has cardinality bounded independently of n, and there are no unramified classes in H^{1}(F_{v}, E(χ^{−1})). If χ(v) = 1, we have:

1. The group H^{1}(F_{v}, E(χ^{−1})) = H^{1}(F_{v}, E) = Hom_{cts}(G_{v}, E) contains a distinguished unramified class: the unique homomorphism

κ_{nr} ∈ Hom(Gal(F_{v}^{nr}/F_{v}), O_{E})

sending the Frobenius element Frob_{v} at v to 1.

2. If v divides p, then the restriction to G_{F_{v}} of the p-adic logarithm of the cyclotomic

character

κ_{cyc} := log_{p}(ϵ_{cyc}) ∈ Hom(G_{F} , E) = H^{1}(F, E)

gives a ramified element of H^{1}(F_{v}, E). The elements κ_{nr} and κ_{cyc} will sometimes be referred to as the unramified and cyclotomic cocycles respectively.

3. For each positive integer n, the connecting homomorphism of Kummer Theory yields

an isomorphism

δ_{v,n} : F_{v}^{×} ⊗ Z/p^{n}Z⟶H^{1}(F_{v}, Z/p^{n}Z(1))

As n varies, the maps δ_{v,n} are compatible with the natural projections on both sides. Passing to the limit with n, and then tensoring with E, we obtain an isomorphism

δ_{v} : F_{v}^{×} ˆ⊗ E⟶H^{1}(F_{v}, E(1)),  (39)

where

F_{v}^{×} ˆ⊗ E := (lim_{←,n} F_{v}^{×} ⊗ Z/p^{n}Z) ⊗_{Z_{p}} E.

4. For each u ∈ F_{v}^{×} ˆ⊗ E, we have

⟨κ_{nr}, δ_{v}(u)⟩_{v} = − ord_{v}(u),  ⟨κ_{cyc}, δ_{v}(u)⟩_{v} = L_{v}(u),  (40)

where ord_{v} and L_{v} are the homomorphisms F_{v}^{×} ˆ⊗ E⟶E defined as in (5) and (6) of the introduction. The equations in (40) are direct consequences of the reciprocity law of local class field theory. In particular, the subspace of H^{1}(F_{v}, E(1)) that is orthogonal to κ_{nr} under the local Tate pairing is equal to δ_{v}(O_{F_{v}}^{×} ˆ⊗ E).

Lemma 1.3. Let v be any place of F. The dimension of H^{1}(F_{v}, E(1)(χ)) is given in the following table:

χ(v) = 1 χ(v) ≠ 1
v ∤ p∞  1  0
v | ∞  0  0
v | p [F_{v} : Q_{p}] + 1 [F_{v} : Q_{p}].

The same is true for H^{1}(F_{v}, E(χ^{−1})).

Proof. For v infinite, the lemma is clear. For v finite, Tate’s local Euler characteristic formula (see the “Corank Lemma” of [8, Chapter 2]) and Tate local duality yield

dim_{E}H^{1}(F_{v}, E(1)(χ))

= dim_{E} H^{0}(F_{v}, E(1)(χ)) + dim_{E} H^{2}(F_{v}, E(1)(χ)) + \begin{cases}0 & v ∤ p \\ [F_{v} : Q_{p}] & v | p \end{cases}

= dim_{E} H^{0}(F_{v}, E(χ^{−1})) + \begin{cases}0 & v ∤ p \\ [F_{v} : Q_{p}] & v | p \end{cases} ,

which explains the remaining values in the table.

1.2 Global cohomology groups

We now define certain subgroups of the global cohomology groups H^{1}(F, E(1)(χ)) and H^{1}(F, E(χ^{−1})) by imposing appropriate local conditions.

Recall that

H_{nr}^{1}(F_{v}, E(χ^{−1})) ≃ H^{1}(G_{v}/I_{v}, E(χ^{−1})^{I_{v}}) = \begin{cases}E · κ_{nr} & if χ(v) = 1; \\ 0 & otherwise. \end{cases}

denotes the subgroup of unramified cohomology classes. We denote by H_{nr}^{1}(F_{v}, E(χ)(1)) the orthogonal complement of H_{nr}^{1}(F_{v}, E(χ^{−1})) under the local Tate pairing, so

H_{nr}^{1}(F_{v}, E(χ)(1)) ≃ \begin{cases}O_{F_{v}}^{×} ˆ⊗ E & if χ(v) = 1; \\ H^{1}(F_{v}, E(χ)(1)) & otherwise. \end{cases}

Let

H_{[𝔭]}^{1}(F, E(χ^{−1})) ⊂ H_{𝔭}^{1}(F, E(χ^{−1}))

denote the subgroups of H^{1}(F, E(χ^{−1})) consisting of classes whose restriction to G_{F_{v}} belong to H_{nr}^{1}(F_{v}, E(χ^{−1})) for all v ≠ 𝔭, and which are trivial/arbitrary, respectively, at the prime 𝔭. Similarly, we denote by

H_{[𝔭]}^{1}(F, E(1)(χ)) ⊂ H_{𝔭}^{1}(F, E(1)(χ))

the subgroups of classes in H^{1}(F, E(1)(χ)) whose restrictions lie in H_{nr}^{1}(F_{v}, E(χ)(1)) for all v ≠ 𝔭, and which are trivial/arbitrary, respectively, at the prime 𝔭.

Recall the group U_{χ} of Gross-Stark units that was defined in equation (4) of the introduction.

Proposition 1.4. The natural map

δ : U_{χ}⟶H_{𝔭}^{1}(F, E(1)(χ))

induced by the connecting homomorphism of Kummer theory is an isomorphism. In particular,

dim_{E} H_{𝔭}^{1}(F, E(1)(χ)) = 1.

Proof. If G = Gal(H/F), then the restriction map

H^{1}(F, E(1)(χ)) ⟶^{res} H^{1}(H, E(1)(χ))^{G}  (41)

induces an isomorphism. Indeed, the kernel and the cokernel of this map are given by H^{i}(G, E(1)(χ)^{G_{H}}) for i = 1, 2 where G_{H} = Gal(F̄/H), and E(1)(χ)^{G_{H}} = 0. Thus

H^{1}(F, E(1)(χ)) ⟶^{≃} H^{1}(H, E(1)(χ))^{G} = H^{1}(H, E(1))^{χ^{−1}} ≃ (H^{×} ˆ⊗ E)^{χ^{−1}},  (42)

where the superscript of χ^{−1} affixed to an E[G]-module denotes the submodule on which G acts via χ^{−1}. The last isomorphism in (42) arises from (39) with F_{v} replaced by H.

Similarly, in the local situation, we have

H^{1}(F_{v}, E(1)(χ)) ≃ (H_{w}^{×} ˆ⊗ E)^{χ^{−1}}.  (43)

where w is some prime of H over v. The result now follows from the fact that the group H_{nr}^{1}(F_{v}, E(1)(χ)) corresponds to (O_{H_{w}}^{×} ˆ⊗ E)^{χ^{−1}} under this identification.

If W is any subspace of the local cohomology group H^{1}(F_{𝔭}, E(χ^{−1})) = Hom(G_{𝔭}, E), define

H_{𝔭,W}^{1}(F, E(χ^{−1})) ⊂ H_{𝔭}^{1}(F, E(χ^{−1}))

to be the subspace consisting of classes whose image under res_{𝔭} belongs to W. The following lemma can be viewed as the global counterpart of Lemma 1.3.

Lemma 1.5. Let χ and 𝔭 satisfy χ(𝔭) = 1. Then

dim_{E} H_{𝔭}^{1}(F, E(χ^{−1})) = [F_{𝔭} : Q_{p}].

More generally, if W ⊂ H^{1}(F_{𝔭}, E) is any subspace containing the unramified cocycle κ_{nr}, then

dim_{E} H_{𝔭,W}^{1}(F, E(χ^{−1})) = dim_{E} W − 1.

Proof. The Poitou-Tate exact sequence in Galois cohomology, applied to the finite modules O_{E}/π^{n}(χ^{−1}), gives rise to the exact sequence

\begin{matrix}0⟶H_{[𝔭]}^{1}(F, O_{E}/π^{n}(χ^{−1}))⟶H_{𝔭}^{1}(F, O_{E}/π^{n}(χ^{−1}))⟶H^{1}(F_{𝔭}, O_{E}/π^{n}(χ^{−1})) \\ ⟶H_{𝔭}^{1}(F, O_{E}/π^{n}(1)(χ))^{∨}, \end{matrix}  (44)

where the last map arises from the local Tate pairing. Now we note that the module H_{[𝔭]}^{1}(F, O_{E}/π^{n}(χ^{−1})) maps (with kernel bounded independently of n) to the group of homomorphisms from G_{H} to O_{E}/π^{n} that are everywhere unramified. Hence H_{[𝔭]}^{1}(F, O_{E}/π^{n}(χ^{−1})) has bounded cardinality as n → ∞. Passing to the limit with n and tensoring with E, we obtain the exact sequence

0⟶H_{𝔭}^{1}(F, E(χ^{−1}))⟶H^{1}(F_{𝔭}, E(χ^{−1}))⟶H_{𝔭}^{1}(F, E(1)(χ))^{∨}.  (45)

We now observe that the element κ_{nr} maps to a non-zero element of the one-dimensional vector space H_{𝔭}^{1}(F, E(χ)(1))^{∨}, by (40) and Proposition 1.4. Hence the last arrow in (45) is surjective, and the same is true for the exact sequence

0⟶H_{𝔭,W}^{1}(F, E(χ^{−1}))⟶W ⟶H_{𝔭}^{1}(F, E(1)(χ))^{∨},

for any W ⊂ H^{1}(F_{𝔭}, E) containing κ_{nr}. The lemma now follows from Proposition 1.4.

1.3 A formula for the ℒ -invariant

Let W_{cyc} be the subspace of H^{1}(F_{𝔭}, E) spanned by the unramified and cyclotomic cocycles κ_{nr} and κ_{cyc}. Write

H_{𝔭,cyc}^{1}(F, E(χ^{−1})) := H_{𝔭,W_{cyc}}^{1}(F, E(χ^{−1})).

By Lemma 1.5, this space is one-dimensional over E. If κ is any non-zero element of this space, we may therefore write

res_{𝔭} κ = x · κ_{nr} + y · κ_{cyc},  (46)

for some x, y ∈ E, and the ratio x/y does not depend on the choice of κ. (Note that y ≠ 0, or else (46) contradicts Lemma 1.3 when W is the 1-dimensional space spanned by κ^{nr}.)

Proposition 1.6. Let ℒ (χ) be the ℒ -invariant introduced in equation (7) of the introduction. Then

ℒ (χ) = −x/y.

Proof. The global reciprocity law of class field theory applied to δ(u_{χ}) ∈ H_{𝔭}^{1}(F, E(χ)(1)) with u_{χ} ∈ U_{χ} and κ ∈ H_{𝔭,cyc}^{1}(F, E(χ^{−1})) implies that

∑_{v}⟨res_{v} κ, δ_{v}(u_{χ})⟩_{v} = 0,

where the sum is taken over all places v of F. By definition, all the terms in this sum vanish except the one corresponding to v = 𝔭. It follows that

⟨res_{𝔭} κ, δ_{𝔭}(u_{χ})⟩_{𝔭} = 0.

Combining the expression (46) for res_{𝔭} κ with (40) yields the proposition.

2 Hilbert modular forms

2.1 Definitions

We briefly describe the basic definitions of classical Hilbert modular forms over a totally real field F. Recall that we have chosen an ordering of the n real embeddings of F. For an element z ∈ C^{n}, an integer k, and elements a, b ∈ F, define

(az + b)^{k} := ∏_{i=1}^{n}(a_{i}z_{i} + b_{i})^{k} ∈ C,

where a_{i}, b_{i} denote the images of a, b, respectively, under the ith real embedding of F.

Let ψ be a narrow ray class character modulo 𝔟 with sign r ∈ (Z/2Z)^{n}. Let α ∈ O_{F} be relatively prime to 𝔟. The map α ↦ sgn(α)^{r}ψ((α)) defines a character

ψ_{f} : (O_{F} /𝔟)^{×} → \overline{Q}^{×}

associated to ψ.

Let k be a positive integer. In [13, Page 649], Shimura defines a space M_{k}(𝔟, ψ) of Hilbert modular forms of level 𝔟 and character ψ. Let h denote the size of the narrow class group Cl^{+}(F). An element f ∈ M_{k}(𝔟, ψ) is an h-tuple of holomorphic functions

f_{λ} : H^{n} → C

for λ ∈ Cl^{+}(F) satisfying certain modularity properties that we now describe (see (2.5a) and (2.15a-c) of [13]).

For each λ ∈ Cl^{+}(F), choose a representative fractional ideal 𝔱_{λ}. Let GL_{2}^{+}(F) denote the group of 2 × 2 invertible matrices over F that have positive determinant at each real place of F. Define

Γ_{λ} := \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ GL_{2}^{+}(F) : a, d ∈ O_{F} , b ∈ 𝔱_{λ}^{−1}𝔡^{−1}, c ∈ 𝔟𝔱_{λ}𝔡, ad − bc ∈ O_{F}^{×}\} ,

where 𝔡 denotes the different of F. Each function f_{λ} satisfies

f_{λ}|_{γ} = ψ_{f}(a)f_{λ}  (47)

for all γ ∈ Γ_{λ}, where

f_{λ}|_{γ}(z) := det(γ)^{k/2}(cz + d)^{−k}f_{λ}(γz).  (48)

In (48), the positive square root is taken in

det(γ)^{k/2} := ∏_{i=1}^{n} det(γ_{i})^{k/2}

and

γz := (\frac{a_{1}z_{1} + b_{1}}{c_{1}z_{1} + d_{1}} , …, \frac{a_{n}z_{n} + b_{n}}{c_{n}z_{n} + d_{n}} ) .

In [13, Page 648], Shimura defines a Hecke operator S(𝔪) for each integral ideal 𝔪 relatively prime to 𝔟 on the space of h-tuples f = (f_{λ}) satisfying (47). We do not recall the definition of S(𝔪) here. The space M_{k}(𝔟, ψ) is defined as the set of f such that

S(𝔪)f = ψ(𝔪)f, for all 𝔪 with (𝔪, 𝔟) = 1.

The modularity property (47) implies that f_{λ} has a Fourier expansion

f_{λ}(z) = a_{λ}(0) + ∑_{\begin{matrix}b∈𝔱_{λ} \\ b≫0 \end{matrix}} a_{λ}(b)e_{F} (bz),

where e_{F} (x) = exp(2πi · Tr_{F/Q}(x)). We call the coefficients a_{λ}(b) the unnormalized Fourier coefficients of f, and define also the normalized Fourier coefficients c(𝔪, f) and c_{λ}(0, f) of f as follows. Any non-zero integral ideal 𝔪 may be written 𝔪 = (b)𝔱_{λ}^{−1} with b totally positive (and b ∈ 𝔱_{λ}) for a unique λ ∈ Cl^{+}(F). Define

c(𝔪, f) := a_{λ}(b) N(𝔱_{λ})^{−k/2}.  (49)

The right side of (49) is easily seen to depend only on 𝔪 and not on the choice of b since f_{λ}(ϵz) Nϵ^{k/2} = f_{λ}(z) for every totally positive unit ϵ of F. Similarly we define

c_{λ}(0, f) := a_{λ}(0) N(𝔱_{λ})^{−k/2}

for each λ ∈ Cl^{+}(F). Note that our normalized Fourier coefficients c(𝔪, f) are denoted C(𝔪, f) in [13]; we have chosen to remain consistent with the notation of [18].

If for each γ ∈ GL_{2}^{+}(F) and λ ∈ Cl^{+}(F), the function f_{λ}|_{γ} has constant term equal to 0, then f is called a cusp form. The space of cusp forms of weight k, level 𝔟 and character ψ is denoted S_{k}(𝔟, ψ).

2.2 Eisenstein Series

The most basic examples of Hilbert modular forms arise from Eisenstein series attached to pairs of narrow ideal class characters of F.

Let 𝔞 and 𝔟 be integral ideals of F, and let η and ψ be (possibly imprimitive) E-valued characters of the narrow ideal class group modulo 𝔞 and 𝔟, respectively, with associated signs q, r ∈ (Z/2Z)^{n}. Let k ≥ 1 be an integer satisfying

q + r ≡ (k, k, …, k) (mod 2Z^{n}).

We view the characters ηψ and ηψ^{−1} as having modulus 𝔞𝔟. Here and elsewhere, an expression like L(ψη^{−1}, 1 − k) without a subscript on the L is assumed to be taken with the set S in equation (1) equal to the set of archimedean primes and those dividing the modulus of the given character.

Proposition 2.1. For all k ≥ 1, there exists an element E_{k}(η, ψ) ∈ M_{k}(𝔞𝔟, ηψ) such that

c(𝔪, E_{k}(η, ψ)) = ∑_{𝔯|𝔪}η (\frac{𝔪}{𝔯}) ψ(𝔯) N𝔯^{k−1},  (50)

for all non-zero ideals 𝔪 of O_{F}. When k > 1, we have

c_{λ}(0, E_{k}(η, ψ)) = \begin{cases}2^{−n}η^{−1}(𝔱_{λ})L(ψη^{−1}, 1 − k) & if 𝔞 = 1, \\ 0 & otherwise. \end{cases}  (51)

When k = 1, we have E_{1}(η, ψ) = E_{1}(ψ, η), and

c_{λ}(0, E_{1}(η, ψ)) = 2^{−n} · \begin{cases}η^{−1}(𝔱_{λ})L(ψη^{−1}, 0) & if 𝔞 = 1 and 𝔟 ≠ 1, \\ ψ^{−1}(𝔱_{λ})L(ηψ^{−1}, 0) & if 𝔟 = 1 and 𝔞 ≠ 1, \\ η^{−1}(𝔱_{λ})L(ψη^{−1}, 0) + ψ^{−1}(𝔱_{λ})L(ηψ^{−1}, 0) & if 𝔞 = 𝔟 = 1, \\ 0 & if 𝔞 ≠ 1 and 𝔟 ≠ 1. \end{cases} (52)

Remark 2.2. This proposition is well-known to the experts, and follows easily, for example, from Katz’s computation of the q-expansion of Eisenstein series in Chapter III of [10]. Since our notations differ somewhat from those of [10], we have supplied the details of the proof of formulae (51) and (52) for the sake of being self-contained. Furthermore, some of the objects that are introduced in our proof of (51) and (52) will be used in later calculations. But the reader willing to take Proposition 2.1 on faith may wish to skip the proof on a first reading; in fact, in the remainder of Section 2.2 we assemble computations of constant terms of Eisenstein series at various cusps, and we recommend that the reader continue on to Section 2.3 and refer back to Section 2.2 as necessary.

Proof of Proposition 2.1. The existence of the Eisenstein series E_{k}(η, ψ) satisfying (50) is given by [13, Proposition 3.4]. We recall the definition of E_{k}(η, ψ) given there. Let

U = {u ∈ O_{F}^{×} : Nu^{k} = 1, u ≡ 1 (mod 𝔞𝔟)}.

For k ≥ 1, we set E_{k}(η, ψ) = (f_{λ}), where f_{λ}(z) is defined via Hecke’s trick as follows. For z ∈ H^{n} and s ∈ C with Re(2s + k) > 2, define

f_{λ}(z, s) := C · \frac{N(𝔱_{λ})^{−k/2}}{N𝔟} ∑_{𝔯∈Cl(F)}N𝔯^{k} ∑_{\begin{matrix}a∈𝔯 \\ b∈𝔡^{−1}𝔟^{−1}𝔱_{λ}^{−1}𝔯 \\ (a,b) mod U, (a,b)≠(0,0) \end{matrix}} \frac{sgn(a)^{q}η(a𝔯^{−1})}{(az + b)^{k}|az + b|^{2s}} ×

∑_{c∈𝔱_{λ}𝔯^{−1}/𝔟𝔱_{λ}𝔯^{−1}} sgn(c)^{r}ψ(c𝔱_{λ}^{−1}𝔯)e_{F} (−bc)

(53)

where

C := \frac{\sqrt{d_{F}} · Γ(k)^{n}}{[O_{F}^{×} : U] N(𝔡)(−2πi)^{kn}} .  (54)

The sums in (53) run over representatives 𝔯 for the wide class group Cl(F), representatives (a, b) for the non-zero elements of the product 𝔯 × 𝔡^{−1}𝔟^{−1}𝔱_{λ}^{−1}𝔯 modulo the action of U (which acts by diagonal multiplication on the two factors), and representatives c for 𝔱_{λ}𝔯^{−1}/𝔟𝔱_{λ}𝔯^{−1}. The term sgn(a)^{q}η(a𝔯^{−1}) in (53) for a = 0 should be interpreted as being equal to 0 if 𝔞 ≠ 1 and equal to η(𝔯^{−1}) if 𝔞 = 1; a similar interpretation holds for c = 0. Let us now assume that ψ is a primitive character, i.e. that the conductor of ψ is 𝔟. From [13, (3.11)], we find that the last sum in (53) may be evaluated as follows:

∑_{c∈𝔱_{λ}𝔯^{−1}/𝔟𝔱_{λ}𝔯^{−1}} sgn(c)^{r}ψ(c𝔱_{λ}^{−1}𝔯)e_{F} (−bc) = sgn(−b)^{r}ψ^{−1}(−b𝔡𝔟𝔱_{λ}𝔯^{−1})τ(ψ).  (55)

Here τ(ψ) is the Gauss sum attached to ψ, defined by

τ(ψ) := ∑_{x∈𝔟^{−1}𝔡^{−1}/𝔡^{−1}} sgn(x)^{r}ψ(x𝔟𝔡)e_{F} (x).  (56)

When 𝔟 = 1, we interpret (56) as τ(ψ) = ψ(𝔡).

For fixed z, the function f_{λ}(z, s) has a meromorphic continuation in s to the entire complex plane. The function f_{λ}(z) is defined as f_{λ}(z, 0). Note that if k > 2 the Hecke regularization process is not necessary; the series in (53) with s = 0 converges and yields the definition of f_{λ}(z).

Suppose that k > 1. By [13, (3.7)], the unnormalized constant term a_{λ}(0) of f_{λ}(z) is equal to 0 if 𝔞 ≠ 1, and is equal to the value of

C · \frac{N(𝔱_{λ})^{−k/2}τ(ψ)}{N𝔟} ∑_{𝔯∈Cl(F)}N𝔯^{k} ∑_{\begin{matrix}b∈𝔡^{−1}𝔟^{−1}𝔱_{λ}^{−1}𝔯/U \\ b≠0 \end{matrix}} \frac{η(𝔯^{−1}) sgn(−b)^{r}ψ^{−1}(−b𝔡𝔟𝔱_{λ}𝔯^{−1})}{Nb^{k}|Nb|^{2s}}  (57)

at s = 0 if 𝔞 = 1. Note that this is the value obtained by sending each z_{i} → ∞i in (53).

The map

(𝔯, b) ↦ 𝔣 = (b)𝔡𝔟𝔱_{λ}𝔯^{−1}

is a surjective [O_{F}^{×} : U]-to-1 map from the set of pairs (𝔯, b) in (57) to the set of non-zero integral ideals of F. We therefore find that a_{λ}(0) is the value at s = 0 of the function

\frac{\sqrt{d_{F}} N(𝔱_{λ})^{k/2} N(𝔡𝔟)^{k−1}Γ(k)^{n}τ(ψ)}{η(𝔡𝔟𝔱_{λ})(2πi)^{kn}} ∑_{𝔣⊂O_{F}} \frac{ηψ^{−1}(𝔣)}{N𝔣^{k} N(𝔡^{−1}𝔟^{−1}𝔱_{λ}^{−1}𝔯𝔣)^{2s}} ,  (58)

where 𝔯 is the chosen representative of Cl(F) equivalent to 𝔡𝔟𝔱_{λ}𝔣^{−1}. The value at s = 0 of the sum in (58) is evidently L(ηψ^{−1}, k). Since τ(ψη^{−1}) = τ(ψ)η^{−1}(𝔡𝔟), the functional equation for Hecke L-series (see [11, Theorem 3.3.1]) yields

a_{λ}(0) = \frac{N(𝔱_{λ})^{k/2}η^{−1}(𝔱_{λ})}{2^{n}}L(ψη^{−1}, 1 − k)

as desired.

When k = 1, [13, (3.7)] shows that the formula (57) requires an extra term equal to

\frac{N(𝔱_{λ})^{1/2}τ(ψ)}{2^{n}[O_{F}^{×} : U] N𝔟} ∑_{\begin{matrix}𝔯∈Cl(F) \\ t∈𝔟^{−1}𝔡^{−1}𝔱_{λ}^{−1}𝔯/𝔡^{−1}𝔱_{λ}^{−1}𝔯 \end{matrix}} sgn(t)^{r}ψ^{−1}(t𝔡𝔟𝔱_{λ}𝔯^{−1}) ∑_{\begin{matrix}b∈𝔯/U \\ b≠0 \end{matrix}} sgn(b)^{q}η(b𝔯^{−1}) · sgn(Nb)|Nb|^{−2s}. (59)

The value of (59) at s = 0 is easily seen to equal 0 if 𝔟 ≠ 1, and to equal

\frac{N(𝔱_{λ})^{1/2}ψ^{−1}(𝔱_{λ})}{2^{n}}L(ηψ^{−1}, 0)

if 𝔟 = 1. This completes the proof when ψ is primitive.

If ψ is imprimitive with conductor 𝔟_{0} | 𝔟, let ψ_{0} denote the associated primitive character. Raising the modulus of ψ_{0} at a prime already dividing 𝔟_{0} affects neither L(ηψ_{0}^{−1}, s) nor E_{k}(η, ψ_{0}). Therefore it suffices to consider the case when the modulus of ψ_{0} is raised at a prime 𝔮 ∤ 𝔟_{0}, that is, when 𝔟 = 𝔟_{0}𝔮. Then L(ηψ^{−1}, s) is obtained from L(ηψ_{0}^{−1}, s) by removing the Euler factor at 𝔮. Meanwhile, E_{k}(η, ψ) is obtained from E_{k}(η, ψ_{0}) as follows. For every modular form f ∈ M_{k}(𝔟_{0}, ψ_{0}), there exists a modular form f|𝔮 ∈ M_{k}(𝔟, ψ) such that for nonzero integral ideals 𝔪 we have

c(𝔪, f|𝔮) = \begin{cases}c(𝔪/𝔮, f) & 𝔮 | 𝔪 \\ 0 & 𝔮 ∤ 𝔪, \end{cases}

and for λ ∈ Cl^{+}(F) we have

c_{λ}(0, f|𝔮) = c_{λ𝔮}(0, f)

(see [13, Proposition 2.3]). Then it is easy to verify from (50) that

E_{k}(η, ψ) = E_{k}(η, ψ_{0}) − ψ_{0}(𝔮) N𝔮^{k−1}E_{k}(η, ψ_{0})|𝔮.

Since

(η^{−1}(𝔱_{λ}) − ψ_{0}(𝔮) N𝔮^{k−1}η^{−1}(𝔱_{λ𝔮})) L(ψ_{0}η^{−1}, 1 − k) = η^{−1}(𝔱_{λ})L(ψη^{−1}, 1 − k)

and

(ψ_{0}^{−1}(𝔱_{λ}) − ψ_{0}(𝔮)ψ_{0}^{−1}(𝔱_{λ𝔮})) L(ηψ_{0}^{−1}, 0) = 0,

we obtain the desired result for imprimitive ψ as well.

We will also need to compute the constant terms in the Fourier expansions of certain Eisenstein series at particular cusps different from ∞. More precisely, let

A = (A_{λ}) = ( \begin{matrix}1 & x_{λ} \\ α_{λ} & y_{λ} \end{matrix} )_{λ∈Cl^{+}(F)} ∈ SL_{2}(F)^{h},  (60)

where x_{λ}, y_{λ}, and α_{λ} are chosen such that

α_{λ} ∈ 𝔭𝔡𝔱_{λ}, x_{λ} ∈ 𝔡^{−1}𝔱_{λ}^{−1}, and y_{λ} ∈ 𝔫.

Recall that for any Hilbert modular form f = (f_{λ}), the slash operator is defined by

f|_{A} := (f_{λ}|_{A_{λ}}).

Proposition 2.3. Fix k > 2. Suppose that 𝔭 is the only prime of F above p. Let η and ψ be narrow ray class characters of conductors 𝔞 and 𝔟, respectively, with 𝔞𝔟 = 𝔫𝔭 and ηψ = χω^{1−k}. Then

c_{λ}(0, E_{k}(η, ψ)|_{A}) = 0

unless 𝔞 = 𝔫, 𝔟 = 𝔭, and there exists a narrow ray class character ν of conductor 1 such that

η = χ · ν^{−1}, ψ = ω^{1−k} · ν.

In this case, we have

c_{λ}(0, E_{k}(η, ψ)|_{A}) = sgn(α_{λ})χ(α_{λ}𝔭^{−1}𝔱_{λ}^{−1}𝔡^{−1}) · \frac{ν(𝔱_{λ}/𝔫^{2})}{⟨N𝔫⟩^{k−1}} · \frac{2^{−n}}{τ(χ^{−1})} · L_{S}(χ^{−1}ω^{1−k}ν^{2}, 1 − k).

Proof. Write E_{k}(η, ψ) = (f_{λ}). For k > 2, the series (53) with s = 0 converges and yields f_{λ}. If for each a ∈ 𝔯 and b ∈ 𝔡^{−1}𝔟^{−1}𝔱_{λ}^{−1}𝔯 we define u and v by the matrix equation

(a b) A_{λ} = (u v) ,  (61)

then

f_{λ}|_{A_{λ}}(z) = C · \frac{N(𝔱_{λ})^{−k/2}}{N𝔟} ∑_{𝔯,a,b}N𝔯^{k} · \frac{sgn(a)^{q}η(a𝔯^{−1})}{(uz + v)^{k}} ∑_{c} sgn(c)^{r}ψ(c𝔱_{λ}^{−1}𝔯)e_{F} (−bc),  (62)

where the indices of the sums are as in (53). To compute the unnormalized constant term of (62), we send each z_{i} → ∞i; the only contribution arises when u = 0. This occurs when a = −bα_{λ} and v = b. By the choice of α_{λ}, the conditions bα_{λ} ∈ 𝔯 and b ∈ 𝔡^{−1}𝔟^{−1}𝔱_{λ}^{−1}𝔯 imply that b ∈ 𝔡^{−1}(𝔭, 𝔟)^{−1}𝔱_{λ}^{−1}𝔯. If 𝔭 ∤ 𝔟 then 𝔭 | 𝔞 and bα_{λ}𝔯^{−1} ⊂ 𝔭; thus η(a𝔯^{−1}) = 0, and the constant term of f_{λ}|_{A_{λ}} is 0. We therefore suppose 𝔭 | 𝔟, and obtain for the constant term

C · \frac{N(𝔱_{λ})^{−k/2} sgn(α_{λ})^{q}η(α_{λ}𝔭^{−1}𝔡^{−1}𝔱_{λ}^{−1})}{N𝔟} ∑_{𝔯∈Cl(F)}N𝔯^{k} ∑_{\begin{matrix}b∈𝔡^{−1}𝔭^{−1}𝔱_{λ}^{−1}𝔯/U \\ b≠0 \end{matrix}} \frac{sgn(−b)^{q}η(−b𝔡𝔭𝔱_{λ}𝔯^{−1})}{Nb^{k}}×

∑_{c∈𝔱_{λ}𝔯^{−1}/𝔱_{λ}𝔯^{−1}𝔟} sgn(c)^{r}ψ(c𝔱_{λ}^{−1}𝔯)e_{F} (−bc).

(63)

If 𝔟 ≠ 𝔭, then a prime 𝔮 ≠ 𝔭 divides the conductor of ψ since 𝔞𝔟 = 𝔫𝔭 and ηψ = χω^{1−k} is primitive away from 𝔭. For c ∈ 𝔱_{λ}𝔯^{−1}/𝔱_{λ}𝔯^{−1}𝔟 in a fixed residue class modulo 𝔱_{λ}𝔯^{−1}𝔭, the value e_{F} (−bc) is constant; the existence of 𝔮 dividing the conductor of ψ implies that the sum of sgn(c)^{r}ψ(c𝔱_{λ}^{−1}𝔯) over the c in each such class is 0.

Therefore, we are reduced to considering the situation 𝔟 = 𝔭, 𝔞 = 𝔫. Now η = χω^{1−k}ψ^{−1} has conductor indivisible by 𝔭. Therefore ψ = ω^{1−k}ν for some narrow ray class character ν of conductor 1, and η = χν^{−1}. When k ≢ 1 (mod (p − 1)), the character ψ is primitive, and we may apply (55). Then an argument following the proof of Proposition 2.1 using the change of variables 𝔣 = (b)𝔡𝔟𝔱_{λ}𝔯^{−1} and the functional equation for L(ηψ^{−1}, k) shows that the value of (63) is equal to

N(𝔱_{λ})^{k/2} · sgn(α_{λ})χ(α_{λ}𝔭^{−1}𝔱_{λ}^{−1}𝔡^{−1}) · \frac{ν(𝔱_{λ}/𝔫^{2})}{⟨N𝔫⟩^{k−1}} · \frac{2^{−n}}{τ(χ^{−1})} · L_{S}(χ^{−1}ω^{1−k}ν^{2}, 1 − k).  (64)

When k ≡ 1 (mod (p − 1)), the sum over c in (63) is easy to evaluate directly, and via the same technique one again arrives at (64).

For weight k = 1, the analogue of Proposition 2.3 requires a separate treatment.

Proposition 2.4. Suppose that 𝔭 is the only prime of F above p and that 𝔫 ≠ 1. Then

c_{λ}(0, E_{1}(1, χ)|_{A}) = sgn(α_{λ})χ(α_{λ}𝔭^{−1}𝔱_{λ}^{−1}𝔡^{−1})\frac{2^{−n}}{τ(χ^{−1})}L_{R}(χ^{−1}, 0).

Proof. Since χ is primitive of conductor 𝔫, the function f_{λ}|_{A_{λ}}(z) is given by the value of the function

C · \frac{N(𝔱_{λ})^{−1/2}}{N𝔫} |α_{λ}z + y_{λ}|^{2s} · ∑_{𝔯,a,b}N𝔯 · \frac{sgn(−b)χ^{−1}(−b𝔡𝔫𝔱_{λ}𝔯^{−1})τ(χ)}{(uz + v)|uz + v|^{2s}}.  (65)

at s = 0. Here the indices of the sum are as in (53) and (u, v) is defined in (61). The map (a, b) ↦ (u, v) induces a bijection between the sets

(𝔯 × 𝔡^{−1}𝔫^{−1}𝔱_{λ}^{−1}𝔯 − {(0, 0)}) / U

and

(𝔯𝔫^{−1} × 𝔡^{−1}𝔱_{λ}^{−1}𝔯 − {(0, 0)}) / U.

Furthermore, since b = −x_{λ}u + v and v𝔡𝔫𝔱_{λ}𝔯^{−1} ∈ 𝔫, we have

sgn(−b)χ^{−1}(−b𝔡𝔫𝔱_{λ}𝔯^{−1}) = sgn(x_{λ}u)χ^{−1}(x_{λ}u𝔡𝔫𝔱_{λ}𝔯^{−1})
= (−1)^{n} sgn(α_{λ})χ(α_{λ}𝔭^{−1}𝔡^{−1}𝔱_{λ}^{−1}) · sgn(u)χ^{−1}(u𝔫𝔯^{−1}),

where the last equality uses x_{λ}α_{λ} ≡ −1 (mod 𝔫) and χ(𝔭) = 1.

Let 𝔯′ = 𝔯𝔫^{−1} and 𝔱′_{λ} = 𝔱_{λ}𝔫^{−1} denote new representatives for Cl(F) and Cl^{+}(F), respectively. Thus the function in (65) may be written as the product of the constant

N𝔫^{−1/2} · (−1)^{n} sgn(α_{λ})χ(α_{λ}𝔭^{−1}𝔡^{−1}𝔱_{λ}^{−1}) · τ(χ)  (66)

and the function

C · N(𝔱′_{λ})^{−1/2}|α_{λ}z + y_{λ}|^{2s} · ∑_{𝔯′,u,v} N𝔯′ · \frac{sgn(u)χ^{−1}(u(𝔯′)^{−1})}{(uz + v)|uz + v|^{2s}},  (67)

where (u, v) ranges over representatives for

(𝔯′ × 𝔡^{−1}(𝔱′_{λ})^{−1}𝔯′ − {(0, 0)}) / U.

The expression in (67) at s = 0 is precisely the λ𝔫^{−1} component of E_{1}(χ^{−1}, 1) given the representatives 𝔱′_{λ} for Cl^{+}(F). Therefore, its unnormalized constant term is equal to

\frac{N(𝔱′_{λ})^{1/2}}{2^{n}} L(χ^{−1}, 0).  (68)

Using the fact that

τ(χ)τ(χ^{−1}) = (−1)^{n} N𝔫

and that N(𝔱′_{λ}) = N(𝔱_{λ})/ N𝔫, the expressions (66) and (68) combine to show that the normalized constant term of f_{λ}|_{A_{λ}} is

sgn(α_{λ})χ(α_{λ}𝔭^{−1}𝔡^{−1}𝔱_{λ}^{−1}) · \frac{2^{−n}}{τ(χ^{−1})}L_{R}(χ^{−1}, 0)

as desired.

When L(ψ, 1 − k) ≠ 0, we define the normalized Eisenstein series

G_{k}(1, ψ) := \frac{2^{n}}{L(ψ, 1 − k)}E_{k}(1, ψ),  (69)

each of whose constant coefficients c_{λ}(G_{k}(1, ψ), 0) is equal to 1. The Eisenstein series G_{k−1}(1, ω^{1−k}) plays an important role in our constructions. The following proposition computes its constant term at the cusp A∞, where A is given in (60).

Proposition 2.5. Suppose that 𝔭 is the only prime of F above p. For k > 2 we have

c_{λ}(0, G_{k−1}(1, ω^{1−k})|_{A}) = 1.

Proof. We must show that

c_{λ}(0, E_{k−1}(1, ω^{1−k})|_{A}) = \frac{1}{2^{n}} L(ω^{1−k}, 2 − k).

The argument used in the proof of Proposition 2.3 with (χ, 𝔫, χω^{1−k}) replaced by (1, 1, ω^{1−k}) gives the desired equality. The details are left to the reader.

2.3 A product of Eisenstein series

Recall that χ is a primitive character of conductor 𝔫 and that 𝔭 | p satisfies χ(𝔭) = 1. Set

S = {λ | 𝔫p∞}, R = S − {𝔭}.

Adjoining a finite prime 𝔮 ∤ 𝔫p to S multiplies L′_{S,p}(χω, 0) and L_{R}(χ, 0) by 1 − χ(𝔮), and leaves ℒ (χ) unchanged when χ(𝔮) ≠ 1, so it suffices to prove Theorem 2 with this minimal choice of S. Let p′ denote the ideal of F given by

p′ = ∏_{𝔮|p, 𝔮≠𝔭} 𝔮,

and define

𝔫_{R} = lcm(𝔫, p′), 𝔫_{S} = lcm(𝔫, p′𝔭).

We will denote by χ_{R} the character χ viewed as having modulus 𝔫_{R} (in particular, we have χ_{R}(𝔮) = 0 for all 𝔮 | p′). We always view the character χω^{1−k} as having modulus 𝔫_{S}, even when it has smaller conductor. In particular, χω^{0} ≠ χ, since χω^{0}(𝔭) = 0 while χ(𝔭) = 1. Occasionally, we will write χ_{S} instead of χω^{0}.

In this section, we consider the modular form

P_{k} := E_{1}(1, χ_{R}) · G_{k−1}(1, ω^{1−k}) ∈ M_{k}(𝔫_{S}, χω^{1−k}).  (70)

It is a fact that every modular form in M_{k}(𝔫_{S}, χω^{1−k}) can be written uniquely as a linear combination of a cusp form and the Eisenstein series E_{k}(η, ψ), where (η, ψ) run over the set J of pairs of (possibly imprimitive) characters of modulus m_{η} and m_{ψ} respectively, satisfying

m_{η}m_{ψ} = 𝔫_{S}, ηψ = χω^{1−k}.

(For k = 2, this is explained in [16, Prop. 1.5], and the case of general k ≥ 2 follows from the same argument.) For each (η, ψ) ∈ J, let a_{k}(η, ψ) ∈ E be the unique constant such that

P_{k} = ( \begin{matrix}A cusp \\ form \end{matrix} ) + ∑_{(η,ψ)∈J} a_{k}(η, ψ)E_{k}(η, ψ).  (71)

We will be particularly interested in the coefficients a_{k}(1, χω^{k−1}) and a_{k}(χ, ω^{k−1}) that appear in this linear combination. It turns out that these coefficients can be expressed as ratios of special values of the classical L-function L_{R}(χ, s) and of the p-adic L-function L_{S,p}(χω, s).

Proposition 2.6. For all integers k ≥ 2, we have

a_{k}(1, χω^{1−k}) = \frac{L_{R}(χ, 0)}{L_{S,p}(χω, 1 − k)} = −ℒ_{an}(χ, k)^{−1}.

Proof. The result follows by comparing the constant terms on both sides of (71). More precisely, since the constant terms of G_{k−1}(1, ω^{1−k}) are equal to 1, it follows that

c_{λ}(0, P_{k}) = c_{λ}(0, E_{1}(1, χ_{R})) = 2^{−n} · \begin{cases}L_{R}(χ, 0) & if 𝔫_{R} ≠ 1 \\ L_{R}(χ, 0) + χ^{−1}(𝔱_{λ})L_{R}(χ^{−1}, 0) & if 𝔫_{R} = 1. \end{cases}  (72)

On the other hand, the constant terms of E_{k}(η, ψ) for (η, ψ) ∈ J are equal to 0 except for the h characters η with m_{η} = 1. For such η, the h-tuple of constant terms c_{λ}(0, E_{k}(η, ψ)) is equal to

2^{−n}L_{S}(η^{−2}χω^{1−k}, 1 − k) × (η^{−1}(𝔱_{λ_{1}}), …, η^{−1}(𝔱_{λ_{h}})).

The linear independence of the characters of Cl^{+}(F) implies that these h-tuples are linearly independent. It follows that a_{k}(η, ψ) = 0 for all unramified η ∉ {1, χ}, and that

a_{k}(1, χω^{1−k}) = \frac{L_{R}(χ, 0)}{L_{S,p}(χω, 1 − k)},

as desired.

Consider now the special case where 𝔭 is the unique prime of F lying above p, so in particular χ_{R} = χ is a primitive character of conductor 𝔫_{R} = 𝔫. We will be interested in the coefficient a_{k}(χ, ω^{1−k}) of E_{k}(χ, ω^{1−k}) in the linear combination (71). If 𝔫_{R} = 𝔫 = 1, observe that the η = χ coefficient in the proof of Proposition 2.6 yields

a_{k}(χ, ω^{1−k}) = \frac{L_{R}(χ^{−1}, 0)}{L_{S,p}(χ^{−1}ω, 1 − k)}.  (73)

The following proposition shows that a similar formula holds more generally whenever 𝔭 is the unique prime of F lying above p.

Proposition 2.7. Suppose that 𝔭 is the unique prime above p in F. For an integer k > 2, we have

a_{k}(χ, ω^{1−k}) = \frac{L_{R}(χ^{−1}, 0)}{L_{S,p}(χ^{−1}ω, 1 − k)} · ⟨N𝔫⟩^{k−1} = −ℒ_{an}(χ^{−1}, k)^{−1} · ⟨N𝔫⟩^{k−1}.

Proof of Proposition 2.7. Since we have proven the result in the case 𝔫 = 1 in equation (73) already, we assume 𝔫 ≠ 1. The computation of a_{k}(χ, ω^{1−k}) will proceed by slashing equation (71) with the collection A = (A_{λ}) of matrices given in (60), and then comparing constant terms.

By Proposition 2.3, we know that c_{λ}(E_{k}(η, ψ)|_{A}) = 0 unless 𝔞 = 𝔫, 𝔟 = 𝔭, and the characters η and ψ are of the form

η = χ · ν^{−1},  ψ = ω^{1−k} · ν,  (74)

where ν is a narrow ray class character of conductor 1. In this remaining case, Proposition 2.3 asserts that

c_{λ}(0, E_{k}(η, ψ)|_{A}) = sgn(α_{λ})χ(α_{λ}𝔭^{−1}𝔱_{λ}^{−1}𝔡^{−1}) · \frac{ν(𝔱_{λ}/𝔫^{2})}{⟨N𝔫⟩^{k−1}} · \frac{2^{−n}}{τ(χ^{−1})} · L_{S}(χ^{−1}ω^{1−k}ν^{2}, 1 − k).

As in the proof of Proposition 2.6, we see that for the h unramified characters ν corresponding to the pairs (η, ψ) of the form (74), the h-tuples c_{λ}(E_{k}(η, ψ)|_{A})) are constant multiples of the vectors (ν(𝔱_{λ_{1}}), …, ν(𝔱_{λ_{h}})). By the linear independence of these h-tuples, the coefficients a_{k}(η, ψ) can once again be read off from the constant terms of P_{k}|_{A}. But it follows from Propositions 2.3 through 2.5 that

c_{λ}(P_{k}|_{A}) = sgn(α_{λ})χ(α_{λ}𝔭^{−1}𝔱_{λ}^{−1}𝔡^{−1}) · \frac{2^{−n}}{τ(χ^{−1})}L_{R}(χ^{−1}, 0).

Therefore we conclude that

a_{k}(χ, ω^{1−k}) = \frac{L_{R}(χ^{−1}, 0)}{L_{S,p}(χ^{−1}ω, 1 − k)} · ⟨N𝔫⟩^{k−1}

and that a_{k}(η, ψ) = 0 for (η, ψ) as in (74) when ν ≠ 1.

2.4 The ordinary projection

Recall that E is a finite extension of Q_{p} containing the values of the character χ, and that we have fixed an embedding \overline{Q}_{p} ⊂ C. Let M_{k}(𝔫_{S}, χω^{1−k}; O_{E}) denote the O_{E}-submodule of M_{k}(𝔫_{S}, χω^{1−k}) consisting of modular forms with all normalized Fourier coefficients lying in O_{E}. The ordinary projection operator

e = lim_{r→∞}(∏_{𝔮|p}U_{𝔮})^{r!}

gives rise to an idempotent in the endomorphism ring of M_{k}(𝔫_{S}, χω^{1−k}; O_{E}). We extend it to M_{k}(𝔫_{S}, χω^{1−k}; E) by E-linearity, using the fact that

M_{k}(𝔫_{S}, χω^{1−k}; E) = M_{k}(𝔫_{S}, χω^{1−k}; O_{E}) ⊗_{O_{E}} E.

The projection e preserves the cusp forms as well as the lines spanned by the various Eisenstein series E_{k}(η, ψ) with (η, ψ) ∈ J. More precisely, if 𝔮 | p, then 𝔮 | 𝔫_{S} = m_{η}m_{ψ}, so we have:

U_{𝔮}E_{k}(η, ψ) = (η(𝔮) + ψ(𝔮)(N𝔮)^{k−1})E_{k}(η, ψ)
= \{ \begin{matrix}η(𝔮) & if 𝔮 ∤ m_{η} \\ ψ(𝔮)(N𝔮)^{k−1} & if 𝔮 | m_{η} \end{matrix} \} × E_{k}(η, ψ).

It follows that for k ≥ 2:

eE_{k}(η, ψ) = \{ \begin{matrix}E_{k}(η, ψ) & if gcd(p, m_{η}) = 1, \\ 0 & otherwise. \end{matrix}  (75)

Let

P_{k}^{o} := eP_{k}  (76)

denote the projection of P_{k} to the ordinary subspace.

Proposition 2.8. The modular form P_{k}^{o} can be written

P_{k}^{o} = ( \begin{matrix}An ordinary \\ cusp form \end{matrix} ) + ∑_{(η,ψ)∈J^{o}} a_{k}(η, ψ)E_{k}(η, ψ),  (77)

where (η, ψ) ranges over the set J^{o} of pairs (η, ψ) ∈ J satisfying

m_{η}m_{ψ} = 𝔫_{S}, ηψ = χω^{1−k}, (p, m_{η}) = 1.

Note that for (η, ψ) = (1, χω^{1−k}) and (χ, ω^{1−k}), the coefficients a_{k}(η, ψ) are the ones given in Propositions 2.6 and 2.7, respectively (the latter only in the case where 𝔭 is the only prime of F above p).

2.5 Construction of a cusp form

We begin by defining a modular form H_{k} as in the introduction, by taking a suitable linear combination of the Eisensein series E_{k}(1, χω^{1−k}) and E_{k}(χ, ω^{1−k}) and the form P_{k}^{o}. It is useful to distinguish two cases.

Case 1. The set R contains a prime above p, i.e. F has a prime above p other than 𝔭. This case is the simplest of the two, but did not arise in the introduction since 𝔭 = (p) when F = Q. We set

H_{k} := u_{k}E_{k}(1, χω^{1−k}) + w_{k}P_{k}^{o},  (78)

where

u_{k} := \frac{1}{1 + ℒ_{an}(χ, k)} ,  w_{k} := \frac{ℒ_{an}(χ, k)}{1 + ℒ_{an}(χ, k)} .  (79)

Note that the vector (u_{k}, w_{k}) is proportional to the vector (a_{k}(1, χω^{1−k}), −1), but has been normalized to that u_{k} + w_{k} = 1.

Case 2. The set R contains no primes above p, i.e. 𝔭 is the unique prime of F above p. In this case the definition of H_{k} involves the Eisenstein series E(χ, ω^{k−1}) as well:

H_{k} := u_{k}E_{k}(1, χω^{1−k}) + v_{k}E_{k}(χ, ω^{1−k}) + w_{k}P_{k}^{o},  (80)

where

u_{k} := \frac{ℒ_{an}(χ, k)^{−1}}{c_{k}} , v_{k} := \frac{ℒ_{an}(χ^{−1}, k)^{−1}⟨N𝔫⟩^{k−1}}{c_{k}} , w_{k} := \frac{1}{c_{k}} ,  (81)

with

c_{k} := ℒ_{an}(χ, k)^{−1} + ℒ_{an}(χ^{−1}, k)^{−1}⟨N𝔫⟩^{k−1} + 1.

In this case, the vector (u_{k}, v_{k}, w_{k}) is proportional to (a_{k}(1, χω^{1−k}), a_{k}(χ, ω^{1−k}), −1) and is normalized so that u_{k} + v_{k} + w_{k} = 1.

It follows that the modular form H_{k} ∈ M_{k}(𝔫_{S}, χω^{1−k}) is a linear combination of a cusp form and the Eisenstein series E_{k}(η, ψ), where (η, ψ) ranges over the elements J^{o} with η ≠ 1 in case 1, and η ≠ 1, χ in case 2.

Lemma 2.9. We have the following:

1. For each (η, ψ) ∈ J^{o} with η ∉ {1, χ}, there is a Hecke operator T_{(η,ψ)} satisfying

T_{(η,ψ)}E_{k}(η, ψ) = 0, T_{(η,ψ)}E_{1}(1, χ_{S}) ≠ 0.

2. In case 1, there is a Hecke operator T_{(χ,ω^{1−k})} satisfying

T_{(χ,ω^{1−k})}E_{k}(χ, ω^{1−k}) = 0, T_{(χ,ω^{1−k})}E_{1}(1, χ_{S}) ≠ 0.

Proof. Let us write η ∼ η′ if the two characters η and η′ agree on all but finitely many primes, i.e., if they come from the same primitive character.

Let (η, ψ) be any element of J^{o}. If η ∼ 1, then ψ ∼ χω^{1−k} and the condition (η, ψ) ∈ J^{o} implies that ψ has conductor 𝔫_{S}. Therefore η has conductor 1 and η = 1. If η ∼ χ, then η is necessarily of modulus 𝔫 when (𝔫, p) = 1, and hence η = χ. Hence, in proving part 1 we may assume that η ≁ 1, χ. Let H_{(η,ψ)} be the abelian extension of F that is cut out by η and ψ by class field theory, and view η and ψ as characters of Gal(H_{(η,ψ)}/F). Let σ be any element of Gal(H_{(η,ψ)}/F) that does not belong to ker(η) ∪ ker(ηχ^{−1}). Such an element exists because a group cannot be the union of two of its proper subgroups. By the Chebotarev density theorem, there is a prime λ ∤ 𝔫_{S} of F whose Frobenius element is equal to σ. For such a prime we have

{η(λ), ψω^{k−1}(λ)} ≠ {1, χ(λ)}, η(λ) × ψω^{k−1}(λ) = 1 × χ(λ).

It follows that

η(λ) + ψω^{k−1}(λ) ≠ 1 + χ(λ).

Now, set

T_{(η,ψ)} := T_{λ} − η(λ) − ψ(λ)(Nλ)^{k−1}  (82)
= T_{λ} − η(λ) − ψω^{k−1}(λ)⟨Nλ⟩^{k−1}.

It is easy to see that the operator T_{(η,ψ)} has the desired properties, in light of the fact that

T_{λ}E_{1}(1, χ_{S}) = (1 + χ(λ))E_{1}(1, χ_{S}),
T_{λ}E_{k}(η, ψ) = (η(λ) + ψ(λ)(Nλ)^{k−1})E_{k}(η, ψ).

This proves part 1. Part 2 is proved by choosing a prime 𝔮 of F above p different from 𝔭, and setting

T_{(χ,ω^{1−k})} := U_{𝔮} − χ(𝔮).  (83)

Since U_{𝔮} acts with eigenvalue χ(𝔮) on E_{k}(χ, ω^{1−k}) and with eigenvalue 1 on E_{1}(1, χ_{S}), the result follows (recall that χ(𝔮) ≠ 1 by Lemma 1.1).

Corollary 2.10. The modular form

F_{k} := ( ∏_{(η,ψ)} T_{(η,ψ)}) H_{k},

where the product is taken over all (η, ψ) ∈ J^{o} with η ≠ 1 in case 1 and η ≠ 1, χ in case 2, belongs to S_{k}(𝔫_{S}, χω^{1−k}).

3 Λ-adic forms

3.1 Definitions

We now recall the basic definitions of Λ-adic forms. Let χ be any totally odd E-valued narrow ray class character of F modulo 𝔫. Recall from the introduction the Iwasawa algebra Λ ≃ O_{E}[[T]], which is topologically generated over O_{E} by functions of the form k ↦ a^{k} with a ∈ 1 + pZ_{p}. For each k ∈ Z_{p}, write

ν_{k} : Λ → O_{E}

for the evaluation map at k. We will write Λ_{(k)} ⊂ F_{Λ} for the localization of Λ at ker ν_{k}, and sometimes view ν_{k} as a homomorphism from Λ_{(k)} to E.

In Section 2, we recalled the definition of the space M_{k}(𝔫_{S}, χω^{1−k}) of classical Hilbert modular forms of weight k, level 𝔫_{S}, and character χω^{1−k}. To each such form f is associated its set of normalized Fourier coefficients c(𝔪, f), indexed by the nonzero integral ideals of F, and normalized constant terms c_{λ}(0, f), indexed by the narrow class group Cl^{+}(F).

Definition 3.1. A Λ-adic form ℱ of level 𝔫 and character χ is a collection of elements of Λ:

\begin{cases}c(𝔪, ℱ) for all nonzero integral ideals 𝔪 of F \\ c_{λ}(0, ℱ) for λ ∈ Cl^{+}(F), \end{cases}

such that for all but finitely many k ≥ 2, there is an element of M_{k}(𝔫_{S}, χω^{1−k}; E) with normalized Fourier coefficients ν_{k}(c(𝔪, ℱ)) for nonzero integral ideals 𝔪, and normalized constant terms ν_{k}(c_{λ}(0, ℱ)) for λ ∈ Cl^{+}(F).

The space of Λ-adic forms of level 𝔫 and character χ is denoted M(𝔫, χ). A Λ-adic form ℱ is said to be a cusp form if ν_{k}(ℱ) belongs to S_{k}(𝔫_{S}, χω^{1−k}) for almost all k ≥ 2, and the space of such cusp forms is denoted S(𝔫, χ). The action of the Hecke operators T_{λ} for λ ∤ 𝔫p and U_{𝔮} for 𝔮 | 𝔫p on M_{k}(𝔫_{S}, χω^{1−k}; E) lifts to a Λ-linear action of these operators on M(𝔫, χ) that preserves the subspace S(𝔫, χ) (see §1.2 of [17]).

Hida’s ordinary projection can still be defined by

e = lim_{r→∞}(∏_{𝔮|p}U_{𝔮})^{r!}.

The image of e is the submodule of ordinary forms:

M^{o}(𝔫, χ) := eM(𝔫, χ), S^{o}(𝔫, χ) := eS(𝔫, χ).

The spaces M^{o}(𝔫, χ) and S^{o}(𝔫, χ) are finitely generated torsion-free Λ-modules. Let T̃ and T denote the Λ-algebras of Hecke operators acting on M^{o}(𝔫, χ) and S^{o}(𝔫, χ) respectively.

By extension, we also call any element of M^{o}(𝔫, χ) ⊗_{Λ} F_{Λ} or S^{o}(𝔫, χ) ⊗_{Λ} F_{Λ} a Λ-adic modular form or Λ-adic cusp form, respectively. Note that if ℱ is such an element, then its weight k specialization ν_{k}(ℱ) is defined for almost all k ∈ Z_{p}.

3.2 Λ-adic Eisenstein series

The most basic examples of Λ-adic forms are the Eisenstein series. Let η and ψ be a pair of narrow ray class characters modulo m_{η} and m_{ψ} respectively, such that ηψ is totally odd.

Proposition 3.2. There exists a Λ-adic modular form

ℰ (η, ψ) ∈ M(m_{η}m_{ψ}, ηψ) ⊗ F_{Λ}

such that

ν_{k}(ℰ (η, ψ)) = E_{k}(η, ψω^{1−k}).

Proof. The Eisenstein series E_{k}(η, ψω^{1−k}) satisfy

c(𝔪, E_{k}(η, ψω^{1−k})) = ∑_{{𝔯|𝔪 \atop (𝔯,p)=1}} η (\frac{𝔪}{𝔯}) ψ(𝔯)⟨N𝔯⟩^{k−1}  (84)

and

c_{λ}(0, E_{k}(η, ψω^{1−k})) = \begin{cases}2^{−n}η^{−1}(𝔱_{λ})L_{p}(η^{−1}ψω, 1 − k) & if m_{η} = 1 \\ 0 & otherwise. \end{cases}  (85)

One sees by inspection that the expressions on the right of (84), viewed as functions of k, belong to the Iwasawa algebra Λ . That the function k ↦ L_{p}(η^{−1}ψω, 1 − k) belongs to F_{Λ} is equivalent to the known assertions about the existence and basic properties of the p-adic L-functions L_{p}(η^{−1}ψω, s) (see, for example, [18, (1.3)]).

We remark in passing that the idea of realizing the special values L_{p}(η^{−1}ψω, 1 − k) as the constant terms of Eisenstein series whose other Fourier coefficients lie in Λ is a key component of the construction of p-adic L-functions for totally real fields due to Serre and Deligne-Ribet.

Recall the normalized Eisenstein series G_{k−1}(1, ω^{1−k}) with constant term 1. Its p-adic interpolation is most conveniently described in terms of the space M′ of Λ-adic forms “with weights shifted by 1”. An element of M′ is a collection of elements c(𝔪, ℱ) and c_{λ}(0, ℱ) of Λ with the property that the specializations {ν_{k}(c(𝔪, ℱ)), ν_{k}(c_{λ}(0, ℱ)} are the normalized Fourier coefficients of an element in M_{k−1}(p, ω^{1−k}; E). There exists an element 𝒢 ∈ M′ ⊗F_{Λ} satisfying

ν_{k}(𝒢) = G_{k−1}(1, ω^{1−k}).

It is defined by the data:

c(𝔪, 𝒢) = 2^{n}ζ_{p}(F, 2 − k)^{−1} ∑_{{𝔯|𝔪 \atop (𝔯,p)=1}}(N𝔯)^{−1}⟨N𝔯⟩^{k−1}, c_{λ}(0, 𝒢) = 1,  (86)

where ζ_{p}(F, s) = L_{p}(1, s) is the p-adic zeta-function attached to F. We see by inspection that all of the coefficients of 𝒢 belong to F_{Λ}, and that ν_{k}(𝒢) = G_{k−1}. The specialization of 𝒢 at k = 1 plays a crucial role in our argument.

Proposition 3.3. Assume Leopoldt’s conjecture for F. The form 𝒢 belongs to M′ ⊗ Λ_{(1)}, and

ν_{1}(𝒢) = 1.

Proof. By a result of Colmez (cf. the main theorem of [2]), the non-vanishing of the p-adic regulator of F is known to imply that the p-adic zeta-function ζ_{p}(F, s) has a simple pole at s = 1. Therefore ζ_{p}(F, 2 − k)^{−1} is regular at k = 1 and vanishes at that point. The result follows.

3.3 A Λ-adic cusp form

In order to invoke Proposition 3.3, we assume Leopoldt’s conjecture for F in the rest of Section 3. In case 2, we also assume that assumption (11) holds.

Recall the classical modular forms P_{k}, P_{k}^{o} and H_{k} in M_{k}(𝔫_{S}, χω^{1−k}) that were defined in Sections 2.3 and 2.4 (see equations (70), (76), (78), and (80)), and the cusp form F_{k} ∈ S_{k}(𝔫_{S}, χω^{1−k}) that was constructed in Corollary 2.10 in Section 2.5.

Proposition 3.4. There exist Λ-adic forms 𝒫 ∈ M(𝔫, χ)⊗Λ_{(1)}, 𝒫^{o}, ℋ ∈ M^{o}(𝔫, χ)⊗Λ_{(1)}, and ℱ ∈ S^{o}(𝔫, χ) ⊗ Λ_{(1)} satisfying, for almost all k ≥ 2:

ν_{k}(𝒫) = P_{k}, ν_{k}(𝒫^{o}) = P_{k}^{o}, ν_{k}(ℋ ) = H_{k}, ν_{k}(ℱ) = F_{k}.  (87)

Proof. The forms 𝒫 and 𝒫^{o} are simply defined by setting

𝒫 = E_{1}(1, χ_{R})𝒢 , 𝒫^{o} = e𝒫.

To define the modular form ℋ , recall first the coefficients u_{k}, v_{k} and w_{k} that were defined in equations (79) and (81). In case 1, we set v_{k} = 0 so that in all cases we have u_{k} +v_{k} +w_{k} = 1. We observe that the coefficients u_{k}, v_{k} and w_{k} can be interpolated by elements u, v, w ∈ F_{Λ} satisfying u(k) = u_{k}, v(k) = v_{k}, and w(k) = w_{k} for almost all k ≥ 2, since the functions ℒ_{an}(χ, k)^{−1} and ℒ_{an}(χ^{−1}, k)^{−1} belong to F_{Λ}. A direct calculation then shows that u, v and w belong to Λ_{(1)}. More precisely,

u(1) = 1;  v(1) = 0;  w(1) = 0 in case 1;
u(1) = \frac{ℒ_{an}^{(t)}(χ^{−1},1)}{ℒ_{an}^{(t)}(χ,1)+ℒ_{an}^{(t)}(χ^{−1},1)} ; v(1) = \frac{ℒ_{an}^{(t)}(χ,1)}{ℒ_{an}^{(t)}(χ,1)+ℒ_{an}^{(t)}(χ^{−1},1)} ; w(1) = 0 in case 2, (88)

where t denotes the common order of vanishing of ℒ_{an}(χ, k) + ℒ_{an}(χ^{−1}, k) and ℒ_{an}(χ^{−1}, k) at k = 1. It is at this stage that we use the hypothesis (11) appearing in Theorem 2. In both case 1 and case 2, it can be seen that the coefficients u, v and w belong to Λ_{(1)} and that u is even invertible in this ring. In particular, the Λ-adic form

ℋ := uℰ (1, χ) + vℰ (χ, 1) + w𝒫^{o}

belongs to M^{o}(𝔫, χ) ⊗ Λ_{(1)} and satisfies ν_{k}(ℋ ) = H_{k}.

Finally, we note that the Hecke operators T_{(η,ψ)} of Lemma 2.9 that are defined in equations (82) and (83) can be viewed as elements of the ordinary Λ-adic Hecke algebra T̃. We may therefore set

ℱ = ∏_{(η,ψ)} T_{(η,ψ)}ℋ

to obtain the desired Λ-adic cusp form satisfying ν_{k}(ℱ) = F_{k}. Proposition 3.4 follows.

From now on, we will write u_{1}, v_{1}, and w_{1} for u(1), v(1), and w(1), respectively. We now analyze the weight 1 specialization of the Λ-adic forms of Proposition 3.4.

Lemma 3.5. We have

ν_{1}(𝒫) = ν_{1}(𝒫^{o}) = E_{1}(1, χ_{R}), ν_{1}(ℋ ) = E_{1}(1, χ_{S}),

and

ν_{1}(ℱ) = t · E_{1}(1, χ_{S}), for some t ∈ E^{×}.

Proof. Proposition 3.3 directly implies that ν_{1}(𝒫) = E_{1}(1, χ_{R}). (It is here that Leopoldt’s conjecture is required in our argument.) To study ν_{1}(𝒫^{o}), we note that the operator U_{𝔭} preserves the two-dimensional subspace W_{χ} of M_{1}(𝔫_{S}, χ) spanned by E_{1}(1, χ_{R}) and E_{1}(1, χ_{S}), and acts non-semisimply on this space, with generalised eigenvalue 1. More precisely, we have

U_{𝔭}E_{1}(1, χ_{S}) = E_{1}(1, χ_{S}),
U_{𝔭}E_{1}(1, χ_{R}) = E_{1}(1, χ_{R}) + E_{1}(1, χ_{S}).

The Hecke operators U_{𝔮} for 𝔮 | p and 𝔮 ≠ 𝔭 act on W_{χ} as the identity. It follows that e acts as the identity on W_{χ}, and in particular eE_{1}(1, χ_{R}) = E_{1}(1, χ_{R}). This shows that ν_{1}(𝒫^{o}) = E_{1}(1, χ_{R}). The calculation of ν_{1}(ℋ ) is a direct consequence of the fact that

ν_{1}(ℰ (1, χ)) = ν_{1}(ℰ (χ, 1)) = E_{1}(1, χ_{S}), u_{1} + v_{1} = 1, w_{1} = 0.

Finally, the conditions satisfied by the Hecke operators T_{(η,ψ)} in Lemma 2.9 and the fact that E_{1}(1, χ_{S}) is an eigenform show that ν_{1}(ℱ) is a non-zero multiple of E_{1}(1, χ_{S}). This concludes the proof.

3.4 The weight 1 + ε specialization

Let ν_{1+ε} : Λ_{(1)}⟶Ẽ = E[ε]/(ε^{2}) be the “weight 1 + ε specialization” that was discussed in the introduction (cf. equation (31)). We now consider the form H_{1+ε} := ν_{1+ε}(ℋ ), and calculate its Fourier coefficients in terms of the homomorphism κ_{cyc} defined in (16). The Fourier coefficients of the form ν_{1}(ℋ ) = E_{1}(1, χ_{S}) are given, for primes 𝔮 ≠ 𝔭, by the sum of the two characters 1 and χ. The key observation is that the Fourier coefficients of the infinitesimal lift H_{1+ε} are still given by the sum of two characters lifting 1 and χ.

Define these two characters ψ_{1}, ψ_{2} : G_{F} → Ẽ^{×} as follows. The character ψ_{1} is unramified outside p and satisfies

ψ_{1}(𝔮) = 1 + v_{1}κ_{cyc}(𝔮)ε  for all 𝔮 ∤ p.  (89)

The character ψ_{2} is unramified outside S and satisfies

ψ_{2}(𝔮) = χ(𝔮)(1 + u_{1}κ_{cyc}(𝔮)ε)  for all 𝔮 ∉ S.  (90)

As usual, we extend the definitions of ψ_{1} and ψ_{2} to totally multiplicative functions defined on the set of ideals of F by the following formulas, for prime 𝔮:

ψ_{1}(𝔮) = 1 if 𝔮 | p,
ψ_{2}(𝔮) = 0 if 𝔮 ∈ S.

Proposition 3.6. The Fourier coefficients of H_{1+ε} satisfy c(1, H_{1+ε}) = 1 and, for each prime ideal 𝔮 of F, we have

c(𝔮, H_{1+ε}) = ψ_{1}(𝔮) + ψ_{2}(𝔮) if 𝔮 ≠ 𝔭  (91)
c(𝔭, H_{1+ε}) = 1 + w′(1)ε.  (92)

Furthermore, H_{1+ε} is a simultaneous eigenform for the Hecke operators T_{𝔮} for 𝔮 ∉ S and U_{𝔮} for 𝔮 ∈ S with eigenvalues given by (91) and (92).

Proof. We will only consider case 2, i.e. the case when 𝔭 is the only prime of F above p. (The analysis of case 1, where v = 0, can be treated by a similar but simpler calculation.) Write ϕ_{1+ε}(u) = u_{1} + u′_{1}ε, and likewise for v and w. Let us also write E_{1+ε}(ψ, η) := ν_{1+ε}(ℰ (ψ, η)). Given an integral ideal 𝔪 of F, write 𝔪 = 𝔟𝔭^{t} with 𝔭 ∤ 𝔟. We note that

c(𝔪, E_{1+ε}(1, χ)) = ∑_{𝔯|𝔟}χ(𝔯)(1 + κ_{cyc}(𝔯)ε),  (93)
c(𝔪, E_{1+ε}(χ, 1)) = ∑_{𝔯|𝔟}χ(𝔯)(1 + κ_{cyc}(𝔟/𝔯)ε).  (94)

It follows from the equations w_{1} = 0 and ν_{1}(𝒫^{o}) = E_{1}(1, χ) that

H_{1+ε} = (u_{1}E_{1+ε}(1, χ) + v_{1}E_{1+ε}(χ, 1)) + (u′_{1}E_{1}(1, χω^{0}) + v′_{1}E_{1}(χ, ω^{0}) + w′_{1}E_{1}(1, χ)) ε
= (u_{1}E_{1+ε}(1, χ) + v_{1}E_{1+ε}(χ, 1)) + w′_{1} (E_{1}(1, χ) − E_{1}(1, χ_{S})) ε.  (95)

We have used the facts E_{1}(1, χω^{0}) = E_{1}(χ, ω^{0}) = E_{1}(1, χ_{S}) and u′_{1} + v′_{1} + w′_{1} = 0 in deriving (95). Using (93) and (94), the 𝔪th coefficient of the first term

u_{1}E_{1+ε}(1, χ) + v_{1}E_{1+ε}(χ, 1)

in (95) may be written:

u_{1} (∑_{𝔯|𝔟}χ(𝔯)(1 + κ_{cyc}(𝔯)ε)) + v_{1} (∑_{𝔯|𝔟}χ(𝔯)(1 + κ_{cyc}(𝔟/𝔯)ε))

= ∑_{𝔯|𝔟}χ(𝔯)(1 + u_{1}κ_{cyc}(𝔯)ε + v_{1}κ_{cyc}(𝔟/𝔯)ε)
= ∑_{𝔯|𝔟}ψ_{1}(𝔟/𝔯)ψ_{2}(𝔯).

Here we have used the fact that u_{1} + v_{1} = 1, by (88).

The remaining term

w′_{1} (E_{1}(1, χ) − E_{1}(1, χ_{S})) ε

in (95) has 𝔪th coefficient equal to

t · w′_{1} · (∑_{𝔯|𝔟}χ(𝔯))ε,

so we obtain

c(𝔪, H_{1+ε}) = ∑_{𝔯|𝔟}ψ_{1}(𝔟/𝔯)ψ_{2}(𝔯) + tw′_{1} ∑_{𝔯|𝔟}χ(𝔯)ε  (96)

= (∑_{𝔯|𝔟}ψ_{1}(𝔟/𝔯)ψ_{2}(𝔯)) × (1 + w′_{1}ε)^{t}.

Equations (91) and (92) follow immediately from (96). We leave to the reader the exercise, using (96) and the definition of the Hecke operators (see for instance equation (1.2.5) of [16]), that H_{1+ε} is indeed an eigenvector for the Hecke operators with the specified eigenvalues.

The eigenform H_{1+ε} determines a Λ_{(1)}-algebra homomorphism valued in the dual numbers:

ϕ_{1+ε} : T̃ ⊗ Λ_{(1)}⟶Ẽ = E[ε]/(ε^{2}),

defined by sending a Hecke operator to its eigenvalue on H_{1+ε}. The homomorphism ϕ_{1+ε} lifts the homomorphism

ϕ_{1} : T̃ ⊗ Λ_{(1)}⟶E

describing the eigenvalues of the Hecke operators on the Eisenstein series ν_{1}(ℋ ) = E_{1}(1, χ_{S}). Geometrically, ϕ_{1} corresponds to a point on the spectrum of T̃, and ϕ_{1+ϵ} describes a tangent vector at that point.

The fact that the cusp form ℱ is obtained by applying a Hecke operator to ℋ , and that ν_{1}(ℱ) ≠ 0 (cf. Lemma 3.5), shows that F_{1+ε} is (up to multiplication by an element of Ẽ^{×}) a normalized cuspidal eigenform with Fourier coefficients in Ẽ and with the same associated system of Hecke eigenvalues as H_{1+ε}. Hence ϕ_{1+ε} factors through the quotient T ⊗ Λ_{(1)} of T̃ ⊗ Λ_{(1)}.

Let T_{(1)} be the localization of T ⊗ Λ_{(1)} at ker ϕ_{1}. We will view ϕ_{1+ε} as a homomorphism on T_{(1)}, which is possible since ϕ_{1+ε} factors through the natural map T ⊗ Λ_{(1)}⟶T_{(1)}. The following theorem summarizes the main results of this section.

Theorem 3.7. Assume Leopoldt’s conjecture for F, and in case 2 assume further that (11) holds. Let ψ_{1}, ψ_{2} be as in (89) and (90). There exists a Λ_{(1)}-algebra homomorphism

ϕ_{1+ε} : T_{(1)}⟶E[ε]/(ε^{2})

satisfying

ϕ_{1+ε}(T_{𝔮}) = ψ_{1}(𝔮) + ψ_{2}(𝔮) if 𝔮 ∉ S;  (97)
ϕ_{1+ε}(U_{𝔮}) = ψ_{1}(𝔮) if 𝔮 ∈ R,  (98)
ϕ_{1+ε}(U_{𝔭}) = 1 + u_{1}ℒ_{an}(χ)ε.  (99)

Proof. Equations (97) and (98) simply restate (91). Equation (99) is a consequence of (92) and the identity (cf. equations (79) and (81))

w(k) = u(k)ℒ_{an}(χ, k),

which implies that w′_{1} = u_{1}ℒ_{an}(χ).

4 Galois representations

In this section, we parlay the homomorphism ϕ_{1+ε} of Theorem 3.7 into the construction of a cohomology class κ ∈ H_{𝔭}^{1}(F, E(χ^{−1})) satisfying

res_{𝔭} κ = −ℒ_{an}(χ)κ_{nr} + κ_{cyc}.  (100)

This class will be extracted from the Galois representations attached to the eigenforms in S^{o}(𝔫, χ).

4.1 Representations attached to ordinary eigenforms

Recall that T_{(1)} is the localization of T⊗Λ_{(1)} at ker ϕ_{1}. Let F_{(1)} be the total ring of fractions of the local ring T_{(1)}. It is isomorphic to a product of fields:

F_{(1)} = F_{1} × ⋯ × F_{t},

where each F_{j} is a finite extension of F_{Λ}. Fix a factor F = F_{j} in this decomposition. We will write T_{𝔮} and U_{𝔮} to denote the images of the corresponding Hecke operators in F under the natural projection F_{(1)}⟶F.

Recall the cyclotomic character ϵ_{cyc} : G_{F} ⟶Z_{p}^{×} that was introduced in equation (13) of the introduction, and satisfies

ϵ_{cyc}(Frob_{𝔮}) = N𝔮, for all 𝔮 ∤ p.

The Λ-adic cyclotomic character

\underline{ϵ}_{cyc} : G_{F} ⟶Λ^{×}

is given by

\underline{ϵ}_{cyc}(Frob_{𝔮})(k) = ⟨N𝔮⟩^{k−1}.

The following key result is proved in [17] (cf. Theorems 2 and 4 of loc. cit.).

Theorem 4.1. There is a continuous irreducible Galois representation

ρ : G_{F} → GL_{2}(F)

satisfying

1. The representation ρ is unramified at all primes 𝔮 ∉ S, and the characteristic polyno-

mial of ρ(Frob_{𝔮}) is

x^{2} − T_{𝔮}x + χ(𝔮)⟨N𝔮⟩^{k−1}.  (101)

In particular, det ρ = χ\underline{ϵ}_{cyc}.

2. The representation ρ is odd, i.e., the image of any complex conjugation in G_{F} under ρ has characteristic polynomial x^{2} − 1.

3. For each 𝔮 | p, let G_{𝔮} ⊂ G_{S} denote a decomposition group at 𝔮. Then

ρ|_{G_{𝔮}} ≅ ( \begin{matrix}χ\underline{ϵ}_{cyc}η_{𝔮}^{−1} & ∗ \\ 0 & η_{𝔮} \end{matrix} )  (102)

where η_{𝔮} is the unramified character of G_{𝔮} satisfying

η_{𝔮}(Frob_{𝔮}) = U_{𝔮}.  (103)

Let V ≅ F^{2} be the representation space attached to ρ. For any choice of F-basis of V there is an associated matrix representation

ρ(σ) = ( \begin{matrix}a(σ) & b(σ) \\ c(σ) & d(σ) \end{matrix} ) , a, b, c, d : G_{F} ⟶F.  (104)

Let R denote the image of T_{(1)} under the projection F_{(1)}⟶F. The ring R is a quotient of T_{(1)} and hence is also a local ring having E as residue field. For x ∈ R, write \overline{x} ∈ E for its reduction modulo the maximal ideal 𝔪 of R. Fix a choice of a complex conjugation δ ∈ G_{F}. Since the representation ρ is totally odd, we may choose an F-basis of V such that

ρ(δ) = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) .

Assume once and for all that the basis of V has been chosen in this way.

Theorem 4.2. The representation ρ satisfies the following properties.

1. For all σ ∈ G_{F}, the entries a(σ) and d(σ) belong to R^{×}, and

\overline{a}(σ) = 1, \overline{d}(σ) = χ(σ).

2. The matrix entry b does not vanish identically on the decomposition group G_{𝔭} at 𝔭.

Proof. The traces of ρ(Frob_{𝔮}) are Hecke operators, and hence lie in R. By the Chebotarev density theorem, the same is true of trace(ρ(σ)) for all σ ∈ G_{F}. The identities

a(σ) = \frac{1}{2} (trace ρ(σ) + trace ρ(σδ))  (105)

and

d(σ) = \frac{1}{2} (trace ρ(σ) − trace ρ(σδ))  (106)

imply that a(σ), d(σ) ∈ R. Furthermore, (101) combined with

ϕ_{1}(T_{𝔮}) = 1 + χ(𝔮) for 𝔮 ∉ S

implies that \overline{trace ρ(σ)} = 1 + χ. Hence part 1 follows from (105) and (106). We now turn to the proof of part 2. Let B denote the R-submodule of F generated by the b(σ) as σ ranges over G_{F}. A standard compactness argument shows that B is a finitely generated R-module. Since d(σ) belongs to R^{×}, the function K(σ) := b(σ)/d(σ) also takes its values in B. Let

\overline{K} : G_{F} ⟶\overline{B}, \overline{B} := B/𝔪B

denote its mod 𝔪 reduction. A direct calculation using the multiplicativity of the representation ρ of equation (104) reveals that the function \overline{K} is a continuous one-cocycle in Z^{1}(G_{F} , \overline{B}(χ^{−1})). Furthermore, part 3 of Theorem 4.1 gives, for each 𝔮 | p, a change of basis matrix

( \begin{matrix}A_{𝔮} & B_{𝔮} \\ C_{𝔮} & D_{𝔮} \end{matrix} ) ∈ GL_{2}(F)  (107)

satisfying

( \begin{matrix}a(σ) & b(σ) \\ c(σ) & d(σ) \end{matrix} ) ( \begin{matrix}A_{𝔮} & B_{𝔮} \\ C_{𝔮} & D_{𝔮} \end{matrix} ) = ( \begin{matrix}A_{𝔮} & B_{𝔮} \\ C_{𝔮} & D_{𝔮} \end{matrix} ) ( \begin{matrix}χ\underline{ϵ}_{cyc}η_{𝔮}^{−1}(σ) & ∗ \\ 0 & η_{𝔮}(σ) \end{matrix} )  (108)

for all σ ∈ G_{𝔮}. Comparing the upper left-hand entries in (108) gives the identity

C_{𝔮}b(σ) = A_{𝔮} [χ\underline{ϵ}_{cyc}η_{𝔮}^{−1}(σ) − a(σ)]  (109)

for all σ ∈ G_{𝔮}. Suppose now that 𝔮 ≠ 𝔭. Since χ ≠ 1 on G_{𝔮}, there is a σ_{𝔮} ∈ G_{𝔮} for which χ(σ_{𝔮}) ≠ 1 and hence for which the expression in square braces on the right of (109), evaluated at σ = σ_{𝔮}, is a unit in R. It follows that C_{𝔮} ≠ 0 and that the ratio w_{𝔮} := A_{𝔮}/C_{𝔮} belongs to B. Equation (109) can be rewritten as the equation in B:

K(σ) = \frac{[χ\underline{ϵ}_{cyc}η_{𝔮}^{−1}(σ) − a(σ)]}{d(σ)}w_{𝔮}, for all σ ∈ G_{𝔮}.  (110)

In order to reduce this equation modulo 𝔪, we note that

ν_{1}(\underline{ϵ}_{cyc}) = 1, ϕ_{1}(η_{𝔮}(Frob_{𝔮})) = c(𝔮, E_{1}(1, χ_{S})) = 1,

and hence ϕ_{1}(η_{𝔮}(σ)) = 1 for all σ ∈ G_{𝔮}. Part 1 of Theorem 4.2 then yields

\overline{K}(σ) = (1 − χ^{−1}(σ))\overline{w}_{𝔮} for all σ ∈ G_{𝔮}.  (111)

It follows that \overline{K} is locally trivial at all 𝔮 | p′, and hence yields a class [\overline{K}] ∈ H_{𝔭}^{1}(F, \overline{B}(χ^{−1})). Suppose that b vanishes on G_{𝔭}. The same is then true of the cocycle \overline{K}, which is therefore trivial at 𝔭. Hence by Lemma 1.5, the cohomology class [\overline{K}] is trivial. We may therefore write \overline{K} as a coboundary:

\overline{K}(σ) = (1 − χ^{−1}(σ))θ,  (112)

for some θ ∈ \overline{B} and all σ ∈ G_{F}. Evaluating (112) at σ = δ shows that in fact θ = 0, and hence \overline{K} = 0 as a cocycle on G_{F}. But the image of K generates the R-module B by definition, and hence \overline{B} = 0. Since B is a finitely generated R-module, Nakayama’s Lemma implies that B = 0, contradicting the irreducibility of ρ. This contradiction proves part 2.

4.2 Construction of a cocycle

In the previous section, we constructed, for each quotient F_{j} of F_{(1)}, a specific Galois representation

ρ_{j} : G_{F} ⟶GL_{2}(F_{j}).

The product of these representations yields a Galois representation

ρ_{(1)} : G_{F} ⟶GL_{2}(F_{(1)}).

To lighten the notations we will write ρ instead of ρ_{(1)} and use the symbols a, b, c and d to denote its matrix entries, with the understanding that in this section, these are now elements of F_{(1)} = F_{1} × ⋯ × F_{t}.

We now suppose we are given an Λ_{(1)}-algebra homomorphism ϕ_{1+ε} : T_{(1)} → Ẽ lifting the homomorphism ϕ_{1} corresponding to the modular form E_{1}(1, χ_{S}). We suppose that ϕ_{1+ε} is given by the sum of two characters ψ_{1}, ψ_{2} : G_{F} → Ẽ^{×} lifting 1, χ, respectively, as in (97) and (98).

Lemma 4.3. The Galois representation ρ satisfies the following properties:

1. For the chosen complex conjugation δ,

ρ(δ) = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) .

2. For all σ ∈ G_{F}, the entries a(σ) and d(σ) belong to T_{(1)}^{×}, and

ϕ_{1+ε} ∘ a = ψ_{1},  (113)
ϕ_{1+ε} ∘ d = ψ_{2}.  (114)

Proof. Part 1 follows directly from the definition of ρ. The proof of part 2 is identical to the proof of part 1 of Theorem 4.2, with R replaced by T_{(1)}.

The following is the main result of this section.

Theorem 4.4. Let ϕ_{1+ε} be as above. Write ψ_{1} = 1 + ψ′_{1}ε, and define a′_{𝔭} ∈ E by

ϕ_{1+ε}(U_{𝔭}) = 1 + a′_{𝔭}ε.

There exists a cohomology class κ ∈ H_{𝔭}^{1}(F, E(χ^{−1})) satisfying

res_{𝔭} κ = −a′_{𝔭}κ_{nr} + κ_{cyc} − res_{𝔭}(ψ′_{1}).

Proof. Consider the change of basis matrices

( \begin{matrix}A_{𝔮} & B_{𝔮} \\ C_{𝔮} & D_{𝔮} \end{matrix} )

with entries in F_{(1)} satisfying (108). Equation (109) with 𝔮 = 𝔭, combined with part 2 of Theorem 4.2, shows that the projection of A_{𝔭} to each factor F_{j} is non-zero. Hence A_{𝔭} belongs to F_{(1)}^{×}. Let

b̃(σ) := \frac{C_{𝔭}}{A_{𝔭}} × b(σ),

and write B for the T_{(1)}-submodule of F_{(1)} generated by the b̃(σ) as σ ranges over G_{F}. Since d(σ) belongs to T_{(1)}^{×}, the function K(σ) := b̃(σ)/d(σ) takes values in B as well. For each 𝔮 | p′, the same reasoning as in the proof of Theorem 4.2 shows that the element C_{𝔮} belongs to F_{(1)}^{×} (i.e. it is not a zero divisor), and that the element x_{𝔮} := \frac{C_{𝔭}}{A_{𝔭}} × \frac{A_{𝔮}}{C_{𝔮}} belongs to B. The restriction of K to G_{𝔮} satisfies

K(σ) = \frac{[χ\underline{ϵ}_{cyc}η_{𝔮}^{−1}(σ) − a(σ)]}{d(σ)}x_{𝔮}, for all σ ∈ G_{𝔮}.  (115)

At the prime 𝔭, we have

K(σ) = \frac{[\underline{ϵ}_{cyc}(σ)η_{𝔭}^{−1}(σ) − a(σ)]}{d(σ)}, for all σ ∈ G_{𝔭}.  (116)

We now claim that the module B is contained in 𝔪 ⊂ T_{(1)} ⊂ F_{(1)}. To see this, let B^{♯} = (B + 𝔪)/𝔪, and let K^{♯} : G_{F} ⟶B^{♯} be the function obtained by composing K with the natural surjection B ⟶ B^{♯}. The function

\overline{K}^{♯} : G_{F} ⟶\overline{B}^{♯} := B^{♯}/𝔪B^{♯}

obtained from K^{♯} by reduction modulo the maximal ideal 𝔪 is a one-cocycle yielding a class

[\overline{K}^{♯}] ∈ H^{1}(F, \overline{B}^{♯}(χ^{−1})).

Equation (116) shows that

K^{♯}(σ) = 0, and hence \overline{K}^{♯}(σ) = 0,  for all σ ∈ G_{𝔭},  (117)

whereas equation (115) shows that [\overline{K}^{♯}] is locally trivial at all 𝔮 | p′. It follows from Lemma 1.3 that [\overline{K}^{♯}] = 0, and arguing with δ as in the proof of Theorem 4.2, we find that in fact \overline{K}^{♯} = 0 as a function. Since the K^{♯}(σ) generate the finitely generated T_{(1)}-module B^{♯}, we have B^{♯} = 𝔪B^{♯} and hence B^{♯} = 0 by Nakayama’s lemma. Thus B ⊂ 𝔪, as desired.

We can now complete the proof of Theorem 4.4. Since the function K takes values in 𝔪 ⊂ T_{(1)}, we may consider its composition with ϕ_{1+ε} and define a function κ : G_{F} → E by

ϕ_{1+ε} ∘ K(σ) = κ(σ)ε.

The function κ is a one-cocycle representing a class [κ] ∈ H^{1}(F, E(χ^{−1})). Since x_{𝔮} ∈ B ⊂ 𝔪,

(115) implies that [κ] is locally trivial at all 𝔮 | p′ and therefore belongs to H_{𝔭}^{1}(F, E(χ^{−1})). Finally, combining (116) with the equations

ϕ_{1+ε} ∘ \underline{ϵ}_{cyc} = 1 + κ_{cyc}ε,
ϕ_{1+ε} ∘ η_{𝔭} = 1 + a′_{𝔭}κ_{nr}ε,
ϕ_{1+ε} ∘ a = 1 + ψ′_{1}ε,
ϕ_{1+ε} ∘ d = χ + ψ′_{2}ε,

(the latter two arising in (113) and (114)) yields

κ(σ) = −a′_{𝔭} · κ_{nr}(σ) + κ_{cyc}(σ) − ψ′_{1}(σ)

for all σ ∈ G_{𝔭}. Theorem 4.4 follows.

Proof of Theorem 2. Applying Theorem 4.4 to the Λ_{(1)}-algebra homomorphism ϕ_{1+ε} constructed in Theorem 3.7, we find that ψ′_{1} = v_{1}κ_{cyc} and a′_{𝔭} = u_{1}ℒ_{an}(χ), and hence obtain a class κ ∈ H_{𝔭}^{1}(F, E(χ^{−1})) satisfying

res_{𝔭} κ = −u_{1}ℒ_{an}(χ)κ_{nr} + (1 − v_{1})κ_{cyc}
= u_{1}(−ℒ_{an}(χ)κ_{nr} + κ_{cyc}).

Recall that u_{1} ≠ 0 by (88). Replacing κ by κ/u_{1}, we obtain (100). This proves part 1 of Theorem 2, and part 2 for the character χ. The proof of part 2 for the character χ^{−1} follows from the observation that, when there is a unique prime of F above p, the roles of the matrix entries b and c can be interchanged in Theorems 4.2 and 4.4. This produces a cohomology class κ ∈ H_{𝔭}^{1}(F, E(χ)) satisfying

res_{𝔭} κ = −ℒ_{an}(χ^{−1})κ_{nr} + κ_{cyc}.

Theorem 2 follows.

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