Adjoint L-Values and Primes of Congruence for Hilbert Modular Forms^{∗}^{†}

Eknath Ghate

October 17, 2002

Abstract

Let f be a primitive Hilbert modular cusp form of arbitrary level and parallel weight k, defined over a totally real number field F. We define a finite set of primes S that depends on the weight and level of f, the field F, and the torsion in the boundary cohomology groups of the Borel-Serre compactification of the underlying Hilbert-Blumenthal variety. We show that, outside S, any prime that divides the algebraic part of the value at s = 1 of the adjoint L-function of f is a congruence prime for f. In special cases we identify the ‘boundary primes’ in terms of expressions of the form N_{F/ℚ}(ϵ^{k−1} −1), where ϵ is a totally positive unit of F.

1 Introduction

The purpose of this paper is to establish a relationship between the primes of congruence of a Hilbert modular cusp form f, and the primes dividing the algebraic part of the value of its adjoint L-function L(s, Ad(f)) at s = 1.

Such matters were first investigated by Doi and Hida [3], Hida [10], [11], and Ribet [19], in the elliptic modular case. Let us state their results. Let f ∈ S_{k}(Γ_{0}(N), ψ) be a normalized newform that is a common eigenform of all the Hecke operators. Let L(s, Ad(f)) denote the imprimitive adjoint Lfunction attached to f, let Γ(s, Ad(f)) denote the associated Γ-factor, and let W (f) be the complex constant that arises in the functional equation of the standard L-function attached to f (see the text for exact definitions). Then, in [10], Hida proved:

∗2000 Mathematics Subject Classification. Primary: 11F33, Secondary: 11F41, 11F67 †Keywords: Hilbert modular forms, congruence primes, adjoint L-values.

Theorem 1 There is a large number field K, such that the following holds: For each prime ℘ | p in K with p > k − 2 and p | 6̸N, there exists periods Ω(f, +) and Ω(f, −) ∈ ℂ^{×}/O_{(℘)}^{×}, such that if

℘ | \frac{W (f) Γ(1, Ad(f)) L(1, Ad(f))}{Ω(f, +)Ω(f, −)},  (1)

then ℘ is a congruence prime for f.^{1}

Here ‘large’ means that K contains the number fields generated by the Fourier coefficients of all the normalized cusp forms in S_{k}(Γ_{0}(N), ψ) which are eigenforms for all the Hecke operators T_{q}, for q | N̸. Also O_{(℘)} is a valuation ring of K, and it is implicit in the statement of the Theorem that the quotient in (1) is an element of O_{(℘)}. Finally, we remind the reader that ℘ is said to be a congruence prime for f = ∑_{m=1}^{∞} a(m, f) q^{m} if there exists another normalized cusp form g = ∑_{m=1}^{∞} b(m, f) q^{m} ∈ S_{k}(Γ_{0}(N), ψ), which is an eigenform for all the Hecke operators T_{q}, such that

a(m, f) ≡ b(m, f) mod ℘,

for all m.

A converse to Theorem 1 was established by Hida [11] and Ribet [19]. They proved:

Theorem 2 Let f be as in Theorem 1, and let ℘ | p be a prime of K, with p > k − 2 and p | 6̸N. If ℘ is a congruence prime for f, then

℘ | \frac{W (f) Γ(1, Ad(f)) L(1, Ad(f))}{Ω(f, +)Ω(f, −)}.

Call a prime ℘|p of K ordinary for f if ℘ | a̸(p, f). In fact in [11] Hida established Theorem 2 under the additional hypothesis that ℘ is ordinary for f. This hypothesis was later removed by Ribet in [19].

1The statement here is slightly different from what is proved in [10]; it is the version exposed, for instance, in Hida’s recent book [17]. Also, in [12], the theorem was extended to include all primes ℘ | p with p ≥ 5 and ℘ ordinary for f.

In this paper we generalize Theorem 1 to the Hilbert modular situation. Let F be a totally real field, and let f be a Hilbert modular cusp form, defined over F, of parallel weight. Assume that f is a normalized newform and a common eigenform of all the Hecke operators. We will define a finite set of primes S in a large number field K and show that:

Theorem 5^{2} For each prime ℘ of K with ℘ ∉ S, there are periods Ω(f, ±ϵ) ∈ ℂ^{×}/O_{(℘)}^{×}, such that if

℘ | \frac{W (f) Γ(1, Ad(f)) L(1, Ad(f))}{Ω(f, ϵ)Ω(f, −ϵ)},

then ℘ is a congruence prime for f.

Though this theorem is rather general, in any specific situation it would be useful to have as much information about the set S as possible. To this end, we shall show that the set S may be decomposed as follows:

S = S_{weight} ∪ S_{level} ∪ S_{elliptic} ∪ S_{F} ∪ S_{invariant} ∪ S_{∂}.

The sets S_{weight}, S_{level} and S_{elliptic} are excluded from Theorem 5 purely for technical reasons (explained in the text). When F = ℚ, these sets consist of the primes p ≤ k − 2, p | N, and p = 2, 3 in Theorem 1. Over more general totally real fields F, we must also exclude the set S_{F}, consisting of the primes dividing the class number and the discriminant of F. Additionally, the set S_{invariant} must be excluded when [F : ℚ] is even and the weight of f is (2, 2, …, 2).

But perhaps the most interesting set of primes in S that must be excluded are the primes in S_{∂}. We will refer to these primes informally as the ‘boundary primes’. They are, more precisely, the primes of torsion in the cohomology groups with integral coefficients of the boundary of the Borel-Serre compactification of the underlying Hilbert-Blumenthal variety. In general these primes are somewhat difficult to describe explicitly, but in the case F = ℚ this is not so: they are the primes p ≤ k − 2 and p dividing N (and so have already been accounted for by the primes in S_{weight} and S_{level} in this case).

The boundary primes arise as obstructions to a key result which we need in order to establish Theorem 5. This result, Theorem 3 in Section 3.3, is a duality theorem for the parabolic cohomology groups of the underlying Hilbert-Blumenthal variety, with integral coefficients. With it in hand, the proof of Theorem 5 follows quickly. In fact our proof essentially imitates the method of proof of Theorem 1 outlined in [10]. We have also benefited from some ideas in [23], where Urban established an analog of Theorem 1 for cusp forms over imaginary quadratic fields.

2The numbering used in the Introduction coincides with the numbering used in the main body of the paper.

The question remains as to how explicitly one can describe the set S_{∂} of boundary primes. As mentioned above, the case F = ℚ is easy, and, in any case, was already treated by Hida in [10]. We are further able to treat some special cases (see Section 3.4). For instance, if [F : ℚ] = 2, F has strict class number 1, and if f has level 1, and parallel weight (2, 2), then we are able to explicitly determine the boundary primes in terms of a generator of the totally positive unit group of F (see Proposition 4). Moreover, when the weight is bigger than 2, we are able to draw on a result of Hida (Theorem 3.12 in [15]) which allows us to describe explicitly the ordinary boundary primes (see Corollary 1). As a result we can obtain the more precise result:

Corollary 2 Let F be a real quadratic field of strict class number 1. Let ϵ_{1} denote a generator of the group of totally positive units of F, and let D_{F} denote the discriminant of F. Let f be a normalized newform, and a common eigenform of the Hecke operators, of level 1 and weight (k, k). Then there exits a large number field K such that the following holds:

If ℘ | p is a prime of K such that p > k − 2, p | 3̸0 · D_{F} · N_{F/ℚ}(ϵ_{1}^{k−1} − 1) and

\begin{cases}℘ ∉ S_{invariant}, & if k = 2, \\ ℘ is ordinary for f, & if k > 2, \end{cases}

then there exist periods Ω(f, ±ϵ) ∈ ℂ^{×}/O_{(℘)}^{×} such that if

℘ | \frac{Γ(1, Ad(f)) L(1, Ad(f))}{Ω(f, ϵ)Ω(f, −ϵ)},

then ℘ is a congruence prime for f.

A similar explicit result for level 1, weight (2, 2, …, 2) cusp forms over totally real fields of strict class number 1 and odd degree is also derived:

Corollary 4 Let F be a totally real field of strict class number 1, odd degree d, and discriminant D_{F}. Let ϵ_{1}, ϵ_{2}, …, ϵ_{d−1} be a basis of the group of totally positive units of F, and let U_{F} denote the product of the primes which divide each N_{F/ℚ}(ϵ_{j} − 1), for j = 1, 2, … d − 1. Let E_{F} denote the product of all the primes p such that the maximal totally real subfield of the p^{th} cyclotomic field is contained in F.

Let f be a normalized newform, and a common eigenform of the Hecke operators, of level 1 and weight (2, 2, …, 2). Then there exits a large number field K such that the following holds:

If ℘ | p is a prime of K such that p | E̸_{F} ·D_{F} ·U_{F}, then there exist periods Ω(f, ±ϵ) ∈ ℂ^{×}/O_{(℘)}^{×} such that if

℘ | \frac{Γ(1, Ad(f)) L(1, Ad(f))}{Ω(f, ϵ)Ω(f, −ϵ)},

then ℘ is a congruence prime for f.

In a sequel to this paper [8] we will investigate the analog of Theorem 2 in the Hilbert modular situation. That is, under certain conditions, we will establish that almost all (ordinary) congruence primes are captured by the adjoint L-value. To do this we use a freeness criterion invented by Taylor and Wiles (see Fujiwara [6] and Diamond [2]) to show that under certain conditions suitable localizations of the above mentioned parabolic cohomology groups are free as Hecke modules. The converse divisibility then follows quickly.

Finally I should mention that one motivation for generalizing Theorems 1 and 2 to the Hilbert modular situation, is that such theorems would provide a possible approach to tackling some conjectures of Doi, Hida and Ishii [4] on twisted elliptic modular adjoint L-values and primes of congruence between base-change and non-base-change Hilbert modular cusp forms (see [7] for more details in the real quadratic case).

Acknowledgments. It is a pleasure to thank H. Hida for many useful discussions and insights regarding the contents of this paper. I thank C. Khare for some helpful remarks, and for the interest he has shown towards this project. I also thank A. Nair, L. Saper, D. Prasad, E. Urban, and T. N. Venkataramana for some useful discussions. Work on this paper began during a visit to the Université de Paris-Nord in the spring of 1998, and I would like to thank the organizers of the Indo-French exchange program for making this visit possible. Lastly, I am indebted to the referee for a careful reading of a preliminary version of this paper, and for numerous comments that have much improved it.

2 Preliminaries

Useful general references for some of the material in this section are the papers [13], [16] and the book [24].

2.1 Some notation

Fix a totally real field F of degree d. Let O_{F} denote the ring of integers of F. Let I_{F} denote the set of embeddings of F into ℝ.

Let G = Res_{F/ℚ}GL_{2}. Then let G_{f} = G(𝔸_{f}) denote the finite part of G(𝔸), where 𝔸 = 𝔸_{f} × ℝ denotes the ring of adeles over ℚ. Let G_{∞} = G(ℝ), and let G_{∞+} denote the set of those elements in G_{∞} which have positive determinant at each place σ ∈ I_{F}. Let G(ℚ)_{+} = G(ℚ) ∩ G_{∞+}.

Let K_{f} be an open, compact subgroup of G(𝔸_{f}), let K_{∞} = ∏ O_{2}(ℝ) denote the standard maximal compact subgroup of G(ℝ), and let K_{∞+} = ∏ SO_{2}(ℝ) ⊂ G_{∞+} denote the connected component of K_{∞} containing the identity element. Let K = K_{f}K_{∞}.

Let Z denote the center of G, and let Z_{∞} denote the center of G_{∞}.

2.2 Hilbert-Blumenthal varieties

We keep the notation of the previous section. Set

Y (K) = G(ℚ)\G(𝔸)/K_{f}K_{∞+}Z_{∞}.

Then Y (K) is the set of complex points of a quasi-projective variety of dimension d defined over ℚ, which, following standard usage, will be referred to as a Hilbert-Blumenthal variety.

By the strong approximation theorem one may find t_{i} ∈ G(𝔸) of the form t_{i} = ( \begin{matrix}a_{i} & 0 \\ 0 & 1 \end{matrix} ) with (a_{i})_{∞} = 1, where (a_{i})_{∞} is the infinite part of the idele a_{i}, such that

G(𝔸) = ∐_{i=1}^{h} G(ℚ) t_{i} K_{f} G_{∞+}.

Note that when det(K_{f}) = Ô_{F}^{×}, with Ô_{F} = ∏_{𝔭} O_{F_{𝔭}}, then h = |Cl_{F}^{+}| is just the strict class number of F, where

Cl_{F}^{+} := F^{×}\𝔸_{F}^{×}/ det(K_{f})F_{∞+}^{×}

is the strict class group of F.

Now set Γ_{i} = G(ℚ)_{+} ∩t_{i}K_{f} G_{∞_{+}}t_{i}^{−1}. Let Z = G_{∞+}/K_{∞+}Z_{∞} denote the dfold product of the Poincaré upper half plane H on which a discrete subgroup Γ of G(ℚ)_{+} acts in the usual way via fractional linear transformations. Let z_{0} = (\sqrt{−1}, \sqrt{−1}, …, \sqrt{−1}) denote the standard ‘base point’ in Z. Then the map γt_{i}u_{f}g_{∞} ↦ g_{∞}(z_{0}), for γ ∈ G(ℚ), u_{f} ∈ K_{f}, and g_{∞} ∈ G_{∞+}, induces a decomposition

Y (K) = ∐_{i=1}^{h} Γ_{i}\Z  (2)

of Y (K) into its connected components.

Let 𝔑 ⊂ O_{F} be an ideal. From Section 2.4 onwards we will restrict to the case when K_{f} = K_{0}(𝔑) is the level 𝔑 compact-open subgroup of G(𝔸_{f}):

K_{0}(𝔑) = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ GL_{2}(Ô_{F}) | c ≡ 0 (mod 𝔑Ô_{F})\}.

In this case we will write Y (𝔑) instead of Y (K), and Γ_{i}(𝔑) instead of Γ_{i}. We have: Γ_{i}(𝔑)

= \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ GL_{2}(F) | a ∈ O_{F} , b ∈ 𝔄_{i}, c ∈ 𝔑𝔄_{i}^{−1}, d ∈ O_{F} , ad − bc ∈ O_{F +}^{×}\},

where 𝔄_{i} = a_{i}O_{F} is the fractional ideal generated by a_{i}, and O_{F +}^{×} is the ring of totally positive units in F.

Note that without loss of generality we may make the following two assumptions on the a_{i} for i = 1, …, h. Firstly we may assume that (a_{i})_{𝔑} = 1, where (a_{i})_{𝔑} is the 𝔑-part of the adele a_{i}, and secondly, that the fractional ideals 𝔄_{i} = a_{i}O_{F} are all integral ideals. We keep these assumptions for the rest of the paper.

2.3 Compactifications of Y (K)

There are various compactifications of Y (K). One such, which we will not have occasion to use in this paper, is the Bailey-Borel-Satake compactification. It is constructed by adding finitely many points, called cusps, to Y (K).

For instance, the number of cusps required to compactify SL_{2}(O_{F})\Z is equal to the class number of F (see Proposition 1.1 of [24]).

Another compactification of Y (K) is the Borel-Serre compactification [1]. It will play an important role in this paper, and, following [15] and [23], we describe it briefly now.

Let X = (ℝ_{+}^{×})^{I_{F}} × ℝ^{I_{F}}. Thus if (y, x) ∈ X then x = (x_{σ}) and y = (y_{σ}) are indexed by the embeddings σ of F into ℝ. Let us set N(y) = ∏_{σ∈I_{F}} y_{σ}. Consider for each t ∈ ℝ with t > 0, the set

X_{t} = {(y, x) ∈ X | N(y) = t}.

Note that for each t > 0 one has X_{t} =^{∼} X_{1} where the identification is induced by the map (y, x) ↦ (\frac{y}{t^{1/d}} , x). Thus X =^{∼} X_{1} × ℝ_{+}^{×}. We now add a copy of X_{1} to X at t = ∞ to get a space X^{∗}.

Let B_{∞} denote the standard Borel of upper triangular matrices in G(ℚ). We first construct a space Z^{∗}, by adding a boundary to Z, whose connected components are in bijection with the set

G(ℚ)/B_{∞} = ℙ^{1}(F)

of Borel subgroups of G(ℚ). Indeed, if B = αB_{∞}α^{−1} is a Borel subgroup of G(ℚ), where α ∈ G(ℚ)_{+} may be taken to be totally positive, then the Iwasawa decomposition corresponding to B_{∞} of α^{−1}g, with g ∈ G_{∞+}, permits us to identify X with Z via the map ι_{α}, where

ι_{α} : (y, x) ↦ α (\begin{matrix}y & x \\ 0 & 1 \end{matrix}) K_{∞_{+}}Z_{∞}.  (3)

The compactification X^{∗} of X described above allows us to adjoin, via ι_{α}, for each α (and hence B) as above, a boundary, say ∂_{B}, to Z. We set ∂Z^{∗} = ∐_{B} ∂_{B}.

If Γ is an arithmetic subgroup of G(ℚ)_{+}, then one may check that Γ acts properly discontinuously on Z^{∗}. Moreover, Γ\Z^{∗} is a compactification of Γ\Z, whose boundary, ∂(Γ\Z^{∗}), is a finite union of spaces Γ_{B}\∂_{B} (where Γ_{B} = Γ ∩ B) indexed by the Γ-conjugacy classes of Borel subgroups, B, of G(ℚ). Now the Borel-Serre compactification of Y (K) is

Y (K)^{∗} := ∐_{i=1}^{h} Γ_{i}\Z^{∗}.

Note that for K_{f} sufficiently deep, this is a smooth manifold, with smooth boundary:

∂(Y (K)^{∗}) = ∐_{i=1}^{h} ∂(Γ_{i}\Z^{∗}).

2.4 Hilbert modular forms

Let n = ∑_{σ∈I_{F}} n_{σ} σ ∈ ℤ[I_{F}] with each n_{σ} ≥ 0. Let t = ∑_{σ∈I_{F}} σ ∈ ℤ[I_{F}] and let k = n + 2t. The data n and k will represent the weights of cusp forms described below. For simplicity, we will assume in this paper that the weights are ‘parallel’, namely that n_{σ} = n_{τ} (and so k_{σ} = k_{τ}), for any σ, τ ∈ I_{F}. Thus n and k will also sometimes be used to denote this common value.

Fix an ideal 𝔑 of O_{F}. We assume from now on that K_{f} = K_{0}(𝔑) (see Section 2.2).

Let 𝔸_{F} = 𝔸 ⊗_{ℚ} F denote the ring of adeles of F. Identify the center Z(𝔸) of G(𝔸) with the idele group 𝔸_{F}^{×}. Similarly identify Z(ℚ) with F^{×}. Let χ : F^{×}\𝔸_{F}^{×} → ℂ be a fixed Hecke character whose conductor divides 𝔑 and whose infinity type is −n.

The character χ, restricted to Ô_{F}^{×}, induces a finite order character χ_{𝔑} of Ô_{F}^{×}/(1 + 𝔑Ô_{F}). We also write χ_{𝔑} for the character of K_{0}(𝔑) defined as follows: for u_{f} = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ K_{0}(𝔑), let

χ_{𝔑}(u_{f}) = χ_{𝔑}(d_{𝔑}) = ∏_{𝔭 | 𝔑} χ(d_{𝔭}).

A Hilbert modular cusp form of weight n, level 𝔑, central character χ, and holomorphy type J ⊂ I_{F} is a function f : G(𝔸) → ℂ satisfying the following properties:

• f(γg) = f(g) for all γ ∈ G(ℚ)

• f(zg) = χ(z)f(g) for all z ∈ Z(𝔸)

• f(gu_{f}u_{∞}) = χ_{𝔑}(u_{f})f(g) exp(2πi(∑_{σ∈J} k_{σ}θ_{σ} − ∑_{σ∉J} k_{σ}θ_{σ})) where

u_{∞} = (\begin{matrix}cos(2πθ_{σ}) & sin(2πθ_{σ}) \\ − sin(2πθ_{σ}) & cos(2πθ_{σ}) \end{matrix}) ∈ K_{∞+},

and where u_{f} ∈ K_{0}(𝔑)

• D_{σ}f = (\frac{n_{σ}^{2}}{2} + n_{σ})f, where D_{σ} is the Casimir operator at σ ∈ I_{F}

• f has vanishing constant terms, that is, for each g ∈ G(𝔸),

∫_{U(ℚ)\U(𝔸)} f(ug) du = 0

where U is the unipotent radical of B_{∞}, the standard Borel subgroup of upper triangular matrices in G.

Let us denote the space of such forms by S_{k,J}(𝔑, χ).

2.5 Fourier expansions

Recall that any f ∈ S_{k,J}(𝔑, χ) has a Fourier expansion which we describe now. Define W : (ℝ_{+}^{×})^{I_{F}} → ℂ by

W (y) = ∏_{σ∈I_{F}} exp(−2πy_{σ}),

where y = (y_{σ})_{σ∈I_{F}}. Let | |_{F} denote the modulus character on 𝔸_{F}^{×}. Let ϑ be an idele which generates the different of F/ℚ. Let e_{F} : F \𝔸_{F} → ℂ denote the usual additive character of 𝔸_{F}. Then

f( \begin{matrix}y & x \\ 0 & 1 \end{matrix} ) = |y|_{F} ∑_{ξ∈F^{×}, [ξ]=J} c(ξyϑ, f)W (ξy_{∞})e_{F} (ξx),

where we have written [ξ] for {σ ∈ I_{F} |ξ^{σ} > 0}. Here, the Fourier coefficients c(g, f) only depend on the fractional ideal generated by the finite part g_{f} of the idele g. Thus if g_{f}O_{F} = 𝔪, then we may write c(𝔪, f) without ambiguity. Moreover, one may check that c(𝔪, f) vanishes outside the set of integral ideals.

There is a natural integral structure on the space of cusp forms coming from Fourier expansion. For a subring A of ℂ containing the values of the finite order character χ_{𝔑}, let us denote by S_{k,J}(𝔑, χ, A) the set of elements of S_{k,J}(𝔑, χ) whose Fourier coefficients lie in A.

2.6 Hecke algebras

Let, momentarily, K_{f} be an arbitrary open compact subgroup of G_{f} = G(𝔸_{f}). Let Δ ⊂ G_{f} be a semigroup, such that Δ ⊃ K_{f}. Let K_{f}\Δ/K_{f} denote the space of double cosets of K_{f} in Δ. Define the Hecke algebra h(Δ, K_{f}) = ℤ[K_{f}\Δ/K_{f}] to be the free abelian group with basis the set of double cosets of K_{f} in Δ. For a double coset K_{f}gK_{f} ∈ K_{f}\Δ/K_{f}, we denote the corresponding basis element by [K_{f}gK_{f}]. The algebra structure on h(Δ, K_{f}) is given by ‘convolution’:

[K_{f}g_{1}K_{f}] · [K_{f}g_{2}K_{f}] = ∑_{m=3}^{M} v_{m}[K_{f}g_{m}K_{f}],

where

K_{f}g_{1}K_{f}g_{2}K_{f} = ∐_{m=3}^{M} K_{f}g_{m}K_{f},

and

v_{m} = |\frac{K_{f}g_{1}K_{f} ∩ g_{m}K_{f}g_{2}^{−1}K_{f}}{K_{f}} | ∈ ℕ

for m = 3, …, M.

Now let

R̂(𝔑) = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ M_{2}(Ô_{F}) | d_{𝔭} ∈ O_{F}^{×}_{𝔭}, c_{𝔭} ∈ 𝔑O_{F}_{𝔭}, for all 𝔭 | 𝔑\},

and let

Δ_{0}(𝔑) = R̂(𝔑) ∩ G(𝔸_{f}).

Let us describe the Hecke algebra h(Δ_{0}(𝔑), K_{0}(𝔑)). For each ideal 𝔪 ⊂ O_{F} let T_{𝔪} = ∑_{g}[K_{0}(𝔑)gK_{0}(𝔑)], where the sum is taken over all distinct double cosets with g ∈ Δ_{0}(𝔑) with det(g)O_{F} = 𝔪. Note that

T_{𝔭} = [K_{0}(𝔑) (\begin{matrix}ϖ_{𝔭} & 0 \\ 0 & 1 \end{matrix}) K_{0}(𝔑)] .

Also, for each 𝔭 with 𝔭 | 𝔑̸ let

S_{𝔭} = [K_{0}(𝔑) (\begin{matrix}ϖ_{𝔭} & 0 \\ 0 & ϖ_{𝔭} \end{matrix}) K_{0}(𝔑)] .

The following Proposition is well known:

Proposition 1 h(Δ_{0}(𝔑), K_{0}(𝔑)) is a commutative algebra with 1. It is generated over ℤ by the Hecke operators T_{𝔭} as 𝔭 varies through all prime ideals of O_{F}, and by S_{𝔭} for 𝔭 | 𝔑̸.

The Hecke algebra h(Δ_{0}(𝔑), K_{0}(𝔑)) acts on the space S_{k,J}(𝔑, χ, A) of cusp forms over A. Let us recall this action. Extend the definition of the character χ_{𝔑} from K_{0}(𝔑) to Δ_{0}(𝔑): for g ∈ Δ_{0}(𝔑), set

χ_{𝔑}(g) = ∏_{𝔭 | 𝔑} χ(d_{𝔭}).

Then, for f : G(𝔸) → ℂ ∈ S_{k,J}(𝔑, χ) and [K_{0}(𝔑)gK_{0}(𝔑)] ∈ h(Δ_{0}(𝔑), K_{0}(𝔑)), define the linear operator [K_{0}(𝔑)gK_{0}(𝔑)] : S_{k,J}(𝔑, χ, A) → S_{k,J}(𝔑, χ, A) by

f|[K_{0}(𝔑)gK_{0}(𝔑)](x) = ∑_{i} χ−1_{𝔑}(g_{i})f(xg_{i}),

where we have decomposed K_{0}(𝔑)gK_{0}(𝔑) = ∐_{i} g_{i}K_{0}(𝔑).

A cusp form f ∈ S_{k,I_{F}} (𝔑, χ) is said to be primitive if it is a newform, it is normalized (c(O_{F} , f) = 1), and it is an eigenfunction of all the Hecke operators (i.e. T_{𝔪}f = c(𝔪, f)f, for all T_{𝔪}). Let B denote the set of normalized cusp forms that are eigenforms for all the Hecke operators T_{𝔪} (thus B contains oldforms as well as newforms).

There is also an action of the group of ‘complex conjugations’ C := K_{∞}/K_{∞+} = {±1}^{I_{F}} on ∑_{J} S_{k,J}(𝔑, χ). In fact, for any two subsets J and J′ of I_{F}, there is a unique element c(J, J′) ∈ C which maps S_{k,J}(𝔑, χ) to S_{k,J′}(𝔑, χ) as an isomorphism of Hecke modules.

Let h_{k}(𝔑, χ, A) be the subalgebra of End_{A}(S_{k,J}(𝔑, χ, A)) generated by T_{𝔭} (for all 𝔭) and S_{𝔭} (for 𝔭 | 𝔑̸). Note that this algebra is independent of J.

2.7 Relation with classical cusp forms

Each f ∈ S_{k,J}(𝔑, χ) may be realized as a tuple of functions (f_{i}) on Z satisfying the usual transformation property with respect to the congruence subgroups Γ_{i}(𝔑) above. To see this let

f′_{i} : G(ℝ) → ℂ

be defined via

f′_{i}(g_{∞}) = f(t_{i}g_{∞}),

for g_{∞} ∈ G(ℝ). Since the map d ↦ d_{𝔑} = ∏_{𝔭 | 𝔑} d induces an isomorphism (O_{F} /𝔑O_{F})^{×} →^{∼} Ô_{F}^{×}/(1 + 𝔑Ô_{F}), we may think of χ_{𝔑} as as a character of (O_{F} /𝔑O_{F})^{×}. Then we see that

• f′_{i}(γg) = χ_{𝔑}(d)^{−1}f(g) for all γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ_{i}(𝔑)

• f′_{i}(zg) = χ_{∞}(z)f(g) for all z ∈ Z_{∞}

• f′_{i}(gu_{∞}) = f′_{i}(g) exp(2πi(∑_{σ∈J} k_{σ}θ_{σ} − ∑_{σ∉J} k_{σ}θ_{σ})) where

u_{∞} = (\begin{matrix}cos(2πθ_{σ}) & sin(2πθ_{σ}) \\ − sin(2πθ_{σ}) & cos(2πθ_{σ}) \end{matrix}) ∈ K_{∞+}

• D_{σ}f′_{i} = (\frac{n_{σ}^{2}}{2} + n_{σ})f′_{i}, where D_{σ} is the Casimir operator at σ ∈ I_{F}

• f′_{i} satisfies a cuspidal condition.

Let us denote the space of cusp forms f′_{i} satisfying the above conditions by S_{k,J}(Γ_{i}(𝔑), χ−1_{𝔑}). Then (cf. [16], Equation 3.5) the map f ↦ (f′_{i})_{i=1}^{h} induces an isomorphism

⨁_{χ} S_{k,J}(𝔑, χ) →^{∼} ⨁_{i=1}^{h} S_{k,J}(Γ_{i}(𝔑), χ−1_{𝔑}),  (4)

where the left hand sum runs over all central characters χ whose restriction to Ô_{F}^{×} is χ_{𝔑}, and whose infinity type is −n.

We now realize the functions in S_{k,J}(Γ_{i}(𝔑), χ−1_{𝔑}) as functions on Z. For

γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ GL_{2}(ℝ) and τ ∈ ℂ let

j(γ, τ) = cτ + d

denote the standard automorphy factor. Now let α = (α_{σ})_{σ∈I_{F}} ∈ G_{∞+} and z = (z_{σ})_{σ∈I_{F}} ∈ Z, and set

j_{k,J}(α, z) = ∏_{σ∈J} j(α_{σ}, z_{σ})^{k_{σ}} ∏_{σ∉J} j(α_{σ}, \overline{z_{σ}})^{k_{σ}}.

Finally define f_{i} : Z → ℂ by

f_{i}(z) = f′_{i}(g_{∞}) j_{k,J}(g_{∞}, z_{0}),

where g_{∞} ∈ G_{∞+} with det(g_{∞}) = 1 is chosen such that

g_{∞}z_{0} = z.  (5)

One may check that f_{i} is well-defined, and that for all γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ_{i}(𝔑),

f_{i}(γz) = χ_{𝔑}(d)^{−1}j_{k,J}(γ, z)f_{i}(z).

Moreover, the fact that f is an eigenfunction of the Casimir operators, along with the fact that f transforms under K_{∞+} in the manner prescribed above, ensures that each f_{i} is holomorphic at σ ∈ J and anti-holomorphic at σ ∉ J (cf. [16], page 460).

Finally, the Fourier expansion of f induces the usual Fourier expansion of classical Hilbert modular cusp forms. Choosing the idele g = \frac{1}{\sqrt{y}}( \begin{matrix}y & x \\ 0 & 1 \end{matrix} ) in (5) above, one may easily compute that each f_{i} has Fourier expansion

f_{i}(z) = |a_{i}|_{F} ∑_{ξ∈F^{×}, [ξ]=J} c(ξa_{i}ϑ, f)W (ξy_{∞})e_{F} (ξx_{∞}),

When J = I_{F} and a_{i} = 1 this reduces to the usual Fourier expansion of holomorphic Hilbert modular cusp forms:

f_{1}(z) = ∑_{ξ∈ϑ^{−1}, ξ>>0} c(ξϑ, f) exp(2πiTr_{F/ℚ}(ξz)).

3 A duality theorem

We now describe the relevant cohomology groups that will play a role in this paper, and discuss duality theorems for these cohomology groups in the middle dimension.

3.1 Cohomology

Let K denote a fixed large number field which contains all the conjugates of

• the totally real field F,

• the number fields generated by the Fourier coefficients of cusp forms f ∈ B (see Section 2.6), and, • the number field generated by the values of the finite order character

χ_{𝔑}.

Fix once and for all an embedding of K into ℂ, via which ℂ can be regarded as a K-algebra. Let O_{K} denote the ring of integers of K, and let Ô_{K} = ∏_{℘} O_{℘} denote the product of all the completions of O_{K}.

Let L(n, A) denote the module of all polynomials with coefficients in A, in the variables {X_{σ}, Y_{σ} | for all σ ∈ I_{F} }, which are homogeneous of degree n_{σ} in each pair X_{σ}, Y_{σ}.

Note that each σ ∈ I_{F} induces a map Ô_{F} → Ô_{K}. Now set L_{i}(n, O_{K}) = t_{i}L(n, Ô_{K}) ∩ L(n, K), where t_{i}L(n, Ô_{K}) denotes all the polynomials of the form P ((X_{σ}, Y_{σ})^{t}(t_{i}^{σ})^{ι}), with P (X_{σ}, Y_{σ}) ∈ L(n, Ô_{K}), and ι is the involution g ↦ det(g)g^{−1}. For an arbitrary O_{K}-algebra A, let L_{i}(n, A) = L_{i}(n, O_{K})⊗_{O_{K}} A.

For us, A will denote either O_{K}, O_{(℘)} (a valuation ring of O_{K} at the prime ℘), O_{℘} (a completion of O_{K} at the prime ℘), K, K_{℘} (a completion of K at the prime ℘), K_{℘}/O_{℘} or ℂ.

Let Γ_{i}(𝔑) be one of the discrete subgroup of G(ℚ)_{+} as in Section 2.2. We make L_{i}(n, A) into a Γ_{i}(𝔑)-module via an action on the factor L_{i}(n, O_{K}) as follows. For γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ_{i}(𝔑) and P (X_{σ}, Y_{σ}) ∈ L_{i}(n, O_{K}), set

(γ · P)(X_{σ}, Y_{σ}) = χ_{𝔑}(d)^{−1}P ((X_{σ}, Y_{σ})^{t}(γ^{σ})^{−1}).  (6)

We will write L_{i}(n, χ, A) for the module L_{i}(n, A) to emphasize the Γ_{i}(𝔑)- action, although note that the action depends only on χ_{𝔑}. If ϵ ∈ O_{F}^{×}, then χ(ϵ) = 1 implies that χ_{𝔑}(ϵ)N_{F/ℚ}(ϵ)^{−n} = 1. This shows that Γ_{i}(𝔑) ∩ F^{×} acts trivially on L_{i}(n, χ^{±1}, A) and so it is in fact a module for \overline{Γ_{i}(𝔑)} = (Γ_{i}(𝔑) ∩ F^{×})\Γ_{i}(𝔑). If one replaces χ_{𝔑}(d)^{−1} in (6) by χ_{𝔑}(a)^{−1}, then one obtains another \overline{Γ_{i}(𝔑)} module which we shall denote by L_{i}(n, χ^{−1}, A).

Assume momentarily that \overline{Γ_{i}(𝔑)} is torsion-free. Give L_{i}(n, χ^{±1}, A) the discrete topology and let L_{i}(n, χ^{±1}, A) denote the sheaf of continuous (therefore locally constant) sections of the covering

Γ_{i}(𝔑)\(Z × L_{i}(n, χ^{±1}, A)) → Γ_{i}(𝔑)\Z.

In this paper we will consider the ordinary sheaf cohomology groups H^{q}(Γ_{i}(𝔑)\Z, L_{i}(n, χ^{±1}, A)), as well as the compactly supported sheaf cohomology groups H_{c}^{q}(Γ_{i}(𝔑)\Z, L_{i}(n, χ^{±1}, A)). Denote the image of the compactly supported cohomology in the ordinary cohomology under the natural map

H_{c}^{q}(Γ_{i}(𝔑)\Z, L_{i}(n, χ^{±1}, A)) →^{ι} H^{q}(Γ_{i}(𝔑)\Z, L_{i}(n, χ^{±1}, A)),

by H_{p}^{q}(Γ_{i}(𝔑)\Z, L_{i}(n, χ^{±1}, A)). This image will be referred to as the parabolic cohomology group, and it will play an important role.

All these cohomology groups may be defined, with some restrictions on the coefficients, when \overline{Γ_{i}(𝔑)} is not necessarily torsion free. We discuss this now. For each i = 1, …, h, choose a normal finite index subgroup \overline{Γ_{i}} ⊂ \overline{Γ_{i}(𝔑)}, with \overline{Γ_{i}} = (Γ_{i}∩F^{×})\Γ_{i} torsion free. For each such Γ_{i}, one has the cohomology groups

H_{?}^{i}(Γ_{i}\Z, L_{i}(n, χ^{±1}, A)),

for ? = ∅, c, or p. as above. Now assume that

[\overline{Γ_{i}(𝔑)} : \overline{Γ_{i}}] is invertible in A,  (7)

and set

H_{?}^{i}(Γ_{i}(𝔑)\Z, L_{i}(n, χ^{±1}, A)) = H_{?}^{i} (Γ_{i}\Z, L_{i}(n, χ^{±1}, A))\overline{Γ_{i}(𝔑)} ,  (8)

for ? = ∅, c, or p. It may be checked that, under (7), the definition (8) is independent of the choice of Γ_{i}.

Note that (7) always holds if A is a field. If A is not a field, then one may still proceed as follows. Let S_{cyclotomic} be the finite set of primes of ℚ defined by

S_{cyclotomic} = {p : ℚ(μ_{p})^{+} ⊂ F },  (9)

where ℚ(μ_{p})^{+} is the maximal totally real subfield of the p-th cyclotomic field ℚ(μ_{p}). We have:

Lemma 1 Let Γ ⊂ GL_{2}(F) be an arithmetic subgroup.

1. Say γ ∈ Γ is a torsion element of prime order ℓ. Then ℓ ∈ S_{cyclotomic}.

2. For each ℓ ∉ S_{cyclotomic}, there is torsion free finite index normal subgroup Γ′ ⊂ Γ such that [Γ : Γ′] is prime to ℓ.

Proof. Since an arbitrary arithmetic subgroup is conjugate, via an element of GL_{2}(F), to a subgroup of GL_{2}(O_{F}), we may assume that Γ ⊂ GL_{2}(O_{F}). For a prime 𝔭 of F, choose a uniformizer π generating O_{𝔭} = (π), the completion of O_{F} at 𝔭. Then for each n ≥ 1 one has:

1 → Γ(π^{n}) → GL_{2}(O_{𝔭}) → GL_{2}(O_{𝔭}/π^{n}) → 1,

where Γ(π^{n}) is the kernel of the reduction mod π^{n} map in the exact sequence above. When n is sufficiently large, it is a fact that Γ(π^{n}) is torsion free, and an easy computation shows that

[GL_{2}(O_{𝔭}) : Γ(π^{n})] = q^{4(n−1)}(q^{2} − q)(q^{2} − 1) =: f(q), say,

where q = p^{f} is the order of the residue field O_{𝔭}/π. In particular, for n sufficiently large, Γ′ = Γ ∩ Γ(π^{n}) is a torsion free normal subgroup of Γ of index dividing f(q).

Now suppose that γ ∈ Γ has order ℓ. Then for n sufficiently large the image of γ in Γ/Γ′ is still an element of order ℓ, so that ℓ | f(q). Thus

N_{F/Q}(𝔭) = q ≡ ±1 mod ℓ,  (10)

for all 𝔭 with 𝔭 | ℓ̸. Now consider the diagram:

F ↪ F (μ_{ℓ})^{+} ↪ F (μ_{ℓ})
| | |
ℚ ↪ ℚ(μ_{ℓ})^{+} ↪ ℚ(μ_{ℓ}),

where F (μ_{ℓ})^{+} is the compositum of F and ℚ(μ_{ℓ})^{+}. Condition (10) says that 𝔭 splits in the extension F (μ_{ℓ})^{+}/F for all but finitely many 𝔭. This implies that F (μ_{ℓ})^{+} = F, and thus ℚ(μ_{ℓ})^{+} ⊂ F, proving the first statement.

For the second, we reverse the above argument. If ℓ ∉ S_{cyclotomic}, then F (μ_{ℓ})^{+}/F is a non-trivial extension of F, and so there are infinitely many primes 𝔭 of F which do not split completely in it. For each such 𝔭 one has q ≢ ±1 mod ℓ. If in addition one takes 𝔭 | ℓ̸, then, for n sufficiently large, the torsion free normal subgroup Γ′ of Γ constructed above has index [Γ : Γ′] prime to ℓ. □

Let us define

S_{elliptic} = {℘ | p : ℚ(μ_{p})^{+} ⊂ F }  (11)

to be the finite set of primes ℘ of K lying above the primes in S_{cyclotomic}. By Lemma 1 we may assume that (7) holds for A = O_{(℘)}, O_{℘}, K_{℘}/O_{℘} etc. whenever ℘ ∉ S_{elliptic}. Thus the cohomology groups with these coefficients considered above are well defined if we assume that 𝔭 ∉ S_{elliptic}.

Let L(n, χ^{±1}, A) be the sheaf on Y (𝔑) which restricts to L_{i}(n, χ^{±1}, A) on the connected component Γ_{i}(𝔑)\Z, for i = 1, 2, …, h. We now define

H_{?}^{q}(Y (𝔑), L(n, χ^{±1}, A)) = ⨁_{i=1}^{h} H_{?}^{i}(Γ_{i}(𝔑)\Z, L_{i}(n, χ^{±1}, A)),  (12)

for ? = ∅, c, or p. Note that, after the discussion above, this cohomology group is defined with suitable restrictions on the coefficients A. In fact the sheaf L(n, χ^{±1}, A) and the corresponding cohomology groups of Y (𝔑) can be defined directly, in which case the decomposition (12) is in fact a statement that can be demonstrated (see Section 6 of [13] for the details).

There is a natural action of the Hecke operator T_{𝔪} on the cohomology groups on the left hand side of (12). For its definition, and its effect on the decomposition (12), the reader may consult Section 4 of [16] or Section 7 of [13]. We also note that the group of complex conjugations C and the strict class group Cl_{F}^{+} act on the cohomology as well (for the former action see [13] page 306, Section 2; the latter action is induced by the action of the center Z(𝔸) = 𝔸_{F}^{×}).

3.2 Cohomology of the boundary

In Section 2.3 we described the Borel-Serre compactification Y (𝔑)^{∗} of Y (𝔑). There are natural inclusions

Y (𝔑) ↪^{ι} Y (𝔑)^{∗} ↩^{j} ∂(Y (𝔑)^{∗}),

where ∂(Y (𝔑)^{∗}) is the boundary of Y (𝔑)^{∗}. Let ι_{∗} denote direct image, and j^{∗} denote inverse image. Then, for the sheaf L(n, χ^{±1}, A) on Y (𝔑) described in the previous section, one obtains the boundary cohomology group:

H^{q}(∂(Y (𝔑)^{∗}), j^{∗}ι_{∗}L(n, χ^{±1}, A)).  (13)

Let us describe the sheaf j^{∗}ι_{∗}L, where L = L(n, χ^{±1}, A), more concretely. It suffices to describe the sheaf j^{∗}ι_{∗}L_{i} on ∂(Γ_{i}(𝔑)\Z^{∗}), where L_{i} = L_{i}(n, χ^{±1}, A). Assume temporarily that \overline{Γ_{i}(𝔑)} is torsion free. Since Γ_{i}(𝔑)\Z^{∗} has fundamental group \overline{Γ_{i}(𝔑)}, a local system construction similar to the one described above yields a sheaf L_{i}^{∗} on Γ_{i}(𝔑)\Z^{∗}. A comparison of stalks shows that this sheaf is nothing but ι_{∗}L_{i}. Now j induces an inclusion on each boundary component Γ_{i}(𝔑)_{B}\∂_{B} ↪ Γ_{i}(𝔑)\Z^{∗}, which, by functoriality of π_{1}, induces a map (the natural inclusion)

\overline{Γ_{i}(𝔑)}_{B} → \overline{Γ_{i}(𝔑)}.  (14)

Thus any local system on Γ_{i}(𝔑)\Z^{∗} gives rise, via (14), to a local system on Γ_{i}(𝔑)_{B}\∂_{B}. A comparison of stalks shows that the local system so obtained from the sheaf ι_{∗}L_{i}, is the sheaf j^{∗}i_{∗}L_{i}. When Γ_{i}(𝔑) is not torsion free, then the usual procedure of ‘taking invariants’ described in the previous section, still allows one to define the boundary cohomology group (13) with some restrictions on the coefficients. In particular these groups are well defined when A = O_{℘}, as long as ℘ ∉ S_{elliptic}.

All the cohomology groups considered so far are related via the boundary long exact sequence:

H_{c}^{q}(Y (𝔑), L)  →  H^{q}(Y (𝔑), L)  →  H^{q}(∂(Y (𝔑)^{∗}), L)
∥  ∥  ∥
⨁_{i=1}^{h} H_{c}^{q}(Γ_{i}(𝔑)\Z, L_{i}) → ⨁_{i=1}^{h} H^{q}(Γ_{i}(𝔑)\Z), L_{i}) → ⨁_{i=1}^{h} H^{q}(∂(Γ_{i}(𝔑)\Z), L_{i}), (15)

where we have written L for L(n, χ^{±1}, A), and L_{i} for L_{i}(n, χ^{±1}, A), and where, for simplicity, we have dropped ι_{∗} and j^{∗}ι_{∗} from the notation.

3.3 Poincaré duality

Fix i ∈ {1, …, h}. Consider the pairing

< , >: L_{i}(n, χ, A) ⊗ L_{i}(n, χ^{−1}, A) → A,

defined via

⟨⨂_{σ} ∑_{l}(u_{σ})_{l}X_{σ}^{n−l}Y_{σ}^{l}, ⨂_{σ} ∑_{l}(v_{σ})_{l}X_{σ}^{n−l}Y_{σ}^{l}⟩ ↦

|a_{i}|_{F}^{n} ∏_{σ∈I_{F}} ∑_{l} \frac{(−1)^{l}(u_{σ})_{l}(v_{σ})_{n−l}}{( \begin{matrix}n \\ l \end{matrix} )} .

Here we assume that A is one of the O_{K}-algebras mentioned above, with the restriction that the primes p ≤ n are invertible in it, so that the expression ( \begin{matrix}n \\ l \end{matrix} )^{−1} makes sense. Thus, when A = O_{(℘)}, O_{℘}, or K_{℘}/O_{℘}, we assume that ℘ ∉ S_{weight}, where

S_{weight} = {℘ | p : p ≤ n = k − 2} .  (16)

If γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ_{i}(𝔑), then ad ≡ ϵ mod 𝔑 for some ϵ ∈ O_{F +}^{×}. Since χ_{𝔑}(ϵ) = N_{F/ℚ}(ϵ)^{n} = 1, we see that χ_{𝔑}(a) = χ_{𝔑}(d)^{−1}. Using this one may check that the pairing < , > is Γ_{i}(𝔑)-equivariant, and that it is perfect. It induces the following pairings on cohomology via cup product, which are again denoted by < , >

< , >: H^{q}(Y (𝔑), L(n, χ, A)) ⊗ H_{c}^{2d−q}(Y (𝔑), L(n, χ^{−1}, A)) → A, (17)
< , >: H_{c}^{q}(Y (𝔑), L(n, χ, A)) ⊗ H^{2d−q}(Y (𝔑), L(n, χ^{−1}, A)) → A. (18)

In order to state the next Proposition, for an O_{℘} module M of finite type, we let M′ denote the maximal torsion free quotient of M. Then we have,

Proposition 2 (Poincaré duality) Say q = d is the middle dimension. Then if A = K_{℘} or ℂ, the pairing (17) is perfect. That is, the morphisms induced by (17):

H^{d}(Y (𝔑), L(n, χ, A)) → Hom_{A}(H_{c}^{d}(Y (𝔑), L(n, χ^{−1}, A)), A),
H_{c}^{d}(Y (𝔑), L(n, χ^{−1}, A)) → Hom_{A}(H^{d}(Y (𝔑), L(n, χ, A)), A),

are isomorphisms. A similar statement holds for the pairing (18).

When A = O_{℘}, and ℘ ∉ S_{weight} ∪ S_{elliptic}, then the morphisms induced by (17) are isomorphisms modulo torsion. That is:

H^{d}(Y (𝔑), L(n, χ, A))′ → Hom_{A}(H_{c}^{d}(Y (𝔑), L(n, χ^{−1}, A))′, A),
H_{c}^{d}(Y (𝔑), L(n, χ^{−1}, A))′ → Hom_{A}(H^{d}(Y (𝔑), L(n, χ, A))′, A)

are isomorphisms. Again, an analogous statement holds for the pairing (18).

Proof. If Y (𝔑)_{/ℚ} is smooth, the statements follow from ordinary Poincaré duality. Even if Y (𝔑) is not smooth, we may still deduce the perfectness of the pairings, with the usual restrictions on the coefficients (cf. [23], Remark after Theorem 1.6). Indeed, if 𝔭 ∉ S_{elliptic}, by Lemma 1, we may choose torsion free finite index normal subgroups \overline{Γ_{i}} of \overline{Γ_{i}(𝔑)} with the property that [\overline{Γ_{i}(𝔑)} : \overline{Γ_{i}}] is invertible in A. Since the perfectness of the pairings is true for the subgroups Γ_{i}, the theorem follows for the Γ_{i}(𝔑) as well by ‘taking invariants’. □

We now derive a duality statement for the parabolic cohomology groups with integral coefficients A = O_{℘}. First note that the two diagrams

H_{c}^{d}(Y (𝔑), L(n, χ, A)) ⊗ H_{c}^{d}(Y (𝔑), L(n, χ^{−1}, A)) → A
↓ ι ∥ ∥
H^{d}(Y (𝔑), L(n, χ, A)) ⊗ H_{c}^{d}(Y (𝔑), L(n, χ^{−1}, A)) → A,

and

H_{c}^{d}(Y (𝔑), L(n, χ, A)) ⊗ H_{c}^{d}(Y (𝔑), L(n, χ^{−1}, A)) → A
∥ ↓ ι ∥
H_{c}^{d}(Y (𝔑), L(n, χ, A)) ⊗ H^{d}(Y (𝔑), L(n, χ^{−1}, A)) → A,

commute. Consequently, we get a (well-defined) induced pairing which we again denote by < , >:

< , >: H_{p}^{d}(Y (𝔑), L(n, χ, A)) ⊗ H_{p}^{d}(Y (𝔑), L(n, χ^{−1}, A)) → A. (19)

Definition. Let S_{∂} be the finite set of primes ℘ of K for which the O_{℘}- modules

H^{d}(∂(Γ_{i}(𝔑)\Z), L_{i})

have torsion, for i varying from 1 through h, and L_{i} = L_{i}(n, χ^{±1}, O_{℘}). Also let

S_{duality} = S_{weight} ∪ S_{elliptic} ∪ S_{∂}.  (20)

Note that S_{duality} is a finite set of primes of K.

Theorem 3 (Poincaré duality for parabolic cohomology) If ℘ is a prime of K such that ℘ ∉ S_{duality}, and A = O_{℘} is the ℘-adic completion of the ring of integers O_{K} of K, then the pairing (19) above is perfect modulo torsion. That is, the induced morphisms

H_{p}^{d}(Y (𝔑), L(n, χ, A))′ → Hom_{A}(H_{p}^{d}(Y (𝔑), L(n, χ^{−1}, A))′, A), (21)
H_{p}^{d}(Y (𝔑), L(n, χ^{−1}, A))′ → Hom_{A}(H_{p}^{d}(Y (𝔑), L(n, χ, A))′, A), (22)

are isomorphisms.

Proof of Theorem 3. Fix ℘ ∉ S_{duality}. First note that since ℘ ∉ S_{elliptic} ∪ S_{weight}, the morphisms (21) and (22) are well defined. We only prove (21) is an isomorphism, since the proof for (22) is similar.

First, one may easily check that the injectivity (modulo torsion) of the morphism (21) follows from the injectivity (modulo torsion) of the maps in Proposition 2 (note that we have assumed ℘ ∉ S_{elliptic}). So we must only show the surjectivity. To this end, let α ∈ Hom_{O_{℘}}(H_{p}^{d}(Y (𝔑), L(n, χ^{−1}, O_{℘})), O_{℘}). Thanks to Proposition 2, on composing α with the map

H_{c}^{d}(Y (𝔑), L(n, χ^{−1}, O_{℘})) →^{i} H_{p}^{d}(Y (𝔑), L(n, χ^{−1}, O_{℘})),

we can find an element x ∈ H^{d}(Y (𝔑), L(n, χ, O_{℘})) such that < x, y >= (α ∘ i)(y), for all y ∈ H_{c}^{d}(Y (𝔑), L(n, χ^{−1}, O_{℘})). Since after tensoring with K_{℘} the morphism (21) is an isomorphism, we may assume that

x ⊗ 1 ∈ H_{p}^{d}(Y (𝔑), L(n, χ, O_{℘})) ⊗ K_{℘} = H_{p}^{d}(Y (𝔑), L(n, χ, K_{℘})).

Let \overline{z} denote taking the class of z ∈ H^{d} modulo H_{p}^{d}, over O_{℘} or K_{℘}. Then

\overline{x} ⊗ 1 = \overline{x ⊗ 1} = 0,

so that \overline{x} is a torsion element in

\frac{H^{d}(Y (𝔑), L(n, χ, O_{℘}))}{H_{p}^{d}(Y (𝔑), L(n, χ, O_{℘}))}.  (23)

By the boundary exact sequence (15) we have

\frac{H^{d}(Y (𝔑), L)}{H_{p}^{d}(Y (𝔑), L)} ⊂ H^{d}(∂(Y (𝔑)^{∗}), L),

with L = L(n, χ, O_{℘}). Thus since ℘ ∉ S_{∂}, (23) is a torsion-free O_{℘}-module. Thus \overline{x} = 0. That is, x ∈ H_{p}^{d}(Y (𝔑), L(n, χ, O_{℘})), as desired. □

Remark 1 One expects that the only maximal ideals of the Hecke algebra that occur in the support of the boundary cohomology group are all Eisenstein. Thus, since for the purpose of investigating congruences it suffices to work locally, one could possibly also work outside the set of primes of K which are Eisenstein. Such matters are currently being investigated by M. Dimitrov in his thesis under the supervision of Prof. J. Tilouine.

3.4 Explicit computation of the set S_{∂}

It would be nice to be able to compute the set S_{∂} explicitly. Here we give a general method for doing this, and then work out some special cases. Recall that S_{∂} is the set of primes ℘ for which the O_{℘}-modules

H^{d}(∂(Γ_{i}(𝔑)\Z), L_{i})

have torsion, as i varies between 1 and h, and L_{i} = L_{i}(n, χ^{±1}, O_{℘}). We only treat the case L_{i}(n, χ, O_{℘}), since the discussion for L_{i}(n, χ^{−1}, O_{℘}) is similar.

Recall that the sheaf L_{i} on the boundary arise from a local system construction (see Section 3.2). Thus, if ℘ ∉ S_{elliptic}, we have

H^{d}(∂(Γ_{i}(𝔑)\Z), L_{i}(n, χ, O_{℘})) = ⊕_{B}H^{d}(\overline{Γ_{i}(𝔑)}_{B}, L_{i}(n, χ, O_{℘})),

where the sum runs over the set of representatives of the Γ_{i}(𝔑)-conjugacy classes of Borel subgroups B, and the bar means image in PGL_{2}(F). Thus, if we ignore the primes in S_{elliptic}, which we will eventually exclude anyway, S_{∂} is exactly the set of primes ℘ for which the O_{℘}-modules

H^{d}(\overline{Γ_{i}(𝔑)}_{B}, L_{i}(n, χ, O_{℘}))

have torsion, for some i and B (of course the B’s depend on i!).

Let now Γ = Γ_{i}(𝔑) for some i, and write L(n, χ, A) for the corresponding \overline{Γ}-module L_{i}(n, χ, A). Let B be a Borel subgroup as above. We can make a reduction. The exact sequence of \overline{Γ}_{B} modules

0 → L(n, χ, O_{℘}) → L(n, χ, K_{℘}) → L(n, χ, K_{℘}/O_{℘}) → 0,

gives the following long exact sequence of group cohomology groups:

⋯ ⟶ H^{d−1}(\overline{Γ}_{B}, L(n, χ, K_{℘})) ⟶ H^{d−1}(\overline{Γ}_{B}, L(n, χ, K_{℘}/O_{℘})) ⟶ H^{d}(\overline{Γ}_{B}, L(n, χ, O_{℘})) ⟶ H^{d}(\overline{Γ}_{B}, L(n, χ, K_{℘})) ⟶ ⋯

From this it is easy to check that H^{d}(\overline{Γ}_{B}, L(n, χ, O_{℘})) is torsion-free if and only if H^{d−1}(\overline{Γ}_{B}, L(n, χ, K_{℘}/O_{℘})) is divisible. Thus it suffices to investigate the divisibility of the cohomology groups H^{d−1}(\overline{Γ}_{B}, L(n, χ, K_{℘}/O_{℘})).

Let C(Γ) denotes the set of cusps of Γ. Then

C(Γ) = Γ\GL_{2}(F)/B_{∞} = Γ\ℙ^{1}(F),

where B_{∞}, the Borel of upper triangular matrices, is the stabilizer of the cusp ∞ = [1 : 0] ∈ ℙ^{1}(F). Fix a set of representatives, g_{1}, …, g_{r} ∈ GL_{2}(F), of the double coset space, where r = r(Γ) is the number of cusps of Γ. Then {g_{j}B_{∞}g_{j}^{−1} | j = 1, …, r} is a set of representatives of the Γ-conjugacy classes of Borel subgroups. Setting B = g_{j}B_{∞}g_{j}^{−1}, we have

H^{d−1}(\overline{Γ}_{B}, L(n, χ, K_{℘}/O_{℘})) = H^{d−1}(\overline{(g_{j}^{−1}Γg_{j})}_{B_{∞}}, L(n, χ, K_{℘}/O_{℘})).

This allows us to restrict our computations to the Borel at ∞.

Now write B_{∞} = T U, where T is the standard torus, and U is the unipotent radical of B_{∞}. Then, for an arbitrary arithmetic subgroup Γ′ ⊂ GL_{2}(F)_{+}, we have the exact sequence

1 → Γ′_{U} → Γ′_{B} → Γ′_{T} → 1,

where Γ′_{T} := Γ′ ∩T and Γ′_{U} := Γ′ ∩U. Projecting this to PGL_{2}(F) and noting that Γ′_{U} ∩ F^{×} = {1}, one obtains

1 → Γ′_{U} → \overline{Γ}′_{B} → \overline{Γ}′_{T} → 1,  (24)

where \overline{Γ}′_{B} = Γ′_{B}/Γ′_{B} ∩ F^{×} and \overline{Γ}′_{T} = Γ′_{T} /Γ′_{T} ∩ F^{×}. The Hochschild-Serre spectral sequence for the \overline{Γ}′_{B}-module L(n, χ, K_{℘}/O_{℘}) is

E_{2}^{p,q} = H^{p}(\overline{Γ}′_{T} , H^{q}(Γ′_{U}, L(n, χ, K_{℘}/O_{℘}))) ⟹ H^{n}(\overline{Γ}′_{B_{∞}}, L(n, χ, K_{℘}/O_{℘})).

In PGL_{2}(F) we have \overline{Γ}′_{T} →^{∼} ℤ^{d−1} and Γ′_{U} →^{∼} ℤ^{d}, so that these groups have cohomological dimension d − 1 and d respectively. Consequently, it is easy to check that all the differentials in the E_{d}-plane are 0: that is the spectral sequence degenerates at E_{d}. Thus, to show that

H^{d−1}(\overline{(g_{j}^{−1}Γ_{i}(𝔑)g_{j})}_{B_{∞}}, L_{i}(n, χ, K_{℘}/O_{℘}))

is divisible, it suffices to show that each term in its filtration

\frac{F^{p}H^{d−1}}{F^{p+1}H^{d−1}} = E_{∞}^{p,d−1−p} = E_{d}^{p,d−1−p}

is divisible.

Clearly, in general, this is a difficult task. However, some special cases can be dealt with, which we discuss now.

3.4.1 F = ℚ

The case F = ℚ has already been dealt with by Hida in [10], but for the sake of completeness we review his computations here.

Note that ℚ has strict class number 1, so there is only one congruence subgroup, Γ_{i}(𝔑), to consider. Moreover we may choose t_{1} = 1. Thus if we let N be the positive integer which generates the ideal 𝔑 ⊂ ℤ, we see that Γ_{1}(𝔑) is just the usual congruence subgroup

Γ_{0}(N) = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(ℤ) | c ≡ 0 mod N\}.

Also, from (11), we have that S_{elliptic} = {℘ | p : p = 2 or 3}, and we ignore these primes in our computations.

The following Proposition has been distilled from the discussion on page 232-233 of [10]:

Proposition 3 If n = 0, then S_{∂} = ∅. For n > 0, let S_{1} to be the set of primes ℘ lying above the set

{p : p ≤ n = k − 2 or p | N}.

Then S_{∂} ⊂ S_{1}.

Proof. Note that we may choose the set of representatives, g_{1}, …, g_{r}, of the cusps of Γ_{0}(N) to lie in SL_{2}(ℤ). Then, up to the center {±1}, the group g_{j}^{−1}Γ_{0}(N)g_{j}∩B_{∞} →^{∼} ℤ is generated by an element of the form π_{j} = ( \begin{matrix}1 & m_{j} \\ 0 & 1 \end{matrix} ), for some m_{j} ∈ ℤ. Write L(n, O_{℘}) for L_{1}(n, χ^{±1}, O_{℘}). Then we must investigate the torsion in

H^{1}(\overline{Γ_{0}(N)}_{B}, L(n, O_{℘})) = \frac{L(n, O_{℘})}{(π_{j} − 1)L(n, O_{℘})},  (25)

for B = g_{j}B_{∞}g_{j}^{−1}.

When n = 0, the action is trivial, and this last module is just O_{℘}, which is clearly torsion-free.

When n > 0 then a direct computation shows that (25) has torsion only for those ℘ whose residue characteristic p satisfies p ≤ n or p | m_{j}. But, since the principal congruence subgroup Γ(N) is a normal subgroup of SL_{2}(ℤ) and Γ(N) ⊂ Γ_{0}(N) ⊂ SL_{2}(ℤ), one may easily check that each m_{j} | N. □

3.4.2 Real quadratic fields

Let F be a real quadratic field. So d = [F : ℚ] = 2. Let ϵ_{0} denote the fundamental unit of F, and let ϵ_{1} denote a generator of the group of totally positive units O_{F +}^{×}. Thus,

ϵ_{1} = \begin{cases}ϵ_{0} & if N_{F/ℚ}(ϵ_{0}) = 1, \\ ϵ_{0}^{2} & if N_{F/ℚ}(ϵ_{0}) = −1. \end{cases}

By (11), we have S_{elliptic} = {℘ | p : p = 2, 3 or 5}, and we ignore such primes as usual! (In fact when F ≠ ℚ(\sqrt{5}), one may take S_{elliptic} = \{℘ | p : p = 2 or 3}).

We first treat the case when the weight n = 0. Even in this case, we make the following restrictive assumption, which will at least allow us to compute.

Assumption 1 Assume that

• F has strict class number h = 1, and,

• the level 𝔑 = 1.

In view of this assumption, we must only consider the congruence subgroup

Γ = Γ_{1}(𝔑) = {γ ∈ GL_{2}(O_{F}) | det(γ) ∈ O_{F +}^{×}}.  (26)

This Γ has only one cusp, and we may choose the representative g_{1} = 1. Now (working modulo the center; we will omit bars from the notation) we see that:

Γ_{T} := Γ ∩ T = \{( \begin{matrix}ϵ & 0 \\ 0 & 1 \end{matrix} ) | ϵ ∈ O_{F +}^{×}\},

and

Γ_{U} := Γ ∩ U = \{( \begin{matrix}1 & b \\ 0 & 1 \end{matrix} ) | b ∈ O_{F} \}.

We have:

Proposition 4 Let F and Γ satisfy the hypothesis of Assumption 1 above. Let n = 0. Let S_{2} be the set of primes of K lying above the set

{p : p | N_{F/ℚ}(ϵ_{1} − 1)}.

Then S_{∂} ⊂ S_{2}.

Proof. Fix ℘ ∉ S_{2}. We show ℘ ∉ S_{∂}. That is, we show H^{2}(Γ_{B_{∞}}, L(n, O_{℘})) is torsion-free, when n = 0. Here, and for the rest of this subsection, we write L(n, A) for L_{1}(n, χ^{±1}, A), noting that χ_{𝔑} = 1. By the general remarks at the beginning of Section 3.4, it suffices to show that the O_{℘}-modules E_{2}^{1,0} and E_{2}^{0,1} are divisible. Note that since we are in the case n = 0, the Γ-action is trivial. We have

E_{2}^{1,0} = H^{1}(Γ_{T} , H^{0}(Γ_{U}, K_{℘}/O_{℘}))
= Hom_{ℤ}(ℤ, K_{℘}/O_{℘})
= K_{℘}/O_{℘},

which is clearly divisible. Moreover

E_{2}^{0,1} = H^{0}(Γ_{T} , H^{1}(Γ_{U}, K_{℘}/O_{℘}))
= Hom_{ℤ}(Γ_{U}, K_{℘}/O_{℘})^{Γ_{T}} ,

where γ ∈ Γ_{T} acts on f ∈ Hom_{ℤ}(Γ_{U}, K_{℘}/O_{℘}) via

(γ · f)(g) = γ · f(γ^{−1} · g) = f(γ^{−1}gγ).

Recall that p is the the residue characteristic of ℘. We claim that, since p | N̸_{F/ℚ}(ϵ_{1} − 1), there are no invariant elements under this action. Indeed if f is such a homomorphism we have (identifying Γ_{U} with O_{F} in an obvious manner) that

f(ϵb) = f(b),  (27)

for all ϵ ∈ O_{F +}^{×} and all b ∈ O_{F}. Fix an integral basis {ω_{1}, ω_{2}} of O_{F}, and write ϵ_{1}ω_{i} = ∑_{j} a_{ij}ω_{j}. Then the equations (27) imply that

(A − Id)F = 0,

where A = (a_{ij}) and F = (f(ω_{1}), f(ω_{2}))^{t}. Thus det(A − Id) = N_{F/ℚ}(ϵ_{1} − 1) annihilates the f(ω_{i}). Since f takes values in the ℘-torsion module K_{℘}/O_{℘}, we see that f = 0, since p | N̸_{F/ℚ}(ϵ_{1} − 1). Thus E_{2}^{0,1} = 0 for such ℘.

In sum, when n = 0, and the residue characteristic p of ℘ satisfies p | N̸_{F/ℚ}(ϵ_{1} − 1), then H^{1}(Γ_{B}, L(n, K_{℘}/O_{℘})) = E_{2}^{1,0} is divisible. □

Remark 2 When the strict class number of F is not 1, the other Γ_{i}(𝔑) in the decomposition (2) must be treated. Moreover, even if the strict class number is 1, and the level 𝔑 is not full, there may be more than one cusp. This must be taken into account, further complicating the computation of the boundary cohomology groups. However the essential idea is already to be found in the above computations.

We now consider the case n > 0, although again, for simplicity, we suppose that Assumption 1 above holds.

Since the computations are still too difficult, we appeal to Hida’s general theory in [15] via which one can extract some information for the ordinary primes. This theory is fairly elaborate and we only make some brief remarks here, referring the reader to Section 3 of [15] for further information.

Let F denote, momentarily, an arbitrary totally real field (of strict class number 1). Fix a prime p of ℚ, and let Σ = {𝔭 | p} denote the set of primes of F lying over p. Let ι = (ι_{𝔭})_{𝔭∈Σ} ∈ {0, ∞}^{Σ} and let r = (r_{𝔭})_{𝔭∈Σ} ∈ ℕ^{Σ}, where r_{𝔭} := rank_{ℤ_{p}}O_{F 𝔭}. Set {ιr} = ∑_{ι_{𝔭} with ι_{𝔭}=∞} r_{𝔭}.

Fix an embedding, j, of \overline{ℚ} into \overline{ℚ}_{p}. Since we have fixed an embedding of K into \overline{ℚ}, such an embedding singles out a prime ℘ of K. Conversely, a choice of a prime ℘ of K yields an embedding of K into \overline{ℚ}_{p} (which may be extended to an embedding of \overline{ℚ} into \overline{ℚ}_{p}).

Each σ ∈ I_{F} determines a unique prime 𝔭 of F which makes

F ⟶^{σ} \overline{ℚ} ⟶^{j} \overline{ℚ}_{p}

a 𝔭-adically continuous embedding of F into \overline{ℚ}_{p}. For each 𝔭 ∈ Σ, let I(𝔭) = {σ ∈ I_{F} | ℘ ∩ σ(F) = σ(𝔭)} ⊂ I_{F} denote the set of all such 𝔭-adically continuous embeddings of F into \overline{ℚ}_{p} (with respect to j).

For each ι as above, let I(ι) = ∪_{𝔭 with ι_{𝔭}=∞}I(𝔭). Set n^{ι} = ∑_{σ∈I(ι)} n_{σ}σ − ∑_{σ∉I(ι)} n_{σ}σ. Let K_{℘}/O_{℘}(n^{ι}) denote the module K_{℘}/O_{℘}, on which ϵ ∈ O_{F}^{×} acts via: ϵ · a = ϵ^{n^{ι}}a.

Let Γ′ = SL_{2}(O_{F}), and let H_{ord}^{q}(∂(Γ′\Z^{∗}), L(n, K_{℘}/O_{℘})) denote the ordinary part of the boundary cohomology group, as defined in [15]. We just recall here that the ordinary part is the maximal O_{℘}-submodule of the boundary cohomology group on which the Hecke operator T_{p} acts as an automorphism. Then Hida proves (cf. [15], Theorem 3.12):

Theorem 4 Fix F and Γ′ as above. Then

H_{ord}^{q}(∂(Γ′\Z^{∗}), L(n, K_{℘}/O_{℘})) =

⨁_{i=1}^{q} H^{q−i} (O_{F}^{×}, ⨁_{ι with {ιr}=i} Hom_{ℤ_{p}}( ⋀_{𝔭∈Σ with i_{𝔭}=∞} ∧^{r_{𝔭}}O_{F 𝔭}, K_{℘}/O_{℘}(n^{ι}))) ,

In the real quadratic situation, we obtain:

Corollary 1 Let F be a real quadratic field of strict class number 1, and let Γ′ = SL_{2}(O_{F}). Suppose that n > 0. Let S′_{∂,ord} denote the set of primes of torsion in the ordinary part of the middle dimensional boundary cohomology group for Γ′. Let S_{3} be the set of primes of K lying above the set

{p : p | N_{F/ℚ}(±ϵ_{0}^{2(n+1)} − 1)},

where the sign is N_{F/ℚ}(ϵ_{0})^{n}. Then S′_{∂,ord} ⊂ S_{3}.

Proof. Let ℘ | p be a prime of K and assume that ℘ ∉ S_{3}. We show that ℘ ∉ S′_{∂,ord}. It is enough to show H_{ord}^{q}(∂(Γ′\Z^{∗}), L(n, K_{℘}/O_{℘})) is divisible when q = d − 1 = 1. By Theorem 4, if p is inert or ramifies in F, H_{ord}^{1} = 0 vacuously. This is because, in either case, if 𝔭 | p, we have r_{𝔭} = 2, and so there is no sum over ι to consider.

Now suppose that p = 𝔭_{1}𝔭_{2} splits in F. Say I_{F} = {σ_{1}, σ_{2}}, and assume that j is chosen so that I(𝔭_{k}) = {σ_{k}}, for k = 1, 2. Then, since i = 1, and r_{_{𝔭}_{1}} = r_{_{𝔭}_{2}} = 1, only ι = (∞, 0) and (0, ∞) contribute to the cohomology, in Theorem 4.

When ι = (∞, 0), this contribution is

H^{0} (O_{F}^{×}, Hom_{ℤ_{p}}(O_{F 𝔭_{1}}, K_{℘}/O_{℘}(n^{ι}))) .  (28)

Note that ϵ^{−nσ_{2}} = ±ϵ^{nσ_{1}}, where the sign is given by N_{F/ℚ}(ϵ)^{n}. Let us identify Hom_{ℤ_{p}}(O_{F 𝔭_{1}}, K_{℘}/O_{℘}(n^{ι})) with K_{℘}/O_{℘} via f ↦ f(1). Note, by page 287 of [15], the action of ϵ^{−1} ∈ O_{F}^{×} on O_{F 𝔭_{1}} is just multiplication by ϵ^{2}. Thus, as an O_{F}^{×}-module, Hom_{ℤ_{p}}(O_{F 𝔭_{1}}, K_{℘}/O_{℘}(n^{ι})) is just K_{℘}/O_{℘}(nσ_{1} − nσ_{2} + 2σ_{1}) = K_{℘}/O_{℘}((2n + 2)σ_{1}), up to the above mentioned sign. Thus since 𝔭 ∉ S_{3} we see that (28) vanishes.

A similar computation for ι = (0, ∞), shows that

H^{0} (O_{F}^{×}, Hom_{ℤ_{p}}(O_{F 𝔭_{2}}, K_{℘}/O_{℘}(n^{ι}))) = 0.

Thus, H_{ord}^{1}(∂(Γ′\Z^{∗}), L(n, K_{℘}/O_{℘})) = 0 is indeed divisible. □

Remark 3 Note that the computations for n > 0 are slightly different since Hida’s theory is for subgroups of SL_{2}(F), whereas we should really be taking Γ as in (26). If, in the case of n = 0, we had taken Γ′ = SL_{2}(O_{F}) instead of Γ = GL_{2}(O_{F})_{+} in Proposition 4, then, since

Γ′_{T} = \{( \begin{matrix}ϵ & 0 \\ 0 & ϵ^{−1} \end{matrix} ) | ϵ ∈ O_{F}^{×}\},

acts on Γ′_{U} = O_{F} by multiplication by ϵ^{2}, we would have gotten the obstruction N_{F/Q}(ϵ_{0}^{2} − 1) there. This matches well with Corollary 1 when n = 0. On the other hand, since it is Γ = GL_{2}(O_{F})_{+} that we really wish to work with, it is clear that the (ordinary) torsion in the boundary for n > 0 will occur at primes ℘|p with p | N_{F/ℚ}(ϵ_{1}^{n+1} − 1) = N_{F/ℚ}(ϵ_{1}^{k−1} − 1).

3.4.3 Totally real cubic fields

Let now [F : ℚ] = 3. We only treat the case n = 0 here. As in the quadratic case, we assume that h = 1 and 𝔑 = O_{F} (cf. Assumption 1). As before, this allows us to deal with only the analog of the the level one congruence subgroup (26), and with only the standard Borel B_{∞}. Note that now \overline{Γ}_{T} = O_{F +}^{×} →^{∼} ℤ^{2} and Γ_{U} = O_{F} →^{∼} ℤ^{3}. Let ϵ_{1} and ϵ_{2} denote a basis of the group of totally positive units.

As usual, we ignore the set of primes in S_{elliptic} which in this case turns out to be {℘ | p : p = 2, 3, or 7}. (If F ≠ ℚ(μ_{7})^{+}, then we may take S_{elliptic} = {℘ | p : p = 2 or 3}). We have:

Proposition 5 Let F be a totally real cubic field of class number 1, and let Γ be as in (26). Let S_{4} be the set of primes of K lying above the set

{p : p | N_{F/ℚ}(ϵ_{1} − 1) and p | N_{F/ℚ}(ϵ_{2} − 1)}.

Then, if n = 0, S_{∂} ⊂ S_{4}.

Proof. Fix O_{℘} with ℘ | p such that ℘ ∉ S_{4}. We show ℘ ∉ S_{∂}. After the remarks at the beginning of Section 3.4, it suffices to show that each of the three term E_{3}^{p,q} (p + q = 2) in the filtration of H^{2}(Γ_{B}, L(n, K_{℘}/O_{℘})), with n = 0, is divisible. Note that again we write L(n, A) for L_{1}(n, χ^{±}, A), since χ_{𝔑} = 1.

We have E_{3}^{2,0} = E_{2}^{2,0}/image(d_{2}^{0,1} : E_{2}^{0,1} → E_{2}^{2,0}). But

E_{2}^{2,0} = Hom_{ℤ}(∧^{2}ℤ^{2}, K_{℘}/O_{℘}) = K_{℘}/O_{℘},

and a quotient of a divisible is divisible, so that E_{3}^{2,0} is divisible.

Secondly, E_{3}^{1,1} = E_{2}^{1,1} = H^{1}(Γ_{T} , M), for M = H^{1}(Γ_{U}, K_{℘}/O_{℘}). Since O_{F +}^{×} = < ϵ_{1} > × < ϵ_{2} >, the inflation-restriction sequence yields:

0 → H^{1}(< ϵ_{1} >, M^{<ϵ_{2}>}) → H^{1}(Γ_{T} , M) → H^{1}(< ϵ_{2} >, M)^{<ϵ_{1}>} → 0. (29)

Now we have that p | N̸_{F/ℚ}(ϵ_{i} − 1), for i = 1 or 2. Suppose i = 2. Then we have M^{<ϵ_{2}>} = 0, just as in the real quadratic case. Thus, the first term of the exact sequence (29) vanishes. Similarly, the last term of (29) vanishes since it is just (M/(ϵ_{2}−1)M)^{<ϵ_{1}>}, and is thus contained in M/(ϵ_{2}−1)M = 0, which vanishes in turn since p | N̸_{F/ℚ}(ϵ_{2} −1). A symmetric argument applies if i = 1. Consequently, E_{3}^{1,1} = 0.

Finally, E_{3}^{0,2} = ker(d_{2}^{0,2} : E_{2}^{0,2} → E_{2}^{2,1}). We claim

E_{2}^{0,2} = H^{0}(Γ_{T} , H^{2}(Γ_{U}, K_{℘}/O_{℘})) = 0.

To see this, note that

H^{2}(Γ_{U}, K_{℘}/O_{℘}) = Hom_{ℤ}(∧^{2}O_{F} , K_{℘}/O_{℘}).

If f is a homomorphism in this latter space, invariant under some ϵ ∈ Γ_{T}, then we must have

(ϵ^{−1} · f)(a ∧ b) = f(ϵa ∧ ϵb) = f(a ∧ b).  (30)

Now a familiar argument applies: let {ω_{1}, ω_{2}, ω_{3}} be an integral basis for O_{F}, and write ϵω_{i} = ∑_{j} a_{ij}ω_{j}. Then if B denotes the matrix of 2 × 2 cofactors (minors with signs) of A = (a_{ij}), the equations (30) show that det(B − Id) annihilates each component of the vector (f(ω_{2} ∧ω_{3}), f(ω_{3} ∧ω_{1}), f(ω_{1} ∧ω_{2}))^{t}. But note that B = A^{−1}, since det(A) = 1. Consequently choosing ϵ to be ϵ_{1} or ϵ_{2}, so that

p | d̸et(A^{−1} − Id) = − det(A − Id) = −N_{F/ℚ}(ϵ − 1),

we see that f = 0. Thus E_{2}^{0,2} = 0, and so E_{3}^{0,2} = 0 as well. □

3.4.4 Totally real fields of arbitrary degree

The referee has suggested that, at least in the case n = 0, our computations of the previous section should generalize. This is indeed the case and we present this now.

So let now [F : ℚ] = d be arbitrary. We assume that n = 0, and as in previous cases, that h = 1 and 𝔑 = O_{F} (cf. Assumption 1). Therefore, as before, Γ is the level one congruence subgroup (26), and we need only work with the standard Borel B_{∞}. Note that now \overline{Γ}_{T} = O_{F +}^{×} →^{∼} ℤ^{d−1} and Γ_{U} = O_{F} →^{∼} ℤ^{d}. Let ϵ_{1}, …, ϵ_{d−1} denote a basis of the group of totally positive units of O_{F}.

Proposition 6 Let F be a totally real field of class number 1, and let Γ be as in (26). Let S_{5} be the set of primes of K lying above the set

⋂_{i=1}^{d−1} {p : p | N_{F/ℚ}(ϵ_{i} − 1)}.

Then, if n = 0, S_{∂} ⊂ S_{5}.

Before we give the proof we start with a simple lemma:

Lemma 2 Let G be a finitely generated group and M a G-module. Say there is a g in the center of G, such that g − 1 : M → M is an automorphism. Then H^{i}(G, M) = 0 for all i ≥ 0.

Proof of Lemma 2. Note that g induces a map on cohomology g : H^{i}(G, M) → H^{i}(G, M) which is well known to be the identity map (cf. [20], Chapter VII, Proposition 3). Thus g − 1 acts as 0 on cohomology. On the other hand, by our hypothesis on g, the map g − 1 is an automorphism on cohomology as well. The Lemma follows. □

Proof of Proposition 6. Fix O_{℘} with ℘ | p such that ℘ ∉ S_{5}. We show ℘ ∉ S_{∂}. As before, after the remarks at the beginning of Section 3.4, it suffices to show that each of the terms E_{d}^{p,q} (p + q = d − 1) in the filtration of H^{d−1}(Γ_{B}, K_{℘}/O_{℘}) (recall n = 0 and χ_{𝔑} = 1), is divisible. We will show that all steps in the filtration vanish, except the top one, which is divisible.

To do this we show that the spectral sequence degenerates at the E_{2} term. Indeed, E_{2}^{p,q} = H^{p}(\overline{Γ}_{T} , M(q)), where

M(q) := H^{q}(Γ_{U}, K_{℘}/O_{℘}) = Hom_{ℤ}(∧^{q}O_{F} , K_{℘}/O_{℘}),

since the action of Γ_{U} on K_{℘}/O_{℘} is trivial. When q > 0, the action of \overline{Γ}_{T} on M(q) is non-trivial. By hypothesis, we can find ϵ = ϵ_{i}, for some i, such that p | N̸_{F/ℚ}(ϵ − 1). Now ϵ − 1 acts on M(q) as a matrix whose determinant is a power of N_{F/Q}(ϵ − 1). This follows from a 19th century theorem attributed to J. J. Sylvester (see, for instance, pages 87-89 of [22]) which says that the determinant of the matrix of s × s minors of a matrix of size t ≥ s is just a power of the original determinant. In particular, ϵ − 1 acts as an automorphism on M(q) for each q > 0. By Lemma 2 we see that for q > 0, E_{2}^{p,q} = 0 for all p ≥ 0. Thus the spectral sequence degenerates at the E_{2} term, and an analysis of the filtration of H^{d}(\overline{Γ}_{B}, O_{℘}) now shows that H^{d−1}(\overline{Γ}_{B}, K_{℘}/O_{℘}) = E_{∞}^{d−1,0} = E_{2}^{d−1,0} = Hom_{ℤ}(∧^{d−1}ℤ^{d−1}, K_{℘}/O_{℘}) = K_{℘}/O_{℘} is divisible, as desired. □

Remark 4 The set

⋂_{i=1}^{d−1} {p : p | N_{F/ℚ}(ϵ_{i} − 1)}.

may depend on the choice of basis ϵ_{1}, …, ϵ_{d−1} of O_{F +}^{×}. However the identity ϵϵ′ − 1 = ϵ(ϵ′ − 1) + (ϵ − 1) shows that the set

⋂_{i=1}^{d−1} {𝔭 ⊂ O_{F} : 𝔭 | ϵ_{i} − 1}.

is independent of the basis. Thus it would probably be better to take for S_{5} the set of primes of K lying over this latter set.

Remark 5 As in the case when F is a real quadratic field, we could possibly use Theorem 4 to compute the torsion in the ordinary part of the boundary cohomology, when the weight n > 0, for arbitrary d. We have, however, refrained from presenting the details.

3.5 A modified pairing

We have investigated in some detail the perfectness of the pairing (19):

< , >: H_{p}^{d}(Y (𝔑), L(n, χ, A)) ⊗ H_{p}^{d}(Y (𝔑), L(n, χ^{−1}, A)) → A,

for various coefficients A. However, note that the sheaves L(n, χ, A) and L(n, χ^{−1}, A) that occur in the modules on both sides are different. For this and other reasons we now modify the cup product pairing (19) slightly.

Fix an idele ν ∈ 𝔸_{F}^{×}, such that νO_{F} = 𝔑 and (ν)_{∞} = 1, and define τ = ( \begin{matrix}0 & −1 \\ ν & 0 \end{matrix} ) ∈ G(𝔸). Then τ induces an isomorphism

τ : H_{p}^{d}(Y (𝔑), L(n, χ, A)) → H_{p}^{d}(Y (𝔑), L(n, χ^{−1}, A))  (31)

defined as follows. Let s_{i} = ( \begin{matrix}1 & 0 \\ 0 & a_{i}^{−1} \end{matrix} ), for the strict class group representatives a_{i} (i = 1, …, h) fixed in Section 2.2. Then

G(𝔸) = ∐_{i=1}^{h} G(ℚ) s_{i} K_{0}(𝔑) G_{∞+}

and, since Γ_{i}(𝔑) = G(ℚ) ∩ s_{i}K_{0}(𝔑)s_{i}^{−1}G_{∞_{+}}, the decomposition (2) remains unchanged using the representatives s_{i}.

Fix i. Since a_{i}^{−1}ν ∈ 𝔸_{F}^{×}, we may pick j ∈ {1, …, h}, q_{i} ∈ F_{+}^{×} and

w_{i} ∈ Ô_{F}^{×} such that a_{i}^{−1}ν = q_{i}a_{j}w_{i}. Let α_{i} = ( \begin{matrix}0 & −1 \\ q_{i} & 0 \end{matrix} ), and u_{i} = ( \begin{matrix}w_{i} & 0 \\ 0 & 1 \end{matrix} ). Then

s_{i}τ = α_{i}t_{j}u_{i} and α_{i}^{−1}Γ_{i}(𝔑)α_{i} = Γ_{j}(𝔑). These choices (for each i) induce the following commutative diagram

Y (𝔑) ⟶ ∐_{i=1}^{h} Γ_{i}(𝔑)\Z
↓  ↓
Y (𝔑) ⟶ ∐_{j=1}^{h} Γ_{j}(𝔑)\Z

where the first vertical map is g ↦ gτ, the second vertical map (on the i^{th} connected component) is z ↦ α_{i}^{−1}z, and the horizontal maps are, as usual, γs_{i}kg_{∞} ↦ g_{∞}(z_{0}), respectively, γ′t_{j}k′g′_{∞} ↦ g′_{∞}(z_{0}), where γ, γ′ ∈ G(ℚ), k, k′ ∈ K_{0}(𝔑), and g_{∞}, g′_{∞} ∈ G_{∞+}.

Now consider the following diagram

Γ_{i}(𝔑) × L_{i}(n, χ, A) ⟶ L_{i}(n, χ, A)
↑ ↓ ↓
Γ_{j}(𝔑) × L_{j}(n, χ^{−1}, A) ⟶ L_{j}(n, χ^{−1}, A)

where the upward pointing arrow is γ ↦ α_{i}γα_{i}^{−1}, the downward pointing arrow is P ↦ α^{−1}P, and the horizontal arrows are the usual actions of Γ_{i}(𝔑), respectively Γ_{j}(𝔑) (see the discussion surrounding equation (6)). In this diagram L_{i} = L_{i}(n, χ, A) is defined using s_{i}. Since t_{i} = a_{i}s_{i}, where a_{i} is considered as a scalar matrix, the modules L_{i} obtained using t_{i} or s_{i} are isomorphic as Γ_{i}(𝔑)-modules, since they differ by a factor of |a_{i}|_{F}^{n}. When A = O_{℘} or K_{℘}/O_{℘} we may always assume that the p-part of the idele a_{i} is trivial ((a_{i})_{p} = 1) so that the two integral structures on L_{i}, arising from the two choices of representatives, are the same.

Now, if γ ∈ Γ_{j}(𝔑), then d_{α_{i}γα_{i}^{−1}} = a_{γ}, so that χ_{𝔑}(d_{α_{i}γα_{i}^{−1}}) = χ_{𝔑}(a_{γ}). This shows that the above diagram is compatible (in the sense of [20], Chapter 7, Section 5). Thus there is an induced map on cohomology

H^{d}(Γ_{i}(𝔑), L_{i}(n, χ, A)) ↦ H^{d}(Γ_{j}(𝔑), L_{j}(n, χ^{−1}, A).

Taking the direct sum over all i, and restricting to the parabolic part, gives us the the map τ in (31). We note here that if the Γ_{i}(𝔑) are not torsion free, we still obtain the map (31) when A = O_{℘} for ℘ ∉ S_{elliptic} by the usual procedure of ‘taking invariants’.

Define a new pairing

[ , ] : H_{p}^{d}(Y (𝔑), L(n, χ, A)) ⊗ H_{p}^{d}(Y (𝔑), L(n, χ, A)) → A,  (32)

by setting

[x, y] =< x, τ(y) > .

The advantage of doing this is two-fold: for one, we may now choose x and y both in H_{p}^{d}(Y (𝔑), L(n, χ, A)). Secondly, this new pairing satisfies the following condition with respect to the Hecke operators T_{𝔪}:

[T_{𝔪}x, y] = [x, T_{𝔪}y].

Also, note that since τ preserves the rational (or integral) structure coming from the coefficients, the paring [ , ] is perfect, exactly when the original cup product pairing < , > is.

The matrix τ also induces a map

τ : S_{k,I_{F}} (𝔑, χ) → S_{k,I_{F}} (𝔑, χ^{−1}),

given by

f|_{τ}(g) = χ^{−1}(det g)f(gτ).

When f is primitive, one has:

f|_{τ} = W (f)f_{c}.  (33)

where f_{c} ∈ S_{k,I_{F}} (𝔑, χ^{−1}) is the cusp form whose Fourier coefficients satisfy c(𝔪, f_{c}) = \overline{c(𝔪, f)}, and W (f) is a complex number, with |W (f)| = 1. Note that W (f) = 1 when f is of level 1.

4 Periods of cusp forms

In this section, we define ‘integral’ periods attached to cusp forms f ∈ S_{k,I_{F}} (𝔑, χ). These are defined by measuring the difference in the two different integral structures put on the cuspidal cohomology groups; one of which arises from the coefficients of these cohomology groups, and the other, via the Eichler-Shimura-Harder isomorphism, from the integral structure defined by Fourier expansion of cusp forms.

4.1 Eichler-Shimura-Harder isomorphism

The following isomorphism, due to Harder [9], and worked out explicitly by Hida in [16] (see also Theorem 1.1 in [15]) is a generalization of the classical Eichler-Shimura isomorphism relating the space of cusp forms to cuspidal cohomology groups:

δ_{i} : ⊕_{J} S_{k,J}(Γ_{i}(𝔑), χ−1_{𝔑}) →^{∼} H_{cusp}^{d}(Γ_{i}(𝔑)\Z, L_{i}(n, χ, ℂ)),

where the sum is over all holomorphy types J. Here we recall that the sheaf L_{i}(n, χ, ℂ) depends only on χ_{𝔑}, the restriction of χ to Ô_{F}^{×}. Taking the direct sum over all i, and using (4), we obtain the isomorphism

δ = ⊕δ_{i} : ⊕_{χ} ⊕_{J} S_{k,J}(𝔑, χ) →^{∼} H_{cusp}^{d}(Y (𝔑), L(n, χ, ℂ)).

Note that there are natural actions of Cl_{F}^{+}, the strict class group of F, the Hecke operators T_{𝔪}, and the group C = K_{∞}/K_{∞+} of complex conjugations, on both sides. All these actions commute, and the isomorphism δ is equivariant with respect to these actions.

When n > 0, or d is odd and n = 0, the cuspidal cohomology groups appearing above are equal to the parabolic cohomology groups, and from now onwards they are identified. But when d is even and n = 0, there are ‘invariant cohomology classes’, denoted by Inv, which are parabolic but not cuspidal: in this case H_{cusp}^{d} is the orthogonal compliment of Inv in H_{p}^{d}. Thus in this case we will have to modify our argument slightly (see Remark 6 below).

4.2 Periods attached to f

4.2.1 Eichler-Shimura periods

Fix a primitive holomorphic cusp form f ∈ S_{k,I_{F}} (𝔑, χ) (that is J = I_{F}). Let

Ĉ = {ϵ : K_{∞}/K_{∞+} → {±1}}.

Let us write [f], respectively [ϵ], respectively [χ], for an eigenspace with respect to the Hecke algebra homomorphism corresponding to f, respectively, the character ϵ ∈ Ĉ, respectively the character χ thought of as a character of Cl_{F}^{+}. Then if δ_{ϵ} denotes δ followed by projection to the ϵ eigenspace, we have the following isomorphism induced by δ

δ_{ϵ} : S_{k,J}(𝔑, χ)[f] →^{∼} H_{p}^{d}(Y (𝔑), L(n, χ, ℂ))[f, ϵ, χ].

It is well known (see [16]) that if A is a p.i.d., then the maximal torsion-free quotient of the A-module

H_{p}^{d}(Y (𝔑), L(n, χ, A))[f, ϵ, χ]

is free of rank 1 over A. We let η(f, ϵ, A) denote a generator. Recall that K is a large Galois extension of ℚ that contains all the conjugates of F as well as all the Hecke fields. Let now A = O_{(℘)} be a valuation ring of K at the prime ℘. Then we define the periods

Ω(f, ϵ, O_{(℘)}) ∈ ℂ^{×}/O_{(℘)}^{×}

by

δ_{ϵ}(f) = Ω(f, ϵ, O_{(℘)}) η(f, ϵ, O_{(℘)}).  (34)

These periods were originally defined by Eichler (in the case F = ℚ) and Shimura, and they govern the behaviour of the standard L-function attached to f within the critical strip. They will also play a role in describing the value of the adjoint L-function attached to f which we will describe in Section 5 below.

4.2.2 Hida periods

We now define another period that was used by Hida in [10] in the case when F = ℚ. However we will not use this period in this paper, and we only mention it for completeness sake. In any case, we show that it is related in a simple manner to the Eichler-Shimura periods defined in Section 4.2.1.

We have, in the notation of Section 4.2.1, the embeddings O_{(℘)} → K → ℂ, which induce maps

H_{p}^{d}(Y (𝔑), L(n, χ, O_{(℘)}))[f, χ] → H_{p}^{d}(Y (𝔑), L(n, χ, ℂ))[f, χ],

whose image we call L_{f}(O_{(℘)}). Now the latter space is 2^{d} dimensional over ℂ, and has as basis the set {δ(f_{J}) | J ⊂ I_{F} }, where f_{J} ∈ S_{k,J}(𝔑, χ) is the cusp form defined by

c(I_{F} , J) · f = f_{J},

where c(I_{F} , J) ∈ C (cf. Section 2.6). On the other hand, L_{f}(O_{(℘)}) is a free O_{(℘)}-module of the same rank, say spanned by the basis {η_{j}(f, O_{(℘)}) | j = 1, …, 2^{d}}. Let U(f, O_{(℘)}) be the change of basis matrix defined via

(δ(f_{J}))_{J⊂I_{F}} = U(f, O_{(℘)}) (η_{j}(f, O_{(℘)}))_{j=1,...,2^{d}},

and let

Ω(f, O_{(℘)}) = det(U(f, O_{(℘)})) ∈ ℂ^{×}/O_{(℘)}^{×},

denote its determinant. The period Ω(f, O_{(℘)}) is sometimes referred to as the Hida period. The following Lemma relates it to the Eichler-Shimura periods:

Lemma 3 Say p ≠ 2. Then

Ω(f, O_{(℘)}) = ∏_{ϵ∈Ĉ} Ω(f, ϵ, O_{(℘)}),

in ℂ^{×}/O_{(℘)}^{×}.

Proof. Up to a power of 2, which can be ignored since we are working in ℂ^{×}/O_{(℘)}^{×} and the residue characteristic p of O_{(℘)} is not equal to 2, we may replace the basis (η_{j}(f, O_{(℘)}))_{j=1,...,2^{d}} by (η(f, ϵ, O_{(℘)}))_{ϵ∈Ĉ} in the definition of Ω(f, O_{(℘)}). The lemma then follows easily. □

5 Adjoint L-functions at s = 1

In this section we introduce the imprimitive adjoint L-function attached to a primitive holomorphic cusp form f ∈ S_{k,I_{F}} (𝔑, χ). Then we quote a well known formula of Shimura relating the value of this L-function at s = 1 to the Petersson inner product of f with itself.

5.1 Adjoint L-functions

We define the imprimitive adjoint L-function L(s, Ad(f)) via its Euler product, following Section 7 of [18]. We note, however, that there are more conceptual descriptions of the adjoint L-function as

• the motivic L-function L(s, Ad(ρ_{f})) attached to the adjoint representation of Gal(\overline{ℚ}/F) on traceless two by two matrices, arising from the two dimensional representation ρ_{f} of Gal(\overline{ℚ}/F) associated to f by Shimura, Ohta, Wiles, Carayol, Taylor and Blasius-Rogawski, or,

• the Langlands L-function L(s, π, r) where π = ⊗π_{𝔭} is the cuspidal automorphic representation of G(𝔸) corresponding to f, and

r : GL_{2}(ℂ) → SL_{3}(ℂ)

is the adjoint representation of the corresponding L-group, GL_{2}(ℂ).

First, fix a prime 𝔭 of F, such that 𝔭 | 𝔑̸. Define α_{𝔭} and β_{𝔭} by

1 − c(𝔭, f)X − χ(𝔭)N_{F/ℚ}(𝔭)X^{2} = (1 − α_{𝔭}X)(1 − β_{𝔭}X).

For such 𝔭 | 𝔑̸ let

L_{𝔭}(s, Ad(f)) = (1 − \frac{α_{𝔭}}{β_{𝔭}} N(𝔭)^{−s})(1 − N(𝔭)^{−s})(1 − \frac{β_{𝔭}}{α_{𝔭}} N(𝔭)^{−s}).

Now suppose that 𝔭 | 𝔑, and that the corresponding local component π_{𝔭} of the automorphic representation π corresponding to f is either a minimal principal series, or a minimal special representation. Here minimal means that the conductor of π_{𝔭} is minimal among all twists π_{𝔭} ⊗ ξ_{𝔭}, where ξ_{𝔮} : F_{𝔭} → ℂ^{×} is a quasi-character. Then for such 𝔭|𝔑 let

L_{𝔭}(s, Ad(f)) = \begin{cases}1 − N(𝔭)^{−s} & if π_{𝔭} is principal series and minimal, \\ 1 − N(𝔭)^{−s−1} & if π_{𝔭} is special and minimal. \end{cases} For all other 𝔭 | 𝔑, set L_{𝔭}(s, Ad(f)) = 1. Now let

L(s, Ad(f)) = ∏_{𝔭} L_{𝔭}(s, Ad(f))^{−1}

denote the imprimitive adjoint L-function attached to f.

This is the definition of the imprimitive adjoint L-function given in Equation (7.1) of [18], after correcting for the following typographical errors. In [18], page 241, line 14 there should be an extra Euler factor of (1 − X), and here L_{𝔮}(X) should be defined not just for π_{𝔮} spherical, but for all minimal principal series representations π_{𝔮}. In particular, if π_{𝔮} is a non-spherical minimal principal series representation, then β_{𝔮} = 0, and we recover the Euler factor defined above. Line 15 of the same page should define L_{𝔮}(X) only for minimal special representations π_{𝔮}.

We also define the associated Γ-factor by

Γ(s, Ad(f)) = Γ_{ℂ}(s + k − 1)^{d}Γ_{ℝ}(s + 1)^{d},

where Γ_{ℂ}(s) = (2π)^{−s}Γ(s) and Γ_{ℝ}(s) = π^{−s/2}Γ(s/2) denote the usual real and complex Γ-factors.

5.2 Residue formula

Let f^{c} denotes the cusp form whose Fourier coefficients satisfy c(𝔪, f^{c}) = \overline{c(𝔪, f)}. It is well known that L(s, f, f^{c}), the Rankin-Selberg L-function of f and f^{c}, has a simple pole at s = k. Following Rankin, its residue was first computed by Shimura in [21] by analyzing the integral expression for L(s, f, f^{c}). Since this residue is essentially L(1, Ad(f)), one obtains (see Theorem 7.1 of [18]):

Γ(1, Ad(f))L(1, Ad(f)) = \frac{2^{a}}{N_{F/ℚ}(𝔑)h_{F} D_{F}^{b}} · (f, f),  (35)

where h_{F} is the class number of F, D_{F} is the discriminant of F, a, b are specifiable integers, and (f, f) is the Petersson inner-product of f with itself. For the definition of (f, f) the reader is referred to Equation (7.1) of [14]. We note that the Petersson inner-product that actually occurs in the residue formula (35) above is (f^{u}, f^{u}), where f^{u} is the ‘unitarization of f’ defined just below (7.2b) in [14]; this is not serious since the two inner products differ by a known power of the discriminant (see [14], (7.2c)). Thus, any ambiguity has been accounted for by introducing the integer b.

6 Main Theorem

Let f ∈ S_{k,I_{F}} (𝔑, χ) be a primitive holomorphic cusp form, with Fourier coefficients c(𝔪, f). Let ℘ be a prime of K. We say that ℘ is a congruence prime for f if there exists another normalized cusp form g ∈ S_{k,I_{F}} (𝔑, χ), that is an eigenform, having Fourier coefficients c(𝔪, g), such that

c(𝔪, f) ≡ c(𝔪, g) (mod ℘),

for all ideals 𝔪 ⊂ O_{F}.

Let S_{F} = {℘ | p : p | D_{F} · h_{F} }, let S_{level} = {℘ | p : p | N_{F/ℚ}(𝔑)}, and when d = [F : ℚ] is even and n = 0, let S_{invariant} be defined as in Remark 6 below. Set

S = S_{duality} ∪ S_{level} ∪ S_{F} ∪ S_{invariant}.  (36)

Note that S is a finite set of primes of K. We have:

Theorem 5 (Congruence number formula) Say that ℘ is a prime of K and that ℘ ∉ S, and say ϵ ∈ Ĉ. If

℘ | \frac{W (f) Γ(1, Ad(f))L(1, Ad(f))}{Ω(f, ϵ, O_{(℘)})Ω(f, −ϵ, O_{(℘)})},

then ℘ is a congruence prime for f.

Proof. For a Dedekind domain O_{M} whose quotient field M contains K, let L(O_{M}) denote the image of the parabolic cohomology group

H_{p}^{d}(Y (𝔑), L(n, χ, O_{M}))[χ]

in the vector space

V (M) := H_{p}^{d}(Y (𝔑), L(n, χ, M))[χ].  (37)

For us M will be either K, with O_{M} = O_{K} or a valuation ring O_{(℘)}; or M will be a completion K_{℘} of K, with O_{M} = O_{℘}.

Note that L(O_{M}) is a lattice in V (M), and we let d(L(O_{M})) denote its discriminant with respect to the non-degenerate modified cup product pairing [ , ] on V (M) (cf. Equation 32). Recall that

℘ | d(L(O_{M})) ⟺ ℘ ∈ Supp(\frac{L^{∗}(O_{M})}{L(O_{M})}),  (38)

where L^{∗}(O_{M}) denotes the lattice which is dual to L(O_{M}) with respect to the pairing [ , ].

Note that V (M) decomposes as

V (M) = V (M)[f] ⊕ W (M),

where

V (M)[f] = H_{p}^{d}(Y (𝔑), L(n, χ, M))[f, χ]

is the eigenspace of the action of Hecke algebra corresponding to f (and χ), and W (M) is the orthogonal compliment of V (M)[f] with respect to the pairing [ , ].

Now define

L_{f}(O_{M}) := L(O_{M}) ∩ V (M)[f], L_{W} (O_{M}) := L(O_{M}) ∩ W (M);
M_{f}(O_{M}) := π_{V [f]}(L(O_{M})), M_{W} (O_{M}) := π_{W} (L(O_{M})),

where π_{V [f]} : V → V [f] and π_{W} : V → W are the two projection maps. Note that

L_{f} ⊂ M_{f} respectively L_{W} ⊂ M_{W}

are lattices in V [f], respectively, lattices in W. Moreover, one has the following isomorphisms of (finite) O_{M}-modules:

\frac{M_{f}}{L_{f}} ⟵^{π_{V [f]}} \frac{L}{L_{f} ⊕ L_{W}} ⟶^{π_{W}} \frac{M_{W}}{L_{W}} .  (39)

Let us compute the discriminant d(L_{f}(O_{(℘)})). Following the notation of Section 4.2.1, we have (cf. proof of Lemma 3)

d(L_{f}(O_{(℘)})) = det([η_{i}(f, O_{(℘)}), η_{j}(f, O_{(℘)})])
= det([η(f, ϵ, O_{(℘)}), η(f, ϵ′, O_{(℘)})]),  (40)

where we really take the principal ideal generated by the determinants. For σ ∈ I_{F}, let c_{σ} = c(I_{F} , I_{F} \ σ) ∈ C. Let a, b ∈ H_{p}^{d}(Y (𝔑), L(n, χ, ℂ)). Then, we have

[c_{σ} · a, b] = −[a, c_{σ} · b].  (41)

Let −ϵ ∈ Ĉ be the character of C defined by (−ϵ)(c_{σ}) = −(ϵ(c_{σ})) for all c_{σ} ∈ C. It follows immediately from (41) that

[η(f, ϵ, O_{(℘)}), η(f, ϵ′, O_{(℘)})] = 0  (42)

unless ϵ′ = −ϵ. Since the residue of characteristic of ℘ is odd we may write

L_{f}(O_{(℘)}) = ⨁_{ϵ∈Ĉ/±1} L_{f,ϵ}(O_{(℘)})

where L_{f,ϵ}(O_{(℘)}) = L_{f}(O_{(℘)}) ∩ (V (O_{(℘)})[ϵ] ⊕ V (O_{(℘)})[−ϵ]). By (42) we see that L_{f,ϵ}(O_{(℘)}) pairs trivially with L_{f,ϵ′}(O_{(℘)}) unless ϵ′ = ϵ. Moreover the determinant in (40) breaks into block diagonal form: there are 2^{d−1} two-by-two blocks of the form

(\begin{matrix}0 & [η(f, ϵ, O_{(℘)}), η(f, −ϵ, O_{(℘)})] \\ [η(f, −ϵ, O_{(℘)}), η(f, ϵ, O_{(℘)})] & 0 \end{matrix}) ,

having determinant (see Equation (34))

d(L_{f,ϵ}(O_{(℘)})) = −\frac{[δ_{ϵ}(f), δ_{−ϵ}(f)]^{2}}{(Ω(f, ϵ, O_{(℘)})Ω(f, −ϵ, O_{(℘)}))^{2}} .

Now let F_{∞} = ∏_{σ∈I_{F}} c_{σ}. Then

[δ_{ϵ}(f), δ(f)_{−ϵ}] = 2^{d} < F_{∞}(δ(f)), τ(δ(f)) >
= 2^{d} W (f) < F_{∞}(δ(f)), δ(f_{c}) >  by (33)
= 2^{d} W (f) (f, f),

which, by (35), is just

= 2^{d−a} W (f) N_{F/ℚ}(𝔑) h_{F} D_{F}^{b} Γ(1, Ad(f)) L(1, Ad(f)).

But, since ℘ ∉ S_{F} ∪ S_{level}, and the residue characteristic of ℘ is odd, we may ignore the factor 2^{d−a}N_{F/ℚ}(𝔑) h_{F} D_{F}^{b}. Therefore

d(L_{f,ϵ}(O_{(℘)})) = (\frac{W (f)Γ(1, Ad(f))L(1, Ad(f)}{Ω(f, ϵ, O_{(℘)}) Ω(f, −ϵ, O_{(℘)})})^{2}.

Thus, putting things together, we obtain:

d(L_{f}(O_{(℘)})) = ∏_{ϵ∈Ĉ/±1}(\frac{W (f)Γ(1, Ad(f))L(1, Ad(f)}{Ω(f, ϵ, O_{(℘)}) Ω(f, −ϵ, O_{(℘)})})^{2}.  (43)

Note that

℘ | d(L_{f}(O_{K})) ⟺ ℘ | d(L_{f}(O_{(℘)})) ⟺ ℘ | d(L_{f}(O_{℘})).  (44)

So we may, and now do, work over the completion K_{℘}.

Lemma 4 Assume ℘ ∉ S. Then

L_{f}^{∗}(O_{℘}) = M_{f}(O_{℘}).

Proof. Since ℘ ∉ S_{duality}, we see by Theorem 3, that the lattice L which is the image of H_{p}^{d}(Y (𝔑), L(n, χ, O_{℘})) in H_{p}^{d}(Y (𝔑), L(n, χ, K_{℘})) is self-dual under the pairing [ , ]. For g ∈ Cl_{F}^{×}, we have [ga, b] = [a, gb], for all a, b ∈ H_{p}^{d}(Y (𝔑), L(n, χ, K_{℘})). This shows that, for the character χ of Cl_{F}^{×} that we have fixed, [ , ] induces a pairing on the χ eigenspace of the cohomology. Moreover, ℘ ∉ S_{F} implies that ℘ | h̸_{F}, so that L(O_{℘}), which is the intersection of the lattice L with the χ eigenspace of the cohomology, is still self dual. It is now a simple exercise in linear algebra to check that the Lemma follows (cf. [10], Equation 4.6). □

By (38), and Lemma 4, we have

℘ | d(L_{f}(O_{℘})) ⟺ ℘ ∈ Supp(\frac{L_{f}^{∗}(O_{℘})}{L_{f}(O_{℘})})

⟺ ℘ ∈ Supp(\frac{M_{f}(O_{℘})}{L_{f}(O_{℘})})

⟺ ℘ ∈ Supp(\frac{M_{f}(O_{K})}{L_{f}(O_{K})}).  (45)

Thus, by (43) and (45), we conclude that for each ϵ ∈ Ĉ,

℘ | \frac{W (f)Γ(1, Ad(f))L(1, Ad(f)}{Ω(f, ϵ, O_{(℘)}) Ω(f, −ϵ, O_{(℘)})} ⟹ ℘ ∈ Supp(\frac{M_{f}(O_{K})}{L_{f}(O_{K})}).  (46)

Now let k = O_{K}/℘ be the residue field attached to ℘. Let h_{f}, respectively h_{W}, denote the images of the Hecke algebra acting on the subspaces V [f], respectively W. By (46) we have that M_{f}(O_{K})/L_{f}(O_{K})) ⊗ k ≠ 0. Now h_{f} acts on (M_{f}(O_{K})/L_{f}(O_{K})) ⊗ k via the character

h_{f} → k
T_{𝔪} ↦ c(𝔪, f) (mod ℘).

By (39) we have

\frac{M_{f}(O_{K})}{L_{f}(O_{K})} ⊗ k = \frac{M_{W} (O_{K})}{L_{W} (O_{K})} ⊗ k,

so that there exists a corresponding homomorphism

h_{W} → k
T_{𝔪} ↦ c(𝔪, f) (mod ℘).

By enlarging K (and thus O_{K}) if necessary we may assume that this homomorphism lifts to a character h_{W} → O_{K}. This shows that there exists a normalized eigenform g ≠ f, such that

c(𝔪, f) ≡ c(𝔪, g) (mod ℘),

for all 𝔪, as desired. □

Remark 6 As mentioned earlier, the parabolic and cuspidal cohomology groups coincide, except when the degree d of F/ℚ is even, and the weight n = 0. In this case we need to modify the argument above slightly. Recall that in this case:

H_{p}^{d}(Y (𝔑), L(0, χ, ℂ)) = H_{cusp}^{d}(Y (𝔑), L(0, χ, ℂ)) ⊕ Inv(ℂ),

where Inv(ℂ) denote the invariant forms. Note further that Inv(ℂ) is the orthogonal compliment to H_{cusp}^{d}(Y (𝔑), L(0, χ, ℂ)) under the pairing [ , ]. Now there is a natural K-structure on H_{cusp}^{d} coming from the Drinfeld-Manin Theorem (see, for instance, [5], Section 7.6), and this is induced by the natural K-structure on H_{p}^{d}. Write Inv(K) for the orthogonal compliment of H_{cusp}^{d}(Y (𝔑), L(0, χ, K)) in H_{p}^{d}(Y (𝔑), L(0, χ, K)) under [ , ]. Then we see that Inv(K) ⊗ ℂ = Inv(ℂ). Now define

H_{cusp}^{d}(Y (𝔑), L(0, χ, O_{(℘)}) =
H_{cusp}^{d}(Y (𝔑), L(0, χ, K)) ∩ H_{p}^{d}(Y (𝔑), L(0, χ, O_{(℘)})).

It is easy to check that the self duality of H_{cusp}^{d}(Y (𝔑), L(0, χ, O_{(℘)}) under [ , ] follows from the self duality of H_{p}^{d}(Y (𝔑), L(0, χ, O_{(℘)})) under [ , ], at the cost of throwing away finitely many additional primes, which we denote by S_{invariant}. Outside the set S_{invariant}, the proof of Theorem 5 goes through as before.

6.1 Some explicit corollaries

We now utilize the computations made in Section 3.4 of S_{∂} to make Theorem 5 more explicit. Recall that we may write (cf. (20) and (36)):

S = S_{weight} ∪ S_{level} ∪ S_{elliptic} ∪ S_{F} ∪ S_{invariant} ∪ S_{∂}.

6.1.1 F = ℚ

Write N for the positive integer that generates the ideal 𝔑 ⊂ ℤ. Then we have: S_{weight} = {℘ | p : p ≤ k − 2}, S_{level} = {℘ | N}, S_{elliptic} = {℘ | 6}, and S_{∂} = S_{weight}∪S_{level} by Proposition 3 in Section 3.4.1. Also S_{F} = S_{invariant} = ∅.

Thus we see that Theorem 5 just reduces to Theorem 1 of the Introduction (proved by Hida in [10]).

6.1.2 Real quadratic fields

Now suppose that F is a real quadratic field of strict class number 1, and assume that the level 𝔑 = 1 (cf. Assumption 1). In particular since the level is 1, W (f) = 1. Under these assumptions we have S_{weight} = {℘ | p : p ≤ k−2} and S_{elliptic} = {℘ | 30}, but now S_{F} = {℘ | D_{F} } and S_{level} = ∅. Moreover, when k = 2, S_{∂} is given by Proposition 4.

When k > 2, Corollary 1 in Section 3.4.2 (see also Remark 3) tells us what the torsion in the ordinary part of the boundary cohomology can be. Note that the proof of Theorem 5 adapts to the ordinary part of the cohomology. Indeed, since the boundary exact sequence (15) remains valid for the ordinary part of the cohomology, one gets an analog of the duality theorem (Theorem 3) for the ordinary part of parabolic cohomology. Thus, if in the proof of Theorem 5, one replaces the vector space (37) by its ordinary part, one obtains an analog of Theorem 5 for primitive forms f that are ordinary at ℘, namely forms for which ℘ | a̸(p, f).

Consequently, we obtain:

Corollary 2 Let F be a real quadratic field of strict class number 1. Let ϵ_{1} denote a generator of the group of totally positive units of F. Let f be a primitive form of level 1 and weight (k, k). Let ℘ | p be a prime of K such that p > k − 2, p | 3̸0 · D_{F} · N_{F/ℚ}(ϵ_{1}^{k−1} − 1) and

\begin{cases}℘ ∉ S_{invariant}, & if k = 2, \\ ℘ is ordinary for f, & if k > 2. \end{cases}

Then, if

℘ | \frac{Γ(1, Ad(f)) L(1, Ad(f))}{Ω(f, ϵ)Ω(f, −ϵ)},

℘ is a congruence prime for f.

Remark 7 Though the restriction ℘ ∉ S_{invariant} when k = 2 is somewhat unpleasant, we remark that this assumption can be avoided in the case when the level is divisible by pO_{F}, by restricting (as described in weight k > 2 above) to the ordinary part of the cohomology. Indeed, since the invariant forms are invariant under the action of SL_{2}(F_{∞}) (cf. [15], (1.19)), the Hecke operator T_{p} acts on the invariant forms by degree, which is a nontrivial power of p. Consequently, the ordinary idempotent kills the invariant forms.

6.1.3 Totally real cubic fields

Let F be a totally real cubic field. We keep Assumption 1 of the real quadratic case, but restrict to the weight (2, 2, 2) situation. All the bad sets of primes are the same as in the real quadratic situation except that now S_{elliptic} = {℘ | 42}, S_{invariant} = ∅, and S_{∂} is described by Proposition 5 in Section 3.4.3. We obtain:

Corollary 3 Let F be a totally real cubic field of strict class number 1 and let ϵ_{1} and ϵ_{2} denote generators of the group of totally positive units of F.

Let U_{F} be the product of all the primes that divide both N_{F/ℚ}(ϵ_{1} − 1) and N_{F/ℚ}(ϵ_{2} − 1). Let f be a normalized newform, that is a common eigenform of the Hecke operators of level 1 and weight (2, 2, 2). Say ℘ | p be a prime of K such that

p | 4̸2 · D_{F} · U_{F} .

Then, if

℘ | \frac{Γ(1, Ad(f)) L(1, Ad(f))}{Ω(f, ϵ)Ω(f, −ϵ)},

℘ is a congruence prime for f.

6.1.4 Totally real fields of arbitrary degree.

Finally, let us consider more generally, the case when F is a totally real field of odd degree d. We make this parity assumption since, when d is odd, S_{invariant} = ∅. All the other assumptions are as in the cubic case. In particular we assume that the weight is (2, 2, …, 2). All the bad sets of primes are the same as in the cubic situation except for S_{elliptic} and S_{∂} (the latter is described by Proposition 6 in Section 3.4.4). We obtain:

Corollary 4 Let F be a totally real field of strict class number 1, odd degree d, and discriminant D_{F}. Let ϵ_{1}, ϵ_{2}, …, ϵ_{d−1} be a basis of the group of totally positive units of F, and let U_{F} denote the product of the primes which divide each N_{F/ℚ}(ϵ_{j} − 1), for j = 1, …, d − 1. Let E_{F} denote the product of all the primes p such that the maximal totally real subfield of the p^{th} cyclotomic field is contained in F.

Let f be a normalized newform, and a common eigenform of the Hecke operators, of level 1 and weight (2, 2, …, 2). Then there exits a large number field K such that the following is true: for each prime ℘ | p of K with p | E̸_{F} · D_{F} · U_{F}, if

℘ | \frac{Γ(1, Ad(f)) L(1, Ad(f))}{Ω(f, ϵ)Ω(f, −ϵ)}

then ℘ is a congruence prime for f.

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