CONGRUENCES AMONG MODULAR FORMS ON U(2, 2) AND THE BLOCH-KATO CONJECTURE

KRZYSZTOF KLOSIN^{1}

Abstract. Let k be a positive integer divisible by 4, ℓ > k a prime, and f an elliptic cuspidal eigenform (ordinary at ℓ) of weight k − 1, level 4, and non-trivial character. Let ρ_{f} be the ℓ-adic Galois representation attached to f. In this paper we provide evidence for the Bloch-Kato conjecture for the motives ad^{0} M_{0}(−1) and ad^{0} M_{0}(2), where M_{0} is the motif attached to f. More precisely, let L(Symm^{2} f, s) denote the symmetric square L-function of f. We prove that (under certain conditions) ord_{ℓ}(L^{alg}(Symm^{2} f, k)) ≤ ord_{ℓ}(#S), where S is the (Pontryagin dual of the) Selmer group attached to the Galois module ad^{0} ρ_{f} |_{G_{K}} (−1), and K = Q(\sqrt{−1}). Our method uses an idea of Ribet [45] in that we introduce an intermediate step and produce congruences between CAP and non-CAP modular forms on the unitary group U(2, 2).

1. Introduction

Let ℓ > 2 be a prime and let λ be a prime of \overline{Q} lying over ℓ. The idea of linking up λ-divisibility of an L-value with the existence of congruences among modular forms and using these congruences to construct elements in a Selmer group goes back to Ribet and his proof of the converse to Herbrand’s theorem [45]. In that paper a special value L(χ, −1) of an even Dirichlet character χ is realized as a constant term of an Eisenstein series E_{χ}. If L(χ, −1) ≡ 0 (mod λ), one shows there exists a cuspidal Hecke eigenform f whose Hecke eigenvalues are congruent to those of E_{χ} (mod λ) and as a result of that the mod λ Galois representation \overline{ρ}_{f} attached to f is reducible (a consequence of the congruence) but can be chosen to be nonsemisimple (a consequence of the irreducibility of the λ-adic Galois representation ρ_{f}), thus giving rise to a non-split extension of one-dimensional Galois modules over \overline{F}_{ℓ}. This extension can be interpreted as a non-zero element in a certain piece (determined by χ) of the class group of Q(μ_{ℓ}).

2 CONGRUENCES AMONG MODULAR FORMS ON U(2, 2)

This strategy can be phrased in the language of automorphic representations suggesting ways to generalize it to other situations. Let π be an automorphic representation of an algebraic group M over Q. Realize M as a Levi subgroup in a maximal parabolic subgroup of a larger algebraic group G over Q and lift π (e.g., by inducing) to an automorphic representation Π of G(A). Assuming one knows how to attach λ-adic Galois representations to automorphic representations of G(A), the one attached to Π will be reducible and semisimple. If λ divides a certain L-value L(π), construct a representation Π′ of G(A), whose Hecke eigenvalues are congruent to those of Π mod λ and whose λ-adic Galois representation is irreducible. These two conditions (respectively) ensure that the mod λ Galois representation attached to Π′ is reducible and non-semisimple, thus giving rise to some non-split extension of Galois modules which one interprets as lying in an appropriate Selmer group related to π. In Ribet’s case, M = GL_{1} × GL_{1}, G = GL_{2}, π = χ ⊗ 1, Π (resp. Π′) is the automorphic representation of GL_{2}(A) attached to E_{χ} (resp. f).

1Cornell University, Department of Mathematics, 310 Malott Hall, Ithaca, NY 14853-4201, USA

Date: April 24, 2008.

Some versions of this approach have been applied by several authors: Mazur, Wiles, Bellaïche-Chenevier, Skinner-Urban, Brown, Berger ([41, 60, 1, 9, 3])) to give lower bounds in terms of L-values on the orders of Selmer groups. The difficult point is the construction of Π′ and different cases require different methods to tackle that point.

Let K = Q(i), f ∈ S_{k−1} (4, (\frac{−4}{·})), a normalized cuspidal Hecke eigenform, ℓ > k a prime such that f is ordinary at ℓ. Write L^{int}(Symm^{2} f, k) for the value at k of the symmetric square L-function of f divided by a suitable “integral” period. In this article we implement the above strategy with M = Res_{K/Q}(GL_{2/K}), G = U(2, 2) - a quasi-split unitary group associated with the extension K/Q, π = base change to K of the automorphic representation associated to f, and L(π) = L^{int}(Symm^{2} f, k). This will allow us to construct elements in the Selmer group of V := ad^{0} ρ_{f}|_{G_{K}} (−1), where ρ_{f} is the λ-adic Galois representation attached to f, G_{K} = Gal(\overline{K}/K), and −1 denotes a Tate twist. We will now describe the construction of Π = the lift of π to G(A) and of the representation Π′.

The representation Π is obtained by lifting f to a Hecke eigenform F_{f} ∈ S_{k}, where S_{k} is the space of (weight k and level 1) hermitian modular forms as defined by Braun [6, 7, 8], using the Maass lifting constructed by Kojima, Gritsenko and Krieg [36, 23, 37]. Denote by S_{k}^{M} ⊂ S_{k} the image of the Maass lift. It is known that the eigenvalues of eigenforms in S_{k} lie in a number field. In fact we always choose a sufficiently large finite extension E of Q_{ℓ} and fix embeddings \overline{Q} ↪ \overline{Q}_{ℓ} ↪ C, so we can view all the algebraic numbers of our interest as lying inside the same field E. From now on λ will denote a uniformizer of E and O its valuation ring. Assuming L^{int}(Symm^{2} f, k) ≡ 0 (mod λ), we need to construct a hermitian modular eigenform F′ ∈ S_{k} orthogonal to the Maass space S_{k}^{M} whose Hecke eigenvalues are congruent to those of F_{f} (mod λ). The form F′ will give rise to the representation Π′ as above. Indeed, the λ-adic Galois representation attached to F_{f} is reducible and semisimple of the form ρ_{f} ⊕(ρ_{f} ⊗ϵ), where ϵ is the ℓ-adic cyclotomic character, while it is conjectured that the λ-adic Galois representation attached to an eigenform orthogonal to the Maass space is irreducible. In proving a bound on the Selmer group we will need to assume this conjecture (see Theorem 1.2).

The construction of F′ is carried out in several steps. Note that, unlike in Ribet’s case, our lift F_{f} is a cusp form, so there is no “constant term” which would “naturally” contain the L-value L^{int}(Symm^{2} f, k). Let χ be a Hecke character of K of infinity type (z/|z|)^{−t} with −k ≤ t ≤ −6. We first define a “nice” hermitian modular form Ξ which is essentially a product of a hermitian Siegel Eisenstein series D and a hermitian theta series θ_{χ} depending on the character χ. Using some results of Shimura [49, 50, 51] on algebraicity of Fourier coefficients of hermitian Eisenstein series we show that Ξ has λ-integral Fourier coefficients. We write

(1.1)  Ξ = C_{F_{f}} F_{f} + F with C_{F_{f}} = (∗)π^{−3} \frac{⟨Dθ_{χ}, F_{f}⟩}{⟨F_{f}, F_{f}⟩},

where (∗) is a λ-adic unit, F is orthogonal to F_{f}, i.e., ⟨F_{f}, F ⟩ = 0, and the inner products are the Petersson inner products on S_{k}. Using results of Shimura [51] and some formulae due to Raghavan and Sengupta [44] we are able to express the inner products by L-functions related to f. More precisely, we get

(1.2) \frac{⟨Dθ_{χ}, F_{f}⟩}{⟨F_{f}, F_{f}⟩} = (∗) η^{−1} \frac{L^{int}(BC(f), 1 + \frac{t+k}{2} , \overline{χ}\overline{ω})L^{int}(BC(f), 2 + \frac{t+k}{2} , \overline{χ}\overline{ω})}{L^{int}(Symm^{2} f, k)},

where (∗) ∈ E with ord_{λ}((∗)) ≤ 0, η is the Hida congruence ideal of f, ω is the unramified Hecke character of K of infinity type (z/|z|)^{−k}, and L^{int}(BC(f), s, ψ) is the L-function of the base change of f to K twisted by the Hecke character ψ. Note that (ignoring the factor η^{−1} for the moment) if one can find χ which makes the numerator of the right-hand side of (1.2) a λ-adic unit, then using (1.1) and λ-integrality of the Fourier coefficients of Ξ and of F_{f} we see that if λ^{n} | L^{int}(Symm^{2} f, k) then we can write C_{F_{f}} = aλ^{−n} with a a λ-adic unit and hence F_{f} ≡ −a^{−1}λ^{n}F (mod λ^{n} ). However F′′ := −a^{−1}λ^{n}F need not be orthogonal to the Maass space. To achieve this last property we modify F′′ appropriately (to obtain F′) using some results of Hida as well as deformation theory of Galois representations.

This way we obtain the first main result of the paper. To simplify the exposition here we omit a certain technical hypothesis on the character χ. For the full statement see Theorem 7.12.

Theorem 1.1. With notation as before, assume that k ∈ Z_{+}, 4 | k, ℓ > k, f is ordinary at ℓ and the mod λ representation \overline{ρ}_{f} restricted to G_{K} is absolutely irreducible. Assume there exists a Hecke character χ of K of conductor prime to ℓ, infinity type (z/|z|)^{t}, −k ≤ t ≤ −6, such that the numerator of the right-hand side of (1.2) is a λ-adic unit. If ord_{λ}(L^{int} Symm^{2} f, k) = n > 0, then there exists F′ ∈ S_{k}, orthogonal to the Maass space, such that F′ ≡ F_{f} (mod λ^{n}).

To be precise, the form F′ in Theorem 1.1 need not be a Hecke eigenform. To measure congruences between F_{f} and eigenforms orthogonal to the Maass space we introduce the notion of a CAP ideal, which is a simple modification of the Eisenstein ideal introduced by Mazur [39]. Theorem 1.1 implies that

(1.3)  ord_{ℓ}(I_{f}) ≥ ord_{ℓ}(#O/λ^{n}),

where I_{f} is the index of the CAP ideal of F_{f} inside the hermitian Hecke O-algebra acting on the orthogonal complement of S_{k}^{M} localized at the maximal ideal corresponding to F_{f}.

We emphasize that the ordinarity assumption on f is essential to our method. It is used to ensure that F′ is orthogonal to the Maass space (see section 8). For a given f it is unknown for how many primes f is ordinary, although one conjectures that for a non-CM form (which is the case here) this set of primes has Dirichlet density one. An analogous statement for elliptic curves is due to Serre [48].

Let V_{Q} be the E[G_{Q}]-module ad^{0} ρ_{f}(−1). Fix a G_{Q}-stable O-lattice T_{Q} ⊂ V_{Q} and set W_{Q} := V_{Q}/T_{Q}. For A = V, T, W, write A_{K} for A_{Q} regarded as an E[G_{K}]- module. Write Σ_{ℓ} for the set of primes of K lying over ℓ. Let Sel_{Σ_{ℓ}}(W_{K}) be the Selmer group of W_{K} in the sense of Bloch-Kato (for definition see section 9) and write S_{Σ_{ℓ}}(W_{K}) for the Pontryagin dual of Sel_{Σ_{ℓ}}(W_{K}). Then our second main theorem (which uses a result of Urban [55]) is the following (again we omit some mild hypotheses on f - see section 9):

Theorem 1.2. Assume that for every eigenform F ∈ S_{k} orthogonal to S_{k}^{M} the λ-adic Galois attached to F is absolutely irreducible. Then

ord_{ℓ}(#S_{Σ_{ℓ}}(W_{K})) ≥ ord_{ℓ}(I_{f}).

The existence of λ-adic Galois representations attached to general automorphic forms on G is at this point only conjectural, so Theorem 1.2 is conditional upon that conjecture. In fact once this conjecture is known in full strength, one should be able to remove the irreducibility assumption from Theorem 1.2, as it is expected that Galois representations attached to non-Maass cuspidal eigenforms are irreducible.

Combining Theorem 1.2 with (1.3) which is a consequence of Theorem 1.1, we obtain the following corollary.

Corollary 1.3. With the same assumptions as in Theorems 1.1 and 1.2 one has

ord_{ℓ}(#S_{Σ_{ℓ}}(W_{K})) ≥ ord_{ℓ}(#O/L^{int}(Symm^{2} f, k)).

Let us briefly explain the relation of Corollary 1.3 to the Bloch-Kato conjecture for M = ad^{0} M_{0}(−1), where M_{0} is the motif attached to f. Let Σ = {2, ℓ} and write S_{Σ}(W_{Q}) for the Pontryagin dual of Sel_{Σ}(W_{Q}) (where we require that the classes be unramified away from ℓ and crystalline at ℓ). Let L(M, s) = ∏_{p} L_{p}(s − 1) be the L-function of M defined by

(1.4) L_{p}(s) := \begin{cases}(1 − α_{p,1}α_{p,2}^{−1}p^{−s})^{−1}(1 − p^{−s})^{−1}(1 − α_{p,1}^{−1}α_{p,2}p^{−s})^{−1} & p ≠ 2 \\ (1 − p^{−s})^{−1} & p = 2, \end{cases}

where α_{p,1}, α_{p,2} are the p-Satake parameters of f. Then the Bloch-Kato conjecture for M can be phrased in the following way:

Conjecture 1.4 (Bloch-Kato). One has

(1.5)  #S_{Σ}(W_{Q}) · Tam_{ω}(T_{Q}) = \frac{L(M, 0)}{Ω_{ω}(T_{Q})}O

as fractional ideals of E, where Tam_{ω}(T_{Q}) is the ℓ-Tamagawa factor and Ω_{ω}(V_{Q}) is a certain period defined with respect to the “integral structures” T_{Q} and ω (for precise definitions see section 9.3).

Given our assumptions (which in particular include the ordinarity assumption on f) Corollary 1.3 falls short of proving that the left-hand side of (1.5) contains the right-hand side of (1.5), but provides some evidence for it. In fact (for an appropriate choice of T_{Q} and ω) the right-hand side of (1.5) equals L^{int}(Symm^{2} f, k)· O. However, the Selmer group on the left-hand side of (1.5) could potentially be smaller than S_{Σ_{ℓ}}(W_{K}) and we do not know if Tam(T_{Q}) = O. For a more detailed discussion see section 9.3. In that section we also explain the relation of Corollary 1.3 to the Bloch-Kato conjecture for the motif ad^{0} M_{0}(2) which is “dual” to ad^{0} M_{0}(−1).

The Bloch-Kato conjecture is currently known only for a few motives - see [33] for a survey of known cases. The most recent result is due to Diamond, Flach and Guo [13] and it concerns the motives ad^{0} M_{0} and ad^{0} M_{0}(1), while our result concerns the motives ad^{0} M_{0}(−1) and ad^{0} M_{0}(2). The method used in [13] is related to the method employed by Wiles and Taylor to prove the Taniyama-Shimura conjecture [61, 54] and so is different from ours.

Let us briefly discuss the organization of the paper. In section 2 we introduce notation which is used throughout the paper. In section 4 we summarize the basic facts concerning the Maass lifting f ↦ F_{f} and compute the Petersson inner product ⟨F_{f}, F_{f}⟩ in terms of L(Symm^{2} f, k). To carry out the calculations we need to first compute the residue of the hermitian Klingen Eisenstein series and this is done in section 3. The inner product ⟨Dθ_{χ}, F_{f}⟩ is computed in section 6. In section 5 we gather the necessary facts concerning the hermitian Hecke algebra which are later used in section 7, where the first main theorem (Theorem 1.1) is proved assuming existence of a certain Hecke operator which allows one to “kill” the “Maass part” of the form F′′ as above, i.e., obtain a form F′ that would be orthogonal to S_{k}^{M}. The existence of such a Hecke operator is proved in section 8 using methods of deformation theory of Galois representations. Finally in section 9 we prove Theorem 1.2 and Corollary 1.3 and discuss the relation of the latter to the Bloch-Kato conjecture.

We also want to mention that it seems possible to extend our result to an arbitrary imaginary quadratic field K provided one knows how to construct a Heckeequivariant Maass lifting in that setting. Such a construction has recently been carried out by the author [35] for K with prime discriminant (see also [30]). We hope to use that construction to prove Theorems 1.1 and 1.2 for such a K in a subsequent paper.

The author would like to thank Joel Bellaïche, Tobias Berger, Jim Brown, and Chris Skinner for many useful and inspiring conversations. We would also like to thank the anonymous referee for suggesting various improvements to the Introduction and section 9.

2. Notation and Terminology

In this section we introduce some basic concepts and establish notation which will be used throughout this paper unless explicitly indicated otherwise.

2.1. Number fields and Hecke characters. Throughout this paper ℓ will always denote an odd prime. Let i = \sqrt{−1}, K = Q(i) and let O_{K} be the ring of integers of K. For α ∈ K, denote by \overline{α} the image of α under the non-trivial automorphism of K. Set Nα := N(α) := α\overline{α}, and for an ideal 𝔫 of O_{K}, set N𝔫 := #(O_{K}/𝔫). As remarked below we will always view K as a subfield of C. For α ∈ C, \overline{α} will denote the complex conjugate of α and we set |α| := \sqrt{α\overline{α}}.

Let L be a number field with ring of integers O_{L}. For a place v of L, denote by L_{v} the completion of L at v and by O_{L,v} the valuation ring of L_{v}. If p is a place of Q, we set L_{p} := Q_{p}⊗_{Q}L and O_{L,p} := Z_{p}⊗_{Z}O_{L}. The letter v will be used to denote places of number fields (including Q and K), while the letter p will be reserved for a (finite or infinite) place of Q. For a finite p, let ord_{p} denote the p-adic valuation on Q_{p}. For notational convenience we also define ord_{p}(∞) := ∞. If α ∈ Q_{p}, then |α|_{Q_{p}} := p^{−ord_{p}(α)} denotes the p-adic norm of α. For p = ∞, | · |_{Q_{∞}} = | · |_{R} = | · | is the usual absolute value on Q_{∞} = R.

In this paper we fix once and for all an algebraic closure \overline{Q} of the rationals and algebraic closures \overline{Q}_{p} of Q_{p}, as well as compatible embeddings \overline{Q} ↪ \overline{Q}_{p} ↪ C for all finite places p of Q. We extend ord_{p} to a function from \overline{Q}_{p} into Q. Let L be a number field. We write G_{L} for Gal(\overline{L}/L). If 𝔭 is a prime of L, we also write D_{𝔭} ⊂ G_{L} for the decomposition group of 𝔭 and I_{𝔭} ⊂ D_{𝔭} for the inertia group of 𝔭. The chosen embeddings allow us to identify D_{𝔭} with Gal(\overline{L}_{𝔭}/L_{𝔭}).

For a number field L let A_{L} denote the ring of adeles of L and put A := A_{Q}. Write A_{L,∞} and A_{L,f} for the infinite part and the finite part of A_{L} respectively. For α = (α_{p}) ∈ A set |α|_{A} := ∏_{p} |α|_{Q_{p}}. By a Hecke character of A_{L}^{×} (or of L, for short) we mean a continuous homomorphism

ψ : L^{×} \ A_{L}^{×} → C^{×}

whose image is contained inside {z ∈ C | |z| = 1}. The trivial Hecke character will be denoted by 1. The character ψ factors into a product of local characters ψ = ∏_{v} ψ_{v}, where v runs over all places of L. If 𝔫 is the ideal of the ring of integers O_{L} of L such that

• ψ_{v}(x_{v}) = 1 if v is a finite place of L, x_{v} ∈ O_{L,v}^{×} and x − 1 ∈ 𝔫O_{L,v}

• no ideal 𝔪 strictly containing 𝔫 has the above property, then 𝔫 will be called the conductor of ψ. If 𝔪 is an ideal of O_{L}, then we set ψ_{𝔪} := ∏ ψ_{v}, where the product runs over all the finite places of L such that v | 𝔪. For a Hecke character ψ of A_{L}^{×}, denote by ψ^{∗} the associated ideal character. Let ψ be a Hecke character of A_{K}^{×}. We will sometimes think of ψ as a character of (Res_{K/Q} GL_{1})(A). We have a factorization ψ = ∏_{p} ψ_{p} into local characters ψ_{p} : (Res_{K/Q} GL_{1}) (Q_{p}) → C^{×}. For M ∈ Z, we set ψ_{M} := ∏_{p≠∞, p|M} ψ_{p}. If ψ is a Hecke character of A_{K}^{×}, we set ψ_{Q} = ψ|_{A^{×}}.

2.2. The unitary group. To the imaginary quadratic extension K/Q one associates the unitary similitude group

GU(n, n) = {A ∈ Res_{K/Q} GL_{n} | AJĀ^{t} = μ(A)J},

where J = [\begin{matrix}& −I_{n} \\ I_{n} \end{matrix}], with I_{n} denoting the n × n identity matrix, the bar over A

standing for the action of the non-trivial automorphism of K/Q and μ(A) ∈ GL_{1}. For a matrix (or scalar) A with entries in a ring affording an action of Gal(K/Q), we will sometimes write A^{∗} for Ā^{t} and Â for (A^{∗})^{−1}. We will also make use of the groups

U(n, n) = {A ∈ GU(n, n) | μ(A) = 1},

and

SU(n, n) = {A ∈ U(n, n) | det A = 1}.

Since the case n = 2 will be of particular interest to us we set G = U(2, 2), G_{1} = SU(2, 2) and G_{μ} = GU(2, 2).

For a Q-subgroup H of G write H_{1} for H ∩ G_{1}. Denote by G_{a} the additive group. In G we choose a maximal torus

T = {[a ; b ; â ; b̂] | a, b ∈ Res_{K/Q} GL_{1}} ,

and a Borel subgroup B = T U_{B} with unipotent radical

U_{B} = \{[\begin{matrix}1 & α & β & γ \\ & 1 & γ̄ − ᾱϕ & ϕ \\ & & 1 \\ & & −ᾱ & 1 \end{matrix}] | α, β, γ ∈ Res_{K/Q} G_{a}, ϕ ∈ G_{a}, β + γᾱ ∈ G_{a}\} .

Let

T_{Q} = \{[\begin{matrix}a \\ & b \\ & & a^{−1} \\ & & & b^{−1} \end{matrix}] | a, b ∈ GL_{1}\}

denote the maximal Q-split torus contained in T. Let R(G) be the set of roots of T_{Q}, and denote by e_{j}, j = 1, 2, the root defined by

e_{j} : [\begin{matrix}a_{1} \\ & a_{2} \\ & & a_{1}^{−1} \\ & & & a_{2}^{−1} \end{matrix}] ↦ a_{j}.

The choice of B determines a subset R^{+}(G) ⊂ R(G) of positive roots. We have

R^{+}(G) = {e_{1} + e_{2}, e_{1} − e_{2}, 2e_{1}, 2e_{2}}.

We fix a set Δ(G) ⊂ R^{+}(G) of simple roots

Δ(G) := {e_{1} − e_{2}, 2e_{2}}.

If θ ⊂ Δ(G), denote the parabolic subgroup corresponding to θ by P_{θ}. We have P_{Δ(G)} = G and P_{∅} = B. The other two possible subsets of Δ(G) correspond to maximal Q-parabolics of G:

• the Siegel parabolic P := P_{{e_{1}−e_{2}}} = M_{P} U_{P} with Levi subgroup

M_{P} = {[A ; Â] | A ∈ Res_{K/Q} GL_{2}} ,

and (abelian) unipotent radical

U_{P} = \{[\begin{matrix}1 & & b_{1} & b_{2} \\ & 1 & \overline{b}_{2} & b_{4} \\ & & 1 \\ & & & 1 \end{matrix}] | b_{1}, b_{4} ∈ G_{a}, b_{2} ∈ Res_{K/Q} G_{a}\}

• the Klingen parabolic Q := P_{{2e_{2}}} = M_{Q}U_{Q} with Levi subgroup

M_{Q} = \{[x ; a  b ; x̂ ; c  d] | x ∈ Res_{K/Q} GL_{1}, [\begin{matrix}a & b \\ c & d \end{matrix}] ∈ U(1, 1)\} ,

and (non-abelian) unipotent radical

U_{Q} = \{[\begin{matrix}1 & α & β & γ \\ & 1 & γ̄ \\ & & 1 \\ & & −ᾱ & 1 \end{matrix}] | α, β, γ ∈ Res_{K/Q} G_{a}, β + γᾱ ∈ G_{a}\}

For an associative ring R with identity and an R-module N we write N_{m}^{n} to denote the R-module of n × m matrices with entries in N. We also set N^{n} := N_{1}^{n}, and

M_{n}(N) := N_{n}^{n}. Let x = [ \begin{matrix}A & B \\ C & D \end{matrix}] ∈ M_{2n}(N) with A, B, C, D ∈ M_{n}(N). Define

a_{x} = A, b_{x} = B, c_{x} = C, d_{x} = D.

For M ∈ Q, N ∈ Z such that MN ∈ Z we will denote by D(M, N) the group G(R) ∏_{p∤∞} K_{0,p}(M, N) ⊂ G(A), where

(2.1) K_{0,p}(M, N) = {x ∈ G(Q_{p}) | a_{x}, d_{x} ∈ M_{2}(O_{K,p}) ,

b_{x} ∈ M_{2}(M^{−1}O_{K,p}), c_{x} ∈ M_{2}(MNO_{K,p})} .

If M = 1, denote D(M, N) simply by D(N) and K_{0,p}(M, N) by K_{0,p}(N). For any finite p, the group K_{0,p} := K_{0,p}(1) = G(Z_{p}) is a maximal (open) compact subgroup of G(Q_{p}). Note that if p ∤ N, then K_{0,p} = K_{0,p}(N). We write K_{0,f}(N) := ∏_{p∤∞} K_{0,p}(N) and K_{0,f} := K_{0,f}(1). Note that K_{0,f} is a maximal (open) compact subgroup of G(A_{f}). Set

K_{0,∞} := \{[\begin{matrix}A & B \\ −B & A \end{matrix}] ∈ G(R) | A, B ∈ GL_{2}(C), AA^{∗} + BB^{∗} = I_{2}, AB^{∗} = BA^{∗}\} .

Then K_{0,∞} is a maximal compact subgroup of G(R). Let

U(m) := {A ∈ GL_{m}(C) | AA^{∗} = I_{m}} .

We have

K_{0,∞} = G(R) ∩ U(4) ⟶^{∼} U(2) × U(2),

where the last isomorphism is given by

[\begin{matrix}A & B \\ −B & A \end{matrix}] ↦ (A + iB, A − iB) ∈ U(2) × U(2).

Finally, set K_{0}(N) := K_{0,∞}K_{0,f}(N) and K_{0} := K_{0}(1). The last group is a maximal compact subgroup of G(A). Let M ∈ Q, N ∈ Z be such that MN ∈ Z. We define the following congruence subgroups of G(Q):

(2.2)  Γ_{0}^{h}(M, N) := G(Q) ∩ D(M, N), Γ_{1}^{h}(M, N) := {α ∈ Γ_{0}^{h}(M, N) | a_{α} − 1 ∈ M_{2}(NO_{K})}, Γ^{h}(M, N) := {α ∈ Γ_{1}^{h}(M, N) | b_{α} ∈ M_{2}(M^{−1}NO_{K})},

and set Γ_{0}^{h}(N) := Γ_{0}^{h}(1, N), Γ_{1}^{h}(N) := Γ_{1}^{h}(1, N) and Γ^{h}(N) := Γ^{h}(1, N). Because we will frequently use the group Γ_{0}^{h}(1) = {A ∈ GL_{4}(O_{K}) | AJA^{∗} = J}, we reserve a special notation for it and denote it by Γ_{Z}. Note that the groups Γ_{0}^{h}(N), Γ_{1}^{h}(N) and Γ^{h}(N) are U(2, 2)-analogues of the standard congruence subgroups Γ_{0}(N), Γ_{1}(N) and Γ(N) of SL_{2}(Z). In general the superscript ‘h’ will indicate that an object is in some way related to the group U(2, 2). The letter ‘h’ stands for ‘hermitian’, as this is the standard name of modular forms on U(2, 2).

2.3. Modular forms. In this paper we will make use of the theory of modular forms on congruence subgroups of two different groups: SL_{2}(Z) and Γ_{Z}. We will use both the classical and the adelic formulation of the theories. In the adelic framework one usually speaks of automorphic forms rather than modular forms and in this case SL_{2} is usually replaced with GL_{2}. For more details see e.g. [22], chapter 3. In the classical setting the modular forms on congruence subgroups of SL_{2}(Z) will be referred to as elliptic modular forms, and those on congruence subgroups of Γ_{Z} as hermitian modular forms.

2.3.1. Elliptic modular forms. The theory of elliptic modular forms is well-known, so we omit most of the definitions and refer the reader to standard sources, e.g. [42]. Let

H := {z ∈ C | Im (z) > 0}

denote the complex upper half-plane. In the case of elliptic modular forms we will denote by Γ_{0}(N) the subgroup of SL_{2}(Z) consisting of matrices whose lowerleft entries are divisible by N, and by Γ_{1}(N) the subgroup of Γ_{0}(N) consisting of matrices whose upper left entries are congruent to 1 modulo N. Let Γ ⊂ SL_{2}(Z) be a congruence subgroup. Set M_{m}(Γ) (resp. S_{m}(Γ)) to denote the C-space of elliptic modular forms (resp. cusp forms) of weight m and level Γ. We also denote by M_{m}(N, ψ) (resp. S_{m}(N, ψ)) the space of elliptic modular forms (resp. cusp forms) of weight m, level N and character ψ. For f, g ∈ M_{m}(Γ) with either f or g a cusp form, and Γ′ ⊂ Γ a finite index subgroup, we define the Petersson inner product

⟨f, g⟩_{Γ′} := ∫_{Γ′\H} f(z)\overline{g(z)}(Im z)^{m−2} dx dy,

and set

⟨f, g⟩ := \frac{1}{[\overline{SL_{2}(Z)} : \overline{Γ}′]} ⟨f, g⟩_{Γ′} ,

where \overline{SL_{2}(Z)} := SL_{2}(Z)/ ⟨−I_{2}⟩ and \overline{Γ}′ is the image of Γ′ in \overline{SL_{2}(Z)}. The value ⟨f, g⟩ is independent of Γ′.

Every elliptic modular form f ∈ M_{m}(N, ψ) possesses a Fourier expansion f(z) = ∑_{n=0}^{∞} a(n)q^{n}, where throughout this paper in such series q will denote e(z) := e^{2πiz}. For γ = [ \begin{matrix}a & b \\ c & d \end{matrix} ] ∈ GL_{2}^{+}(R), set j(γ, z) = cz + d.

In this paper we will be particularly interested in the space S_{m} (4, (\frac{−4}{·})), where (\frac{−4}{·}) is the non-trivial character of (Z/4Z)^{×}. Regarded as a function Z → \{1, −1\}, it assigns the value 1 to all prime numbers p such that (p) splits in K and the value −1 to all prime numbers p such that (p) is inert in K. Note that since the character (\frac{−4}{·}) is primitive, the space S_{m} (4, (\frac{−4}{·})) has a basis consisting of primitive normalized eigenforms. We will denote this (unique) basis by N. For f = ∑_{n=1}^{∞} a(n)q^{n} ∈ N, set f^{ρ} := ∑_{n=1}^{∞} \overline{a(n)}q^{n} ∈ N.

Fact 2.1. ([42], section 4.6) One has a(p) = (\frac{−4}{p}) \overline{a(p)} for any rational prime p ∤ 2.

This implies that a(p) = \overline{a(p)} if (p) splits in K and a(p) = −\overline{a(p)} if (p) is inert in K.

For f ∈ N and E a finite extension of Q_{ℓ} containing the eigenvalues of T_{n}, n = 1, 2, … we will denote by ρ_{f} : G_{Q} → GL_{2}(E) the Galois representation attached to f by Deligne (cf. e.g., [11], section 3.1). We will write \overline{ρ}_{f} for the reduction of ρ_{f} modulo a uniformizer of E with respect to some lattice Λ in E^{2}. In general \overline{ρ}_{f} depends on the lattice Λ, however the isomorphism class of its semisimplification \overline{ρ}_{f}^{ss} is independent of Λ. Thus, if \overline{ρ}_{f} is irreducible (which we will assume), it is well-defined.

2.3.2. Hermitian modular forms. For a systematic treatment of the theory of hermitian modular forms see [6, 7, 8] as well as [23, 37, 36]. We begin by defining the hermitian upper half-plane

H = {Z ∈ M_{2}(C) | − i(Z − Z^{∗}) > 0},

where i = [ \begin{matrix}i \\ & i \end{matrix}]. Set Re Z = \frac{1}{2}(Z + Z^{∗}) and Im Z = −\frac{1}{2}i(Z − Z^{∗}). Let

G_{μ}^{+}(R) := {g ∈ G_{μ}(R) | μ(g) > 0}.

The group G_{μ}^{+}(R) acts on H by γZ = (a_{γ}Z + b_{γ})(c_{γ}Z + d_{γ})^{−1}, with γ ∈ G_{μ}^{+}(R). For a holomorphic function F on H, an integer m and γ ∈ G_{μ}^{+}(R) put

F |_{m}γ = μ(γ)^{2m−4}j(γ, Z)^{−m}F (γZ),

with the automorphy factor j(γ, Z) = det(c_{γ}Z + d_{γ}).

Let Γ^{h} be a congruence subgroup of Γ_{Z}. We say that a holomorphic function F on H is a hermitian modular form of weight m and level Γ^{h} if

F |_{m}γ = F for all γ ∈ Γ^{h}.

The group Γ^{h} is called the level of F. If Γ^{h} = Γ_{0}^{h}(N) for some N ∈ Z, then we say that F is of level N. Forms of level 1 will sometimes be referred to as forms of full level. One can also define hermitian modular forms with a character. Let Γ^{h} = Γ_{0}^{h}(N) and let ψ : A_{K}^{×} → C^{×} be a Hecke character such that for all finite p, ψ_{p}(a) = 1 for every a ∈ O_{K,p}^{×} with a − 1 ∈ NO_{K,p}. We say that F is of level N and character ψ if

F |_{m}γ = ψ_{N}(det a_{γ})F for every γ ∈ Γ_{0}^{h}(N).

A hermitian modular form of level Γ^{h}(M, N) possesses a Fourier expansion

F (Z) = ∑_{τ∈S(M)} c(τ)e(tr τZ),

where S(M) = {x ∈ S | tr xL(M) ⊂ Z} with S = {h ∈ M_{2}(K) | h^{∗} = h} and L(M) = S ∩ M_{2}(MO_{K}). As we will be particularly interested in the case when M = 1, we set

S := S(1) = \{[\begin{matrix}t_{1} & t_{2} \\ \overline{t_{2}} & t_{3} \end{matrix}] ∈ M_{2}(K) | t_{1}, t_{3} ∈ Z, t_{2} ∈ \frac{1}{2}O_{K}\} .

Denote by M_{m}(Γ^{h}) the C-space of hermitian modular forms of weight m and level Γ^{h}, and by M_{m}(N, ψ) the C-space of hermitian modular forms of weight m, level N and character ψ. For F ∈ M_{m}(Γ^{h}) and α ∈ G_{μ}^{+}(R) one has F |_{m}α ∈ M_{m}(α^{−1}Γ^{h}α) and there is an expansion

F |_{m}α = ∑_{τ∈S} c_{α}(τ)e(tr τZ).

We call F a cusp form if for all α ∈ G_{μ}^{+}(R), c_{α}(τ) = 0 for every τ such that det τ = 0. Denote by S_{m}(Γ^{h}) (resp. S_{m}(N, ψ)) the subspace of cusp forms inside M_{m}(Γ^{h}) (resp. M_{m}(N, ψ)). If ψ = 1, set M_{m}(N) := M_{m}(N, 1) and S_{m}(N) := S_{m}(N, 1).

Theorem 2.2 (q-expansion principle, [28], section 8.4). Let ℓ be a rational prime and N a positive integer with ℓ ∤ N. Suppose all Fourier coefficients of F ∈ M_{m}(N, ψ) lie inside the valuation ring O of a finite extension E of Q_{ℓ}. If γ ∈ Γ_{Z}, then all Fourier coefficients of F |_{m}γ also lie in O.

If F and F′ are two hermitian modular forms of weight m, level Γ^{h} and character ψ, and either F or F′ is a cusp form, we define for any finite index subgroup Γ_{0}^{h} of Γ^{h}, the Petersson inner product

⟨F, F′⟩_{Γ_{0}^{h}} := ∫_{Γ_{0}^{h}\H} F (Z)\overline{F′(Z)}(det Y)^{m−4}dXdY,

where X = Re Z and Y = Im Z, and

⟨F, F′⟩ = [\overline{Γ}_{Z} : \overline{Γ}_{0}^{h}]^{−1} ⟨F, F′⟩_{Γ_{0}^{h}} ,

where \overline{Γ}_{Z} := Γ_{Z}/ ⟨i⟩ and \overline{Γ}_{0}^{h} is the image of Γ_{0}^{h} in \overline{Γ}_{Z}. The value ⟨F, F′⟩ is independent of Γ_{0}^{h}.

There exist adelic analogues of hermitian modular forms. For F ∈ M_{m}(N, ψ), the function φ_{F} : G(A) → C defined by

φ_{F} (g) = j(g_{∞}, i)^{−m}F (g_{∞}, i)ψ^{−1}(det d_{k}),

where g = g_{Q}g_{∞}k ∈ G(Q)G(R)K_{0,f}(N), is an automorphic form on G(A).

3. Eisenstein series

The goal of this section is to compute the residue of the hermitian Klingen Eisenstein series (cf. Definition 3.1 and Theorem 3.10). This computation will be used in the next section.

3.1. Siegel, Klingen and Borel Eisenstein series. Siegel and Klingen Eisenstein series are induced from the maximal parabolic subgroups P and Q of G = U(2, 2) respectively. (For the definitions of P and Q see section 2.2.) Let

δ_{P} : P (A) → R_{+}

be the modulus character of P (A),

(3.1)  δ_{P} ([A ; Â] u) = | det A \overline{det A}|_{A}^{2},

with A ∈ Res_{K/Q} GL_{2}(A), u ∈ U_{P} (A), and

δ_{Q} : Q(A) → R_{+}

the modulus character of Q(A),

(3.2)  δ_{Q} ([x ; a  b ; x̂ ; c  d] u) = |x\overline{x}|_{A}^{3},

with x ∈ Res_{K/Q} GL_{1}(A), [ \begin{matrix}a & b \\ c & d \end{matrix} ] ∈ U(1, 1)(A) and u ∈ U_{Q}(A). As before, K_{0} = K_{0,∞}K_{0,f} will denote the maximal compact subgroup of G(A). Using the Iwasawa decomposition G(A) = P (A)K_{0} we extend both characters δ_{P} and δ_{Q} to functions on G(A) and denote these extensions again by δ_{P} and δ_{Q}.

Definition 3.1. For g ∈ G(A), the series

E_{P} (g, s) := ∑_{P (Q)\G(Q)} δ_{P} (γg)^{s}

is called the (hermitian) Siegel Eisenstein series, while the series

E_{Q}(g, s) := ∑_{Q(Q)\G(Q)} δ_{Q}(γg)^{s}

is called the (hermitian) Klingen Eisenstein series.

Properties of E_{P} (g, s) were investigated by Shimura in [50]. We summarize them in the following proposition.

Proposition 3.2 (Shimura). The series E_{P} (g, s) is absolutely convergent for Re (s) > 1 and can be meromorphically continued to the entire s-plane with only a simple pole at s = 1. One has

(3.3)  res_{s=1} E_{P} (g, s) = \frac{45L (2, (\frac{−4}{·}))}{4πL (3, (\frac{−4}{·}))},

where L(·, ·) denotes the Dirichlet L-function.

Properties of the Klingen Eisenstein series were investigated by Raghavan and Sengupta in [44]. The only difference is that instead of E_{Q}(g, s), [44] uses an Eisenstein series that we will denote by E_{s}(Z). The connection between E_{Q}(g, s) and E_{s}(Z) is provided by Lemma 4.6. After the connection has been established the following proposition follows from Lemma 1 in [44].

Proposition 3.3 (Raghavan-Sengupta). The series E_{Q}(g, s) converges absolutely for Re (s) > 1 and can be meromorphically continued to the entire s-plane. The possible poles of E_{Q}(g, s) are at most simple and are contained in the set {0, 1/3, 2/3, 1}.

In section 3.4 we will show that E_{Q}(g, s) has a simple pole at s = 1 and calculate the residue.

Both E_{P} (g, s) and E_{Q}(g, s) have their classical analogues, i.e., series in which g is replaced by a variable Z in the hermitian upper half-plane H. Let g_{∞} ∈ G(R) be such that Z = g_{∞}i and set g = (g_{∞}, 1) ∈ G(R) × G(A_{f}). Define

E_{P} (Z, s) := E_{P} (g, s)

and

E_{Q}(Z, s) = E_{Q}(g, s).

We will show in Lemma 4.6 that

E_{Q}(Z, s) = ∑_{γ∈Q(Z)\Γ_{Z}} (\frac{det Im (γZ)}{(Im (γZ))_{2,2}} )^{3s} ,

where for any matrix M we denote its (i, j)-th entry by M_{i,j}.

Remark 3.4. Note that we use the same symbols E_{P} (·, s) and E_{Q}(·, s) to denote both the adelic and the classical Eisenstein series. We distinguish them by inserting g ∈ G(A) or Z ∈ H in the place of the dot. We will continue this abuse of notation for other Eisenstein series we study.

We now turn to the Eisenstein series which is induced from the Borel subgroup B of G, which we call the Borel Eisenstein series. It is a function of two complex variables s and z, defined by

E_{B}(g, s, z) := ∑_{γ∈B(Q)\G(Q)} δ_{Q}(γg)^{s}δ_{P} (γg)^{z}.

Note that as the Levi subgroup of B is abelian (it is the torus T), the character δ_{Q}^{s}δ_{P}^{z} is a cuspidal automorphic form on T (A). Thus the following proposition follows from [43], Proposition II.1.5.

Proposition 3.5. The series E_{B}(g, s, z) is absolutely convergent for

(s, z) ∈ {(s′, z′) ∈ C × C | Re (s′) > 2/3, Re (z′) > 1/2}.

It can be meromorphically continued to all of C × C.

Remark 3.6. It follows from the general theory (cf. [38], chapter 7) that by taking iterated residues of Eisenstein series induced from minimal parabolics one obtains Eisenstein series on other parabolics. These series are usually referred to as residual Eisenstein series. In fact E_{P} and E_{Q} are residues of E_{B} taken with respect to the variable s and z respectively. We will prove this fact in section 3.5, but see also [32], Remark 5.6.

3.2. Siegel Eisenstein series with positive weight. In this section we define an Eisenstein series induced from the Siegel parabolic, having positive weight, level and non-trivial character. For notation refer to section 2. Let m, N be integers with m ≥ 0 and N > 0. Note that K_{0,∞} is the stabilizer of i in G(R). Let ψ : K^{×} \ A_{K}^{×} → C^{×} be a Hecke character of A_{K}^{×} with local decomposition ψ = ∏_{p} ψ_{p}, where p runs over all the places of Q. Assume that

ψ_{∞}(x_{∞}) = (\frac{x_{∞}}{|x_{∞}|})^{m}

and

ψ_{p}(x_{p}) = 1 if p ≠ ∞, x_{p} ∈ O_{K,p}^{×}, and x_{p} − 1 ∈ NO_{K,p}.

As before we set ψ_{N} = ∏_{p|N} ψ_{p}. Let δ_{P} denote the modulus character of P. We define

μ_{P} : M_{P} (Q)U_{P} (A) \ G(A) → C

by setting

μ_{P} (g) = \begin{cases}0 & g ∉ P (A)K_{0}(N) \\ ψ(det d_{q})^{−1}ψ_{N}(det d_{κ})^{−1}j(κ_{∞}, i)^{−m} & g = qκ ∈ P (A)K_{0}(N). \end{cases}

Note that μ_{P} has a local decomposition μ_{P} = ∏_{p} μ_{P,p}, where

(3.4)  μ_{P,p}(q_{p}κ_{p}) = \begin{cases}ψ_{p}(det d_{q_{p}})^{−1} & if p ∤ N∞, \\ ψ_{p}(det d_{q_{p}})^{−1}ψ_{p}(det d_{κ_{p}}) & if p | N, p ≠ ∞, \\ ψ_{∞}(det d_{q_{∞}})^{−1}j(κ_{∞}, i)^{−m} & if p = ∞ \end{cases}

and δ_{P} has a local decomposition δ_{P} = ∏_{p} δ_{P,p}, where

(3.5)  δ_{P,p} ([A ; Â] uκ) = | det A det \overline{A}|_{Q_{p}}.

Definition 3.7. The series

E(g, s, N, m, ψ) := ∑_{γ∈P (Q)\G(Q)} μ_{P} (γg)δ_{P} (γg)^{s/2}

is called the (hermitian) Siegel Eisenstein series of weight m, level N and character ψ.

The series E(g, s, N, m, ψ) converges for Re (s) sufficiently large, and can be continued to a meromorphic function on all of C (cf. [50], Proposition 19.1). It also has a complex analogue E(Z, s, m, ψ, N) defined by

E(Z, s, m, ψ, N) := j(g_{∞}, i)^{m}E(g, s, N, m, ψ)

for Z = g_{∞}i, g = g_{Q}g_{∞}κ_{f} ∈ G(Q)G(R)K_{0,f}(N). It follows from Lemma 18.7(3) of [50] and formulas (16.40) and (16.48) of [51], together with the fact that K has class number one that

E(Z, s, m, ψ, N) = ∑_{γ∈(P (Q)∩Γ_{0}^{h}(N))\Γ_{0}^{h}(N)} ψ_{N}(det d_{γ})^{−1}(det Im Z)^{s−m/2}|_{m}γ =

= ∑_{γ∈(P (Q)∩Γ_{0}^{h}(N))\Γ_{0}^{h}(N)} ψ_{N}(det d_{γ})^{−1} det(c_{γ}Z + d_{γ})^{−m}×

× | det(c_{γ}Z + d_{γ})|^{−2s+m}(det Im Z)^{s−m/2}.

(3.6)

3.3. The Eisenstein series on U(1, 1). Let B_{1} denote the upper-triangular Borel subgroup of U(1, 1) with Levi decomposition B_{1} = T_{1}U_{1}, where

T_{1} := {[a ; â] | a ∈ Res_{K/Q} GL_{1}}

and

U_{1} = \{\left[{1 x \atop 1}\right] | x ∈ G_{a}\} .

Let δ_{1} : B_{1}(A) → R_{+} be the modulus character given by

δ_{1} ([a ; â] u) = |a\overline{a}|_{A}

for u ∈ U_{1}(A). Let K_{1} = K_{1,∞}K_{1,f} denote the maximal compact subgroup of U(1, 1)(A) with

K_{1,∞} = \{[\begin{matrix}α & β \\ −β & α \end{matrix}] ∈ GL_{2}(C) | |α|^{2} + |β|^{2} = 1, α\overline{β} ∈ R\}

being the maximal compact subgroup of U(1, 1)(R) and K_{1,f} = ∏_{p≠∞} U(1, 1)(Z_{p}). As usually we extend δ_{1} to a map on U(1, 1)(A) using the Iwasawa decomposition. For g ∈ U(1, 1)(A), set

(3.7)  E_{U(1,1)}(g, s) = ∑_{γ∈B_{1}(Q)\U(1,1)(Q)} δ_{1}(γg)^{s}.

The following proposition follows from [50], Theorem 19.7.

Proposition 3.8. The series E_{U(1,1)}(g, s) converges absolutely for Re (s) > 1 and continues meromorphically to all of C. It has a simple pole at s = 1 with residue 3/π.

We now define a complex analogue of E_{U(1,1)}(g, s). As SL_{2}(R) acts transitively on H, so does U(1, 1)(R) ⊃ SL_{2}(R). Hence for every z_{1} ∈ H there exists g_{∞} ∈ U(1, 1)(R) such that z_{1} = g_{∞}i. Set g = (g_{∞}, 1) ∈ U(1, 1)(R) × U(1, 1)(A_{f}). An easy calculation shows that

(3.8)  δ_{1}(g) = Im (z_{1}).

For z_{1} and g as above, define the complex Eisenstein series corresponding to E_{U(1,1)}(g, s) by

(3.9)  E_{U(1,1)}(z_{1}, s) := E_{U(1,1)}(g, s).

CONGRUENCES AMONG MODULAR FORMS ON U(2, 2) 15

It is easy to see that

(3.10)  E_{U(1,1)}(z_{1}, s) = ∑_{γ∈B_{1}(Z)\U(1,1)(Z)}(Im (γz_{1}))^{s}.

The series E_{U(1,1)}(z_{1}, s) possesses a Fourier expansion of the form

E_{U(1,1)}(z_{1}, s) = ∑_{n∈Z} c_{n}(y_{1}, s)e^{2πinx_{1}},

where x_{1} := Re (z_{1}) and y_{1} := Im (z_{1}). Lemma 3.9. Let z_{1} and g be as before, i.e., z_{1} = g_{∞}i. Then

c_{0}(s, y_{1}) = y_{1}^{s} + \frac{ζ(2s − 1)}{ζ(2s)} \frac{Γ (s − \frac{1}{2})}{Γ(s)}\sqrt{π} y_{1}^{1−s},

where ζ(s) denotes the Riemann zeta function. Proof. This is a standard argument. See, e.g., [10], the proof of Theorem 1.6.1. □ 3.4. Residue of the Klingen Eisenstein series. Let E_{Q}(g, s) be the Klingen Eisenstein series defined in section 3.1. This section and section 3.5 are devoted to proving the following theorem. Theorem 3.10. The series E_{Q}(g, s) has a simple pole at s = 1 and one has

(3.11)  res_{s=1} E_{Q}(g, s) = \frac{5π^{2}L (2, (\frac{−4}{·}))}{4ζ_{K}(2)L (3, (\frac{−4}{·}))},

where ζ_{K}(s) denotes the Dedekind zeta function of K.

Theorem 3.10 is a consequence of the following proposition. Proposition 3.11. The following statements hold:

(i) For any fixed s ∈ C with Re (s) > 2/3 the function E_{B}(g, s, z) has a simple

pole at z = 1/2 and

(3.12)  res_{z=1/2}E_{B}(g, s, z) = \frac{3}{2π} E_{Q}(g, s + 1/3).

(ii) For any fixed z ∈ C with Re (z) > 1/2 the function E_{B}(g, s, z) has a simple

pole at s = 2/3 and

(3.13)  res_{s= \frac{2}{3}} E_{B}(g, s, z) = \frac{π^{2}}{6ζ_{K}(2)}E_{P} (g, z + 1/2) .

Indeed, using Proposition 3.11 and interchanging the order of taking residues we obtain:

res_{s= \frac{2}{3}} E_{Q} (g, s + \frac{1}{3}) = \frac{2π}{3} \frac{π^{2}}{6ζ_{K}(2)}res_{z= \frac{1}{2}} E_{P} (g, \frac{1}{2} + z) .

By Proposition 3.2,

res_{z= \frac{1}{2}} E_{P} (g, \frac{1}{2} + z) = \frac{45L (2, (\frac{−4}{·}))}{4πL (3, (\frac{−4}{·}))},

and thus we finally get

res_{s=1} E_{Q}(g, s) = \frac{5π^{2}L (2, (\frac{−4}{·}))}{4ζ_{K}(2)L (3, (\frac{−4}{·}))},

which proves Theorem 3.10.

We now prepare for the proof of Proposition 3.11, which will be completed in section 3.5.

Let [ x ; a b ; x̂ ; c d ] ∈ M_{Q}(A). Since [ \begin{matrix}a & b \\ c & d \end{matrix} ] ∈ U(1, 1)(A), we can use the Iwasawa

decomposition for U(1, 1)(A) with respect to the upper-triangular Borel to write [ \begin{matrix}a & b \\ c & d \end{matrix} ] = [ α β ; α̂ ] κ with κ ∈ K_{1}, where K_{1} is as in section 3.3. Note that if κ = [ \begin{matrix}κ_{1} & κ_{2} \\ κ_{3} & κ_{4} \end{matrix} ],

then [ \begin{matrix}1 \\ & κ_{1} & & κ_{2} \\ & & 1 \\ & κ_{3} & & κ_{4} \end{matrix} ] ∈ K_{0}. Define a character

ϕ_{Q} : M_{Q}(A) → R_{+}

by

ϕ_{Q} ([x ; a  b ; x̂ ; c  d]) = ϕ_{Q} ([x ; 1 ; x̂ ; 1] [1 ; α  β ; 1 ; α̂]) = |α\overline{α}|_{A},

and a character

ϕ_{P} : M_{P} (A) → R_{+}

by:

(3.14) ϕ_{P} ([A ; Â]) =

ϕ_{P} ([x ∗ ; y ; x̂ ; ∗ ŷ] [\begin{matrix}κ_{1} & κ_{2} \\ κ_{3} & κ_{4} \\ & & κ′_{1} & κ′_{2} \\ & & κ′_{3} & κ′_{4} \end{matrix}]) = |xy^{−1}\overline{(xy^{−1})}|_{A},

where we used the Iwasawa decomposition for GL_{2}(A_{K}) = Res_{K/Q} GL_{2}(A) with respect to its upper-triangular Borel B_{R}, and its maximal compact subgroup K_{R} = U(2) ∏_{v∤∞} GL_{2}(O_{K,v}) to write A ∈ GL_{2}(A_{K}) as

A = \left[{x ∗ \atop y}\right] [\begin{matrix}κ_{1} & κ_{2} \\ κ_{3} & κ_{4} \end{matrix}] ∈ B_{R}(A)K_{R}.

We again have [ \begin{matrix}κ_{1} & κ_{2} \\ κ_{3} & κ_{4} \\ & & κ′_{1} & κ′_{2} \\ & & κ′_{3} & κ′_{4} \end{matrix} ] ∈ K_{0}.

Extend ϕ_{Q} and ϕ_{P} as well as δ_{Q} and δ_{P} to functions on G(A) using the Iwasawa decompositions

(3.15)  G(A) = B(A) K_{0} = P (A) K_{0} = Q(A) K_{0}.

A simple calculation shows that

(3.16)  δ_{Q}^{s}δ_{P}^{z} = δ_{Q}^{s+ \frac{2}{3} z}ϕ_{Q}^{2z} = δ_{P}^{\frac{3}{4} s+z}ϕ_{P}^{\frac{3}{2} s}

for any complex numbers s and z. Let E_{B}(g, s, z) be the Borel Eisenstein series defined in section 3.1. By Proposition 3.5 the series is absolutely convergent if Re (s) > 2/3 and Re (z) > 1/2 and admits meromorphic continuation to all of C^{2}. Using identity (3.16) and rearranging terms we get:

E_{B}(g, s, z) := ∑_{γ∈Q(Q)\G(Q)} δ_{Q}(γg)^{s+ \frac{2}{3} z} ∑_{α∈B(Q)\Q(Q)} ϕ_{Q}(αγg)^{2z} =

(3.17)  = ∑_{γ∈P (Q)\G(Q)} δ_{P} (γg)^{\frac{3}{4} s+z} ∑_{α∈B(Q)\P (Q)} ϕ_{P} (αγg)^{\frac{3}{2} s}.

Let E_{U(1,1)}(g, s) be the Eisenstein series defined by formula (3.7). We also define an Eisenstein series on Res_{K/Q} GL_{2}(A) by:

(3.18)  E_{Res_{K/Q} GL_{2}}(g, s) = ∑_{γ∈B_{R}(Q)\Res_{K/Q} GL_{2}(Q)} δ_{R}(γg)^{s},

where δ_{R} denotes the modulus character on B_{R} defined by:

δ_{R} : B_{R} → R_{+}
(3.19)  δ_{R} (\left[{a ∗ \atop b}\right]) = |a\overline{a}b^{−1}\overline{b}^{−1}|_{A}^{1/2}.

The following maps

\begin{matrix}π_{Q} : M_{Q}U_{Q} & → U(1, 1) \\ ([x ; a  b ; x̂ ; c  d] , u) & ↦ [\begin{matrix}a & b \\ c & d \end{matrix}] , \end{matrix} ; (3.20)

and

\begin{matrix}π_{P} : P & → Res_{K/Q} GL_{2} \\ [A X ; Â] & ↦ A \end{matrix} ; (3.21)

give bijections

B(Q) \ Q(Q) ≅ B_{1}(Q) \ U(1, 1)(Q)

and

B(Q) \ P (Q) ≅ B_{R}(Q) \ Res_{K/Q} GL_{2}(Q),

respectively. On the A-points we can extend π_{Q} to a map G(A) → U(1, 1)(A)/K_{1} and π_{P} to a map G(A) → Res_{K/Q} GL_{2}(A)/K_{R} by declaring them to be trivial on K_{0}. Hence we can rewrite (3.17) as

(3.22) E_{B}(g, s, z) := ∑_{γ∈Q(Q)\G(Q)} δ_{Q}(γg)^{s+ \frac{2}{3} z}E_{U(1,1)}(π_{Q}(γg), 2z) =

= ∑_{γ∈P (Q)\G(Q)} δ_{P} (γg)^{\frac{3}{4} s+z}E_{Res_{K/Q} GL_{2}}(π_{P} (γg), \frac{3}{2}s).

3.5. E_{Q}(g, s) as a residual Eisenstein series. In this section we complete the proof of Proposition 3.11. We will only present a proof of part (i) of the proposition as the proof of (ii) is completely analogous. (In part (ii) the role of E_{U(1,1)} (see below) is played by E_{Res_{K/Q} GL_{2}} for which an easy computation shows that res_{s=1} E_{Res_{K/Q} GL_{2}}(g, s) = π^{2}/(4ζ_{K}(2)).) In what follows Z will denote a variable in the hermitian upper half-plane H, and z_{1} a variable in the complex upper half-plane H. Otherwise we use notation from sections 3.1-3.4. Write g = g_{Q}g_{∞}κ ∈ G(A) with g_{Q} ∈ G(Q), g_{∞} ∈ G(R) and κ ∈ K_{0,f}. We have E_{B}(g, s, z) = E_{B}(g_{∞}, s, z) and E_{Q}(g, s) = E_{Q}(g_{∞}, s), hence it is enough to prove (3.12) for g = (g_{∞}, 1) ∈ G(R) × G(A_{f}). Let K_{1} denote the maximal compact subgroup of U(1, 1)(A) and let π_{Q} : G(A) → U(1, 1)(A)/K_{1} be as in formula (3.20). Lemmas 3.12 and 3.14 are easy.

Lemma 3.12. If g = (g_{∞}, 1) ∈ G(A), then Im (π_{Q}(g)_{∞}i) = Im (g_{∞}i)_{2,2}.

Remark 3.13. Note that for any 2 × 2 matrix M with entries in C one has Im (M_{2,2}) = (Im (M))_{2,2}. Hence the conclusion of Lemma 3.12 can also be written as Im (π_{Q}(g)_{∞}i) = Im ((g_{∞}i)_{2,2}).

Lemma 3.14. For any Z ∈ H, there exists γ ∈ Q(Z) such that (Im γZ)_{2,2} > \frac{1}{2}.

The next lemma is just a simple adaptation to the case of hermitian modular forms of the proof of Hilfsatz 2.10 of [21].

Lemma 3.15. For every Z ∈ H, we have

sup_{γ∈Γ_{Z}} det Im (γZ) < ∞.

Proposition 3.16. Let δ > 0 and g = (g_{∞}, 1) ∈ G(R) × G(A_{f}). For every s ∈ C with Re (s) > 1 + δ and every z ∈ C with |z − \frac{1}{2}| < δ, the series

(3.23)  D := |z − 1/2| ∑_{γ∈Q(Q)\G(Q)} |δ_{Q}(γg)^{s+2z/3}E_{U(1,1)}(π_{Q}(γg), 2z)|

converges.

Proof. Using the same arguments as in the proof of Lemma 4.6 (cf. section 4.2) one shows that

D = ∑_{γ∈Q(Z)\Γ_{Z}} |(\frac{det Im (γZ)}{(Im (γZ))_{2,2}} )^{3s+2z}| |z − 1/2| |E_{U(1,1)}(π_{Q}(γg)_{∞}i, 2z)|.

(Note that z′ := π_{Q}(γg)_{∞}i is a complex variable.) As g = (g_{∞}, 1) and γ ∈ Γ_{Z} ⊂ K_{0,f}, we have π_{Q}(γg)_{∞} = π_{Q}((γg_{∞}, 1))_{∞}. By Lemmas 3.12 and 3.14 we can find a set S of representatives of Q(Z) \ Γ_{Z} such that for every γ ∈ S we have

(3.24)  Im (π_{Q}(γg)_{∞}i) = Im ((γg_{∞}i)_{2,2}) > \frac{1}{2}.

The series E_{U(1,1)}(z_{1}, 2z) has a Fourier expansion of the form

E_{U(1,1)}(z_{1}, 2z) = ∑_{n∈Z} c_{n}(2z, Im (z_{1}))e^{2πinRe (z_{1})},

and E_{U(1,1)}(z_{1}, 2z) − c_{0}(2z, Im (z_{1})) for every fixed z_{1} continues to a holomorphic function on the entire z-plane and for every fixed z is rapidly decreasing as Im (z_{1}) → ∞. It follows that for any given N > 0 there exists a constant M(N) (independent of z_{1} and independent of z as long as |z − 1/2| < δ) such that |E_{U(1,1)}(z_{1}, 2z) − c_{0}(2z, Im (z_{1}))| < M(N) as long as Im (z_{1}) > N. Set x_{γ} := Re (π_{Q}(γg)_{∞}i) and y_{γ} := Im (π_{Q}(γg)_{∞}i) = Im ((γg_{∞}i)_{2,2}) . Taking N = 1/2, we see by formula (3.24) that there exists a constant M (independent of γ) such that |E_{U(1,1)}(x_{γ} + iy_{γ}, 2z)| ≤ M + |c_{0}(2z, y_{γ})|. Using (3.8) and Lemma 3.9 one sees that there exists a positive constant C independent of z and of γ such that

|z − 1/2||c_{0}(2z, y_{γ})| < C + |y_{γ}|^{1+2δ}.

Thus we conclude that there exists a positive constant A (independent of z and γ) such that

\begin{aligned}|(z − \frac{1}{2}) E_{U(1,1)} (π_{Q}(γg_{∞})i, 2z)| & ≤ A(1 + Im (π_{Q}(γg_{∞})i)^{1+2δ}) = \\ & = A(1 + Im (γg_{∞}i)_{2,2}^{1+2δ}). \end{aligned} ; (3.25)

For s′ ∈ C lying inside the region of absolute convergence of E_{s′}(Z) let

|E|_{s′}(Z) := ∑_{γ∈Q(Z)\Γ_{Z}} |(\frac{det Im (γZ)}{(Im (γZ))_{2,2}} )^{s′} |

denote the majorant of E_{s}(Z). By formula (3.25) we have

(3.26)  D ≤ A|E|_{3s+2z}(Z) + A ∑_{γ∈S} |(\frac{det Im (γZ)}{(Im (γZ))_{2,2}} )^{3s+2z}| (Im (γZ))_{2,2}^{1+2δ}.

Note that |E|_{3s+2z}(Z) is well-defined (i.e., 3s + 2z is in the region of absolute convergence of E_{s′}(Z)) by our assumption on s and z. Denote the second term of the right-hand side of formula (3.26) by D_{2}. Then

D_{2} = A ∑_{γ∈S} |(\frac{det Im (γZ)}{(Im (γZ))_{2,2}} )^{3s+2z−(1+2δ)}| (det Im (γZ))^{1+2δ}.

By Lemma 3.15 there exists a constant M(Z) such that det Im (γZ) ≤ M(Z) for every γ ∈ S and hence

D_{2} ≤ AM(Z)^{1+2δ}|E|_{3s+2z−(1+2δ)} < ∞

as Re (3s + 2z − (1 + 2δ)) > 3 by our assumptions on z and s. This finishes the proof. □

Proof of Proposition 3.11. We need to show that for a fixed s ∈ C with Re (s) > 2/3 and for every ϵ > 0 there exists δ > 0 such that |z − 1/2| < δ implies

(3.27) D(z) := |(z − \frac{1}{2}) ∑_{γ∈Q(Q)\G(Q)} δ_{Q}(γg)^{s+2z/3}E_{U(1,1)}(π_{Q}(γg), 2z)−

− \frac{3}{2π} ∑_{γ∈Q(Q)\G(Q)} δ_{Q}(γg)^{s+1/3}| < ϵ.

As remarked at the beginning of the section we can assume without loss of generality that g = (g_{∞}, 1) ∈ G(R) × G(A_{f}). We first show that (3.27) holds for s with Re (s) > 1. Fix s ∈ C with Re (s) > 1 and δ′ > 0 such that 0 < δ′ < Re (s) − 1. From now on assume |z − 1/2| < δ′. Fix a set S of representatives of Q(Q) \ G(Q). By Proposition 3.16 and the fact that E_{Q}(g, s′) converges absolutely for s′ with Re (s′) > 1, there exists a finite subset S_{1} of S such that the following two inequalities:

(3.28)  ∑_{γ∈S_{2}} |(δ_{Q}(γg))^{s+1/3}| < \frac{πϵ}{6},

(3.29)  ∑_{γ∈S_{2}} |z − \frac{1}{2}| | δ_{Q}(γg)^{s+2z/3}E_{U(1,1)}(π_{Q}(γg), 2z)| < \frac{ϵ}{4}

are simultaneously satisfied. Here S_{2} denotes the complement of S_{1} in S. We have D(z) ≤ D_{1}(z) + D_{2}(z), where

D_{j}(z) := |(z − \frac{1}{2}) ∑_{γ∈S_{j}} δ_{Q}(γg)^{s+2z/3}E_{U(1,1)}(π_{Q}(γg), 2z) − \frac{3}{2π} ∑_{γ∈S_{j}} δ_{Q}(γg)^{s+1/3}| .

Note that if we replace δ′ with a smaller δ′′ > 0, then estimates (3.28) and (3.29) remain true as long as |z −1/2| < δ′′ for the same choice of S_{1}. Hence we find δ > 0 with δ < δ′ such that D_{1}(z) < \frac{ϵ}{2}. This is clearly possible as D_{1}(z) is a finite sum and it follows from Proposition 3.8 that 3/2π is the residue of E_{U(1,1)}(π_{Q}(γg), 2z) at z = 1/2. On the other hand D_{2}(z) ≤ D_{3}(z) + D_{4}(z), where

D_{3}(z) := ∑_{γ∈S_{2}} |z − \frac{1}{2}| | δ_{Q}(γg)^{s+2z/3}E_{U(1,1)}(π_{Q}(γg), 2z)|

and

D_{4}(z) := \frac{3}{2π} ∑_{γ∈S_{2}} |(δ_{Q}(γg))^{s+1/3}| .

Formulas (3.28) and (3.29) imply now that D_{3}(z) < ϵ/4 and D_{4}(z) < ϵ/4. Hence

D(z) ≤ D_{1}(z) + D_{2}(z) ≤ D_{1}(z) + D_{3}(z) + D_{4}(z) < ϵ

as desired.

We have thus established the equality res_{z=1/2}E_{B}(g, s, z) = \frac{3}{2π} E_{Q}(g, s+1/3) for s with Re (s) > 1. However, both sides are meromorphic functions in s and since the right-hand side is holomorphic for Re (s) > 2/3, so must be the left-hand side. Hence they agree for Re (s) > 2/3. □

4. The Petersson norm of a Maass lift

The goal of this section is to express the denominator of C_{F_{f}} in formula (1.1) by a special value of the symmetric square L-function of f.

4.1. Maass lifts. Let H, as before, denote the complex upper half-plane. The space H×C×C affords an action of the Jacobi modular group Γ^{J} := SL_{2}(Z)⋉O_{K}^{2},

under which ([ \begin{matrix}a & b \\ c & d \end{matrix} ] , λ, μ) takes (τ, z, w) ∈ H × C × C to (\frac{aτ+b}{cτ+d}, \frac{z}{cτ+d}, \frac{w}{cτ+d}).

Definition 4.1. A holomorphic function

ϕ : H × C × C → C

is called a Jacobi form of weight k and index m if for every [ \begin{matrix}a & b \\ c & d \end{matrix} ] ∈ SL_{2}(Z) and λ, μ ∈ O_{K},

ϕ = ϕ|_{k,m} [\begin{matrix}a & b \\ c & d \end{matrix}] := (cτ + d)^{−k}e (−m\frac{czw}{cτ + d}) ϕ_{m} (\frac{aτ + b}{cτ + d}, \frac{z}{cτ + d}, \frac{w}{cτ + d})

and

ϕ = ϕ|_{m}[λ, μ] := e(mλ\overline{λ}t + \overline{λ}z + λw) ϕ_{m}(τ, z + λτ + μ, w + λ̄t + μ̄).

Let k be a positive integer divisible by 4 and F a hermitian cusp form of weight k and full level. By rearranging the Fourier expansion F (Z) = ∑_{B∈S} c(B)e(tr BZ) of F we obtain

(4.1)  F (Z) = ∑_{m∈Z_{>0}} ϕ_{m}(τ, z, w)e(mτ′)

where Z = [ \begin{matrix}τ & z \\ w & τ′ \end{matrix}] ∈ H and

ϕ_{m}(τ, z, w) = ∑_{\begin{matrix}l∈Z_{≥0},t∈ \frac{1}{2} O_{K} \\ t\overline{t}≤lm \end{matrix}} c ([\begin{matrix}l & t \\ t̄ & m \end{matrix}]) e(lτ + t̄z + tw)

is a Jacobi form of weight k and index m. The expansion (4.1) is called the Fourier-Jacobi expansion of F.

Definition 4.2. The Maass space denoted by S_{k}^{M}(Γ_{Z}) is the C-linear subspace of S_{k}(Γ_{Z}) consisting of those F ∈ S_{k}(Γ_{Z}) which satisfy the following condition: there exists a function c_{F}^{∗} : Z_{≥0} → C such that

c_{F} (B) = ∑_{d∈Z_{>0},d|ϵ(B)} d^{k−1}c_{F}^{∗} (4 det B/d^{2})

for all B ∈ S, where ϵ(B) := max \{q ∈ Z_{>0} | \frac{1}{q}B ∈ S\} . We call F ∈ S_{k}^{M}(Γ_{Z}) a Maass form or a CAP form.

Theorem 4.3 (Raghavan-Sengupta [44]). There exists a C-linear isomorphism between the Maass space and the space

(4.2) S_{k−1}^{+} (4, (\frac{−4}{·})) :=

= \{ϕ ∈ S_{k−1} (4, (\frac{−4}{·})) | ϕ = ∑_{n=1}^{∞} b(n)q^{n}, b(n) = 0 if (\frac{−4}{n}) = 1\} .

We will describe this isomorphism in more detail. Any Jacobi form ψ of weight k and index 1 can be written as a finite linear combination:

(4.3)  ψ(τ, z, w) = ∑_{t∈A} f_{t}(τ)θ_{t}(τ, z, w),

where A = \{0, \frac{1}{2}, \frac{i}{2}, \frac{i+1}{2}\}, θ_{t}(τ, z, w) := ∑_{λ∈t+O_{K}} e(λ\overline{λ}τ + l̄z + w) and

f_{t}(τ) = ∑_{l≥0,l≡−4nt (mod 4)} c_{F}^{∗} (l)e(lτ/4).

The map ψ(τ, z, w) ↦ f_{0}(τ) gives an injection of J_{k,1}, the space of Jacobi forms of weight k and index 1, into S_{k−1} (4, (\frac{−4}{·})). If we put ψ = ϕ_{1} and define ϕ by ϕ|_{k−1} [ \begin{matrix}& −1 \\ 4 \end{matrix} ] = f_{0}, the composite F ↦ ϕ_{1}(τ, z, w) ↦ f_{0}(τ) ↦ ϕ gives the isomorphism alluded to in Theorem 4.3. Denoting this isomorphism by Ω, we can map any normalized Hecke eigenform f = ∑_{n≥1} b(n)q^{n} ∈ S_{k−1} (4, (\frac{−4}{·})) to the element F_{f} := Ω^{−1}(f − f^{ρ}) ∈ S_{k}^{M}(Γ_{Z}). Here f^{ρ} = ∑_{n≥1} \overline{b(n)}q^{n}. This lifting is Hecke equivariant in a sense, which will be explained in section 5.4. Note that F_{f} = −F_{f^{ρ}} and F_{f} ≠ 0 if and only if f ≠ f^{ρ}.

Definition 4.4. If f ≠ f^{ρ}, then F_{f} is called the Maass lift of f or the CAP lift of f.

Proposition 4.5. If f = ∑_{n=1}^{∞} b(n)q^{n} ∈ S_{k−1} (4, (\frac{−4}{·})) is a normalized eigenform, then

(4.4)  c_{F_{f}}^{∗} (n) = \begin{cases}\frac{−2i}{u(n)}(b(n) − \overline{b(n)}) & if n ≢ 1 (mod 4) \\ 0 & if n ≡ 1 (mod 4) \end{cases} ,

where u(n) := #{t ∈ A | 4N(t) ≡ −n (mod 4)}.

Proof. This follows from formula (4) on page 670 in [37]. □

4.2. The Petersson norm of F_{f}. To express ⟨F_{f}, F_{f}⟩ by an L-value we will use an identity proved in [44] that involves a variant E_{s}(Z) (defined below) of the Klingen Eisenstein series E_{Q}(g, s) (which was defined in section 3.1). For a matrix M, denote by M_{i,j} the (i, j)-th entry of M. Let C be the subgroup of Γ_{Z} consisting of all matrices whose last row is [0 0 0 1]. Set

E_{s}(Z) = ∑_{γ∈C\Γ_{Z}} (\frac{det Im γZ}{(Im γZ)_{1,1}} )^{s} .

The series converges for Re (s) > 3 ([44], Lemma 1).

Lemma 4.6. Let g = (g_{∞}, 1) ∈ G(A) and Z = g_{∞}i. Then

(4.5)  E_{Q}(g, s) = \frac{1}{4}E_{3s}(Z).

Proof. First note that

E_{s}(Z) = 4 ∑_{γ∈C′\Γ_{Z}} (\frac{det Im γZ}{(Im γZ)_{1,1}} )^{s} ,

where C′ is the subgroup of Γ_{Z} consisting of matrices whose last row is of the form

[0 0 0 α] with α ∈ O_{K}^{×}. Moreover we have C′ = wQ(Z)w^{−1} with w = [ \begin{matrix}& 1 \\ 1 \\ & & & 1 \\ & & 1 \end{matrix} ].

This gives

∑_{γ∈C′\Γ_{Z}} (\frac{det Im γZ}{(Im γZ)_{1,1}} )^{s} = ∑_{γ∈Q(Z)\Γ_{Z}} (\frac{det Im wγw^{−1}Z}{(Im wγw^{−1}Z)_{1,1}} )^{s} = ∑_{γ∈Q(Z)\Γ_{Z}} (\frac{det Im γZ}{(Im γZ)_{2,2}} )^{s} ,

as w ∈ Γ_{Z}.

Now for γ ∈ Γ_{Z} we have δ_{Q}(γg) = δ_{Q}(q), where q = (q_{∞}, 1) and γg_{∞} = q_{∞}κ_{∞}

with q_{∞} ∈ Q(R), κ_{∞} ∈ K_{0,∞}. If q_{∞} = um with m = [ x ; a b ; x̂ ; c d ] ∈ M_{Q}(R) and

u ∈ U_{Q}(R), then

δ_{Q}(γg) = δ_{Q}(um) = δ_{Q}(m(m^{−1}um)) = δ_{Q}(m) = |x\overline{x}|_{A}^{3}.

Moreover

Im γZ = Im γg_{∞}i = Im q_{∞}i = Im um i.

A direct calculation shows that det Im u(mi) = det Im m i and that (Im u(mi))_{2,2} = (Im mi)_{2,2}. On the other hand

Im m i = [x\overline{x}_{\frac{1}{(ci+d)\overline{(ci+d)}}}] ,

hence we have

\frac{det Im γZ}{(Im γZ)_{2,2}} = δ_{Q}(γg)^{1/3}.

The lemma now follows from the fact that the natural injection

Q(Z) \ Γ_{Z} → Q(Q) \ G(Q)

is a bijection. This is a consequence of the identity Q(A) = Q(Q) Q(R) Q(∏_{p∤∞} Z_{p}), which follows from Lemma 8.14 of [50]. □

Set

(4.6)  E_{s}^{∗}(Z) := π^{−2s}Γ(s)Γ(s − 1)ζ(2s − 2)ζ_{K}(s)E_{s}(Z).

In [44] Raghavan and Sengupta prove that E_{s}^{∗}(Z) can be analytically continued in s to the entire complex plane except for possible simple poles at s = 0, 1 , 2 , 3. Using Lemma 4.6 and Theorem 3.10 we conclude that E_{s}^{∗}(Z) has a simple pole at s = 3 and

(4.7)  res_{s=3} E_{s}^{∗}(Z) = \frac{2}{π^{2}} ζ(3).

Combining results of section 3 of [44] with a formula on page 200 in [loc. cit.] we get

\begin{aligned}⟨F_{f}, E_{s−k+3}^{∗}F_{f}⟩ & = 4^{−3s}π^{−3s+2k−6}Γ(s)Γ(s − k + 2)Γ(s − k + 3)× \\ & × (∏_{j=1}^{3} ζ(s − k + j)) L(Symm^{2} f, s) ⟨ϕ_{1}, ϕ_{1}⟩ . \end{aligned} ; (4.8)

Here we define L(Symm^{2} f, s) for a normalized eigenform f = ∑_{n=1}^{∞} a(n)q^{n} as an Euler product:

\begin{aligned}L(Symm^{2} f, s) & = (1 − a(2)^{2}2^{−s})^{−1}(1 − \overline{a(2)}^{2}2^{−s})^{−1}× \\ & × ∏_{p≠2} [(1 − α_{p,1}^{2}p^{−s})(1 − α_{p,1}α_{p,2}p^{−s})(1 − α_{p,2}^{2}p^{−s})]^{−1} \end{aligned} ; (4.9)

where the complex numbers α_{p,1} and α_{p,2} are the p-Satake parameters of f defined by the equation

1 − a(p)x + (\frac{−4}{p}) p^{k−2}x^{2} = (1 − α_{p,1}x)(1 − α_{p,2}x).

Combining formulas (4.7) and (4.8) we obtain:

(4.10)  ⟨F_{f}, F_{f}⟩ = 2^{−2k−3}Γ(k) · π^{−k−2} ⟨ϕ_{1}, ϕ_{1}⟩ L(Symm^{2} f, k).

Finally, to relate ⟨ϕ_{1}, ϕ_{1}⟩ to ⟨f, f⟩, in the next subsection we will prove the following lemma.

Lemma 4.7. The following identity holds:

(4.11)  ⟨ϕ_{1}, ϕ_{1}⟩ = 2 ⟨f, f⟩_{Γ_{1}(N)} = 24 ⟨f, f⟩ .

Combining Lemma 4.7 with formula (4.10) we finally obtain:

Theorem 4.8. The following identity holds:

(4.12)  ⟨F_{f}, F_{f}⟩ = 2^{−2k+2} · 3 · Γ(k) · π^{−k−2} ⟨f, f⟩ L(Symm^{2} f, k).

4.3. Inner product formula for Jacobi forms. This section is devoted to proving Lemma 4.7.

Proof of Lemma 4.7. Let ψ_{1} and ψ_{2} denote two Jacobi forms of weight k and index m. It is easy to show that (4.13)

⟨ψ_{1}, ψ_{2}⟩ = ∫_{F} v^{k} (∫_{F_{τ}} ψ_{1}(τ, z, w)\overline{ψ_{2}(τ, z, w)}e^{\frac{−π|z−\overline{w}|^{2}}{v}} dz_{0} dz_{1} dw_{0} dw_{1}) du dv,

where F is the standard fundamental domain for the action of SL_{2}(Z) on the complex upper half-plane and F_{τ} ⊂ {τ} × C × C is a fundamental domain for the action of the matrices [−1 0 0] and [1 λ μ] (λ, μ ∈ O_{K}) on C × C. After performing a change of variables on C × C (keeping τ fixed)

z′ = z + w w′ = z − w,

and denoting by F′_{τ} the fundamental domain F_{τ} in the new variables, the integral over F_{τ} in (4.13) becomes

\frac{1}{8} ∫_{F′_{τ}} ψ_{1}(τ, z′, w′)\overline{ψ_{2}(τ, z′, w′)}v^{k}e^{\frac{−π| \frac{z′+w′}{2} − \frac{\overline{z′}−\overline{w′}}{2} |^{2}}{v}} dz′_{0} dz′_{1} dw′_{0} dw′_{1}.

Set ψ_{1} = ψ_{2} = ϕ_{1}, where ϕ_{1} is the first Fourier-Jacobi coefficient of the CAP form F_{f}. Using formula (4.3) we can write:

(4.14)  ⟨ϕ_{1}, ϕ_{1}⟩ = \frac{1}{8} ∑_{t∈A} ∑_{t′∈A} ∫_{F} f_{t}(τ)\overline{f_{t′}(τ)}v^{k−4}I(t, t′, τ) du dv

with

\begin{aligned}I(t, t′, τ) & = ∫_{F′_{τ}} ( ∑_{a∈t+O_{K}} ∑_{b∈t′+O_{K}} e(N(a)τ + \overline{a}z + aw) e(N(b)\overline{τ} + b\overline{z} + \overline{b}\overline{w})× \\ & × e^{− \frac{π}{v} ((Im (z′))^{2}+(Re (w′))^{2})}) dz′_{0} dz′_{1} dw′_{0} dw′_{1}. \end{aligned} ; (4.15)

Changing variables again we get

(4.16)  I(t, t′, τ) = ∑_{a∈t+O_{K}} ∑_{b∈t′+O_{K}} e(N(a)τ − N(b)\overline{τ})I_{1} I_{2}

with

I_{1} = 4 ∫_{Ω_{1}} e(2x′Re (a) − 2\overline{x′}Re (b)) e^{\frac{π}{2} (2x′_{1})}dx′_{0} dx′_{1},

where Ω_{1} is the parallelogram in C spanned by the two R-linearly independent complex numbers 1 and τ, and x′ = x′_{0} + ix′_{1} ∈ C, with x′_{0}, x′_{1} ∈ R. Before we define I_{2} we note that I_{1} can be written as

(4.17)  I_{1} = 4 ∫_{0}^{Im (τ)} e^{− \frac{π}{v} ·4(x′_{1})^{2}} (∫_{0}^{1} e(2x′Re (a) − 2\overline{x′}Re (b)) dx′_{0}) dx′_{1}.

Now the integral inside the parantheses in (4.17) equals e^{−8πRe (a) x′_{1}} if Re (a) = Re (b) and 0 otherwise. Hence

(4.18)  I_{1} = \{ \begin{matrix}4 ∫_{0}^{Im (τ)} e^{−4 \frac{π}{v} (x′_{1})^{2}} e^{−8πRe (a) x′_{1}} dx′_{1} if & Re (a) = Re (b) \\ 0  if & Re (a) ≠ Re (b) \end{matrix}

The integral

I_{2} := 4 ∫_{Ω_{2}} e(−2y′ Im (a) + 2\overline{y′} Im (b))e^{− \frac{π}{v} ·4(y′_{1})} dy′_{0} dy′_{1},

where Ω_{2} denotes the region in the complex plane spanned by the two R-linearly independent complex numbers 1 and −τ and y′ = y′_{0} + iy′_{1} ∈ C with y′_{0}, y′_{1} ∈ R, can be handled in a similar way. In fact one gets:

(4.19)  I_{2} = \{ \begin{matrix}4 ∫_{0}^{Im (τ)} e^{−4 \frac{π}{v} (y′_{1})^{2}} e^{8πIm (a) x′_{1}} dx′_{1} if & Im (a) = Im (b) \\ 0  if & Im (a) ≠ Im (b) \end{matrix} .

Substituting (4.18) and (4.19) into (4.16) one sees that I(t, t′, τ) = 0 if t ≠ t′, and that after rearranging terms

I(t, t, τ) = 16 (  ∑_{Re (a)∈Re (t)+Z} ∫_{0}^{v} e^{− \frac{4π}{v} ((Re (a))v+x′_{1})^{2}}dx′_{1}) ×

× ( ∑_{Im (a)∈Im (t)+Z} ∫_{0}^{v} e^{− \frac{4π}{v} ((Im (a))v+y′_{1})^{2}}dy′_{1}) =

= 16 ∫_{R} e^{− \frac{4π}{v} (Re (t)+x′_{1})^{2}} dx′_{1} ∫_{R} e^{− \frac{4π}{v} (Im (t)+y′_{1})^{2}} dy′_{1} = 4v, (4.20)

where τ = u + iv. Hence

(4.21)  ⟨ϕ_{1}, ϕ_{1}⟩ = ∫_{F} ∑_{t∈A} f_{t}(τ)\overline{f_{t}(τ)}v^{k−4} · 4v du dv.

From this it follows that ∑_{t∈A} f_{t}(τ)\overline{f_{t}(τ)}v^{k−1} is “invariant” under SL_{2}(Z). We want to relate (4.21) to

⟨f, f⟩′ := ∫_{Γ_{1}(4)\H} f(τ)\overline{f(τ)}v^{k−3} du dv.

Denote by ⟨f_{t}, f_{t}⟩′ the integral ∫_{Γ_{1}(4)\H} f_{t}(τ)\overline{f_{t}(τ)}v^{k−3} du dv. We will use calculations carried out in [36]. In particular one has f_{1/2} = f_{i/2} and f_{(i+1)/2} = f_{0}|_{k−1} \left[{1 \atop 2 1}\right], hence we conclude that the quantities ⟨f_{t}, f_{t}⟩′ are well-defined, since f_{0}|_{k−1}α = f_{0} for all α ∈ Γ_{1}(4). Moreover, we have

\begin{aligned}∑_{t∈A} ⟨f_{t}, f_{t}⟩′ & = ⟨f_{0}, f_{0}⟩′ + ⟨f_{(i+1)/2}, f_{(i+1)/2}⟩′ + 2 ⟨f_{1/2}, f_{1/2}⟩′ = \\ & = ⟨f_{0}, f_{0}⟩′ + ⟨f_{(i+1)/2}|_{k−1} \left[{1 \atop 2 1}\right] , f_{(i+1)/2}|_{k−1} \left[{1 \atop 2 1}\right]⟩′ + 2 ⟨f_{1/2}, f_{1/2}⟩′ = \\ & = 2 ⟨f_{0}, f_{0}⟩′ + 2 ⟨f_{1/2}, f_{1/2}⟩′ . \end{aligned} (4.22)

We use formula (3.5’) from [36], which is erroneously stated there, and should read

f_{1/2}(τ) = −\frac{i}{2} f_{0}|_{k−1} [ \begin{matrix}& −1 \\ 1 \end{matrix} ] (τ) − \frac{i}{2}f_{0}|_{k−1} \left[{−1 \atop 1 −2}\right] (τ),

hence

⟨f_{1/2}, f_{1/2}⟩′ = \frac{1}{2} ⟨f_{0}, f_{0}⟩′ + \frac{i}{2} (⟨f_{0}, f_{0}|_{k−1} \left[{1 \atop 2 1}\right]⟩′ − ⟨f_{0}|_{k−1} \left[{1 \atop 2 1}\right] , f_{0}⟩′) = \frac{1}{2} ⟨f_{0}, f_{0}⟩′

as f_{0}|_{k−1} \left[{1 \atop 4 1}\right] = f_{0}. Thus we obtain

∑_{t∈A} ⟨f_{t}, f_{t}⟩′ = 3 ⟨f_{0}, f_{0}⟩′ = 3 ⟨f_{0}|_{k−1} [ \begin{matrix}& −1 \\ 4 \end{matrix} ] , f_{0}|_{k−1} [ \begin{matrix}& −1 \\ 4 \end{matrix} ]⟩′ = 3 ⟨ϕ, ϕ⟩′ .

Since ϕ = f − f^{ρ} , and ⟨f, f^{ρ}⟩′ = 0, we get ⟨ϕ, ϕ⟩′ = 2 ⟨f, f⟩′, so finally

⟨ϕ_{1}, ϕ_{1}⟩ = \frac{4}{[SL_{2}(Z) : Γ_{1}(4)]} ∑_{t∈A} ⟨f_{t}, f_{t}⟩′ = \frac{24}{[SL_{2}(Z) : Γ_{1}(4)]} ⟨f, f⟩ = 2 ⟨f, f⟩′ .

□

5. Hecke operators

5.1. Elliptic Hecke algebra. The theory of Hecke operators acting on the space of elliptic modular forms is well-known, so we refer the reader to standard sources (e.g., [42], [14]) for definitions of most of the objects as well as their basic properties used in this subsection.

Definition 5.1. Let k be a positive integer divisible by 4, and A a Z-algebra. Denote by T_{Z} the Z-subalgebra of End_{C} (S_{k−1} (4, (\frac{−4}{·}))) generated by the Hecke operators T_{n}, n = 1, 2, . . .. We set

(1) T_{A} := T_{Z} ⊗_{Z} A;

(2) T′_{A} to be the A-subalgebra of T_{A} generated by the set

Σ′ := {T_{p}}_{p split in K} ∪ {T_{p^{2}}}_{p inert in K};

(3) T_{A}^{(2)} to be the A-subalgebra of End_{C} (S_{k−1} (4, (\frac{−4}{·}))) generated by T_{A} and the (A-linear) operator Tr T_{2} which multiplies any normalized eigenform g = ∑ a(n)q^{n} by a(2) + \overline{a(2)}.

Suppose f = ∑_{n=1}^{∞} a_{f}(n)q^{n} ∈ S_{k−1} (4, (\frac{−4}{·})) is a primitive normalized eigenform. Recall that we denote the set of such forms by N. For T ∈ T_{C}, set λ_{f,C}(T) to denote the eigenvalue of T corresponding to f. It is a well-known fact that λ_{f,C}(T_{n}) = a_{f}(n) for all f ∈ N and that the set {a_{f}(n)}_{n∈Z_{>0}} is contained in the ring of integers of a finite extension L_{f} of Q. Let E be a finite extension of Q_{ℓ} containing the fields L_{f} for all f ∈ N. Denote by O the valuation ring of E and by λ a uniformizer of O. Then {a_{f}(n)}_{f∈N ,n∈Z_{>0}} ⊂ O. Moreover, one has

(5.1)  T_{E} = ∏_{f∈N} E

and

(5.2)  T_{O} = ∏_{𝔪} T_{O,𝔪},

where T_{O,𝔪} denotes the localization of T_{O} at 𝔪 and the product runs over all maximal ideals of T_{O}. Every f ∈ N gives rise to an O-algebra homomorphism T_{O} → O assigning to T the eigenvalue of T corresponding to f. We denote this homomorphism by λ_{f} and its reduction mod λ by \overline{λ}_{f}. If 𝔪 = ker \overline{λ}_{f}, we write 𝔪_{f} for 𝔪 or if we want to emphasize the ring 𝔪 lives in, we write 𝔪_{T_{O},f}. The algebra T′_{Z} is studied in detail in section 8.1.

5.2. Hermitian Hecke algebra. The theory of Hecke operators acting on the space S_{k}(Γ_{Z}) is discussed in [23] and [37]. We summarize it here to the degree that we need it. For the formulation of the theory which is valid for hermitian modular forms of level higher than one (as well as the non-holomorphic ones) see [34]. See also [35] for a theory of Hecke operators acting on the space of adelic hermitian modular forms.

Set Δ := G_{μ}^{+}(Q) ∩ M_{4}(O_{K}). For a ∈ Δ, the double coset space Γ_{Z}aΓ_{Z} decomposes into a finite disjoint union of right cosets

Γ_{Z}aΓ_{Z} = ∐_{j} Γ_{Z}a_{j}

with a_{j} ∈ Δ. For F ∈ S_{k}(Γ_{Z}) set F |_{k}[Γ_{Z}aΓ_{Z}] := ∑_{j} F |_{k}a_{j}.

Definition 5.2. The hermitian Hecke algebra (over C), denoted by T_{C}^{h} is the subalgebra of End_{C}(S_{k}(Γ_{Z})) generated by the double cosets of the form Γ_{Z}aΓ_{Z} for a ∈ Δ. We call F ∈ S_{k}(Γ_{Z}) an eigenform if it is an eigenfunction for all T ∈ T_{C}^{h}. We will denote the eigenvalue of T corresponding to F by λ_{F,C}(T).

For a rational prime p we define an operator

(5.3)  T_{p}^{h} := Γ_{Z} [ \begin{matrix}1 \\ & 1 \\ & & p \\ & & & p \end{matrix} ] Γ_{Z},

if p is inert in K we additionally define

T_{1,p}^{h} := Γ_{Z} [ \begin{matrix}1 \\ & p \\ & & p^{2} \\ & & & p \end{matrix} ] Γ_{Z},

and if p = π\overline{π} splits or ramifies in K we define

(5.4)  T_{π}^{h} := Γ_{Z} [ \begin{matrix}1 \\ & π \\ & & p \\ & & & π \end{matrix} ] Γ_{Z}.

We now describe the action of the operators T_{p}^{h}, T_{1,p}^{h} and T_{π}^{h} on the Fourier coefficients of hermitian modular forms. As before let S := {h ∈ M_{2}(K) | h^{∗} = h}. To shorten our notation we define the following elements of GL_{2}(K):

α_{a} = \left[{1 \atop ã p}\right] , a ∈ O_{K}/pO_{K}, p inert, ã any lift of a to O_{K}
α_{p} = [\begin{matrix}p \\ & 1 \end{matrix}] , p inert;
β_{a} = \left[{1 \atop a π}\right] , a = 0, 1, …, p − 1, p = π\overline{π} split;
β_{p} = [\begin{matrix}π \\ & 1 \end{matrix}] , p = π\overline{π} split,

(5.5)

and for a 2 × 2 matrix M, we set M̃ = [ \begin{matrix}& 1 \\ 1 \end{matrix} ] M [ \begin{matrix}& 1 \\ 1 \end{matrix} ]. Moreover, if B ∈ S, we set

s(B) := \begin{cases}p & ord_{p}(det(B)) = 0; \\ −p(p − 1) & ord_{p}(det(B)) > 0, ord_{p}(ϵ(B)) = 0; \\ p^{2}(p − 1) & ord_{p}(ϵ(B)) > 0, \end{cases}

where ϵ(B) is as in Definition 4.2. Finally, if p is inert we write P_{p}^{1} for the disjoint union of O_{K}/pO_{K} and p.

Lemma 5.3. Let F ∈ S_{k}(Γ_{Z}) with Fourier expansion

F (Z) = ∑_{B∈S} c_{F} (B)e^{2πitr (BZ)},

and let T ∈ T_{C}^{h}. Then

T F (Z) = ∑_{B∈S} c_{T F} (B)e^{2πitr (BZ)},

with

(5.6) c_{T F} (B) =

p^{2k−4}c_{F} (p^{−1}B) + c_{F} (pB) + p^{k−3} ∑_{a∈P_{p}^{1}} c_{F} (p^{−1}α_{a}^{∗}Bα_{a})  T = T_{p}^{h}, p inert;
p^{2k−4}c_{F} (p^{−1}B) + c_{F} (pB) + p^{k−3} ∑_{a,b=0}^{p} c_{F} ((β_{a}β̂_{b})^{∗}Bβ_{a}β̂_{b}) T = T_{p}^{h}, p split;
p^{k−2}π^{−k} ∑_{a=0}^{p} (c_{F} (β̃_{a}^{∗}Bβ̃_{a}) + p^{k−2}c_{F} ( ˆβ̃_{a}^{∗}B ˆβ̃_{a}))  T = T_{π}^{h}, p split;
p^{2k−4}s(B) + p^{k−6} ∑_{a∈P_{p}^{1}} (c_{F} (α̃_{a}^{∗}Bα̃_{a}) + p^{2k−2}c_{F} (α̂_{a}^{∗}Bα̂_{a})) T = T_{1,p}^{h}, p inert.

Proof. This follows easily from the right coset decomposition of each of the Hecke operators. The decomposition of T_{p}^{h} was computed by Krieg in [37], p.677. The decomposition of T_{p}^{h} for split p and that of T_{π}^{h} was computed by the author in [35], Lemmas 6.5, 6.8, but see also Lemmas 6.6 and 6.9 in loc. cit. Finally, one can show that T_{1,p} decomposes in the following way:

T_{1,p}^{h} := Γ_{Z} [ \begin{matrix}1 \\ & p \\ & & p^{2} \\ & & & p \end{matrix} ] Γ_{Z} =

= Γ_{Z} [ \begin{matrix}p^{2} \\ & p \\ & & 1 \\ & & & p \end{matrix} ] ⊔ ∐_{α∈O_{K}/p} Γ_{Z} [ \begin{matrix}p & pα \\ & p^{2} \\ & & p \\ & & −\overline{α} & 1 \end{matrix} ]

⊔ ∐_{\begin{matrix}α,γ∈O_{K}/p \\ β∈Z/p^{2} \end{matrix}} Γ_{Z} [ \begin{matrix}1 & α & β+α\overline{γ} & γ \\ & p & p\overline{γ} \\ & & p^{2} \\ & & −\overline{α}p & p \end{matrix} ] ⊔ ∐_{\begin{matrix}δ∈O_{K}/p \\ ϕ∈Z/p \end{matrix}} Γ_{Z} [ \begin{matrix}p & & & pδ \\ & 1 & \overline{δ} & ϕ \\ & & p \\ & & & p^{2} \end{matrix} ] ⊔

⨆_{\begin{matrix}β,ϕ∈Z/pZ \\ βϕ≡0 (mod p) \end{matrix}} Γ_{Z} [ \begin{matrix}p & & β \\ & p & & ϕ \\ & & p \\ & & & p \end{matrix} ] ⊔ ⨆_{\begin{matrix}β∈(Z/pZ)^{×} \\ γ∈(O_{K}/pO_{K})^{×} \end{matrix}} Γ_{Z} [ \begin{matrix}p & & β & γ \\ & p & \overline{γ} & |γ|^{2}β^{−1} \\ & & p \\ & & & p \end{matrix} ] .

(5.7)

This can be deduced from the calculations in [23]. □

Remark 5.4. Note that in Lemma 5.3, we have c_{F} (B) = 0 unless B ∈ S.

For any split or ramified prime p = π\overline{π} set Σ′_{p} := \{T_{π}^{h}, T h ; \overline{π}, T_{p}\} and for any inert prime p, set Σ′_{p} := {T_{p}^{h}, T_{1,p}^{h}}.

Proposition 5.5 (Gritsenko, [23]). The Hecke algebra T_{C}^{h} is generated as a Calgebra by the set ⋃_{p} Σ′_{p}.

Proposition 5.6. The space S_{k}(Γ_{Z}) has a basis consisting of eigenforms.

Proof. This is a standard argument, which uses the fact that T_{C}^{h} is commutative and all T ∈ T_{C}^{h} are self-adjoint. □

5.3. Integral structure of the hermitian Hecke algebra. For a split or ramified prime p = π\overline{π} set

Σ_{p} = \{T_{p}^{h}, π^{k}p^{2−k}T_{π}^{h}, \overline{π}^{k}p^{2−k}T h ; \overline{π}\}

and for an inert prime p set

Σ_{p} := {T_{p}^{h}, T_{1,p}^{h}}.

Definition 5.7. Set T_{Z}^{h} (resp. T_{Z}^{h,(2)}) to be the Z-subalgebra of T_{C}^{h} generated by ⋃_{p} Σ_{p} (respectively by ⋃_{p≠2} Σ_{p}). For any Z-algebra A, set T_{A}^{h} := T_{Z}^{h} ⊗_{Z} A and

T_{A}^{h,(2)} := T_{Z}^{h,(2)} ⊗_{Z} A.

Note that T_{Z}^{h} is a finite free Z-algebra.

Lemma 5.8. Let ℓ > 2 be a rational prime, E a finite extension of Q_{ℓ} and O the valuation ring of E. Suppose that F (Z) = ∑_{B∈S} c_{F} (B)e^{2πitr (BZ)} ∈ S_{k}(Γ_{Z}) with c_{F} (B) ∈ O for all B ∈ S. Let T ∈ T_{O}^{h}. Then T F (Z) = ∑_{B∈S} c_{T F} (B)e^{2πitr (BZ)} with c_{T F} (B) ∈ O for every B ∈ S.

Proof. This follows directly from Lemma 5.3 and the assumption that ℓ be odd. (The latter implies that the operators T_{2}^{h} and (i + 1)^{k}2^{2−k}T_{i+1}^{h} preserve the Ointegrality of the Fourier coefficients of F.) □

From now on N^{h} will denote a fixed basis of eigenforms of S_{k}(Γ_{Z}).

Theorem 5.9. Let F ∈ N^{h}. There exists a number field L_{F} with ring of integers O_{L_{F}} such that λ_{F,C}(T) ∈ O_{L_{F}} for all T ∈ T_{O_{L_{F}}}^{h} .

Proof. This is similar to the Eichler-Shimura isomorphism in the case of elliptic modular forms. □

Let ℓ be a rational prime and E a finite extension of Q_{ℓ} containing the fields L_{F} from Theorem 5.9 for all F ∈ N^{h}. Denote by O the valuation ring of E and by λ a uniformizer of O. As in the case of elliptic modular forms, F ∈ N^{h} gives rise to an O-algebra homomorphism T_{O}^{h} → O assigning to T the eigenvalue of T corresponding to the eigenform F. We denote this homomorphism by λ_{F} and its mod λ reduction by \overline{λ}_{F}. Theorem 5.9 implies that we have

T_{E}^{h} ≅ ∏_{F ∈N^{h}} E.

Moreover, as in the elliptic modular case, we have

(5.8)  T_{O}^{h} ≅ ∏_{𝔪} T_{O,𝔪}^{h},

where the product runs over the maximal ideals of T_{O}^{h} and T_{O,𝔪}^{h} denotes the localization of T_{O}^{h} at 𝔪. A similar description holds for T_{O}^{h,(2)}. As before, if 𝔪 = ker \overline{λ}_{F}, we write 𝔪_{F} for 𝔪 or if we want to emphasize what ring 𝔪 lives in, we write 𝔪_{T_{O}^{h},F} or 𝔪_{T_{O}^{h,(2)},F} accordingly.

5.4. Action on the Maass space.

Theorem 5.10 (Gritsenko, [23], section 2). The action of the Hecke algebra T_{C}^{h} respects the decomposition of S_{k}(Γ_{Z}) into the Maass space and its orthogonal complement.

Theorem 5.11 (Gritsenko, [23], section 3). There exists a C-algebra map

Desc : T_{C}^{h} → T_{C}^{(2)}

such that for every T ∈ T_{C}^{h} the diagram

\xymatrix{S_{k}^{Maass}(Γ_{Z}) \ar[r]^{T} & S_{k}^{Maass}(Γ_{Z}) \\ S_{k−1} (4, (\frac{−4}{·})) \ar[r]^{Desc(T)} \ar[u]^{f↦F_{f}} & S_{k−1} (4, (\frac{−4}{·})) \ar[u]_{f↦F_{f}}}

commutes. In particular one has

Desc(T_{p}^{h}) = p^{k−1} + p^{k−2} + p^{k−3} + T_{p^{2}} for all p ≠ 2,
Desc(T_{1,p}^{h}) = p^{k−4}(1 + p^{2})T_{p^{2}} + p^{2k−8}(p^{3} + p^{2} + p − 1) if p is inert in K,
Desc(T_{π}^{h}) = p^{k−2}π^{−k}(1 + p)T_{p} if p = π\overline{π} is split in K,
Desc(T_{1+i}^{h}) = 3 · 2^{k−4}(1 + i)^{−k}Tr T_{2}
Desc(T_{2}^{h}) = 2^{k−4}(1 + i)^{−k}((Tr T_{2})^{2} − 2^{k−1}).

(5.9)

Here T_{n} is as in section 5.1, and Tr T_{2} denotes the operator from Definition 5.1.

Corollary 5.12. If f ∈ S_{k−1} (4, (\frac{−4}{·})) is an eigenform, then so is F_{f}.

Remark 5.13. Let f ∈ N, f ≠ f^{ρ}. We will always assume that either F_{f} or F_{f^{ρ}} belongs to N^{h}. Hence we can write N^{h} = N^{M} ⊔N^{NM}, where N^{M} consists of Maass lifts F_{f} with f ∈ N and N^{NM} consists of eigenforms orthogonal to those in N^{M}.

5.5. Lifting Hecke operators to the Maass space. Let E and O be as before. We will now prove a result regarding the map Desc, which will be used in section 7.3. Let T_{Z} and T′_{Z} be as in Definition 5.1. It is clear from Theorem 5.11 and the definition of T_{Z}^{h,(2)} that Desc(T_{A}^{h,(2)}) = T′_{A} for any O-algebra A. Moreover, we have the following diagram

(5.10)  \xymatrix{T_{O}^{h,(2)} \ar[r]^{Desc} \ar[d]^{≀} & T′_{O} \ar[d]_{≀} \\ ∏_{𝔪^{(2)}} T_{O,𝔪^{(2)}}^{h,(2)} \ar[r] & ∏_{𝔪′} T′_{O,𝔪′}}

with the lower horizontal arrow defined so that the diagram commutes. It is clear that Desc respects the direct product decomposition in diagram (5.10). In particular, for f ∈ N, Desc : T_{O}^{h,(2)} ↠ T′_{O} factors through T_{O,𝔪_{F_{f}}^{(2)}}^{h,(2)} ↠ T′_{O,𝔪′_{f}}. Let T_{O}^{M}

be the image of T_{O}^{h,(2)} in End_{C}(S_{k}^{M}(Γ_{Z})). The horizontal arrows in diagram (5.10) factor through T_{O}^{M} ≅ ∏_{𝔪^{M}} T_{O,𝔪^{M}}^{M} and the following diagram

(5.11)  \xymatrix{T_{O}^{h,(2)} \ar[r] \ar[d]^{≀} & T_{O}^{M} \ar[r] \ar[d]^{≀} & T′_{O} \ar[d]_{≀} \\ ∏_{𝔪^{(2)}} T_{O,𝔪^{(2)}}^{h,(2)} \ar[r] & ∏𝔪̃ T_{O,𝔪^{M}}^{M} \ar[r] & ∏_{𝔪′} T′_{O,𝔪′}}

commutes. All the horizontal arrows in diagram (5.11) are surjections and the lower ones are induced from the upper ones, which respect the direct product decompositions. In particular we have

T_{O,𝔪_{F_{f}}^{(2)}}^{h,(2)} ↠ T_{O,𝔪_{F_{f}}^{M}}^{M} ↠ T′_{O,𝔪′_{f}} .

Let N_{F_{f}}^{M} := {F ∈ N^{M} | 𝔪_{F_{f}}^{M} = 𝔪_{F}^{M}}. The goal of this section is to prove the following proposition.

Proposition 5.14. If f ∈ N, f ≠ f^{ρ} is ordinary at ℓ, and ℓ ∤ (k −1)(k −2)(k −3), then for every split prime p = π\overline{π}, p ∤ ℓ, there exists T^{M}(p) ∈ T_{O,𝔪_{F_{f}}^{M}}^{M} such that

Desc(T^{M}(p)) ∈ T′_{O,𝔪′_{f}} equals the image of T_{p} ∈ T′_{O} under the canonical projection T′_{O} ↠ T′_{O,𝔪′_{f}}.

As will be discussed in section 9.1, to every eigenform F ∈ S_{k}(Γ_{Z}) one can attach a 4-dimensional ℓ-adic Galois representation ρ_{F}. Moreover, if F = F_{g}, for some g ∈ N, then the Galois representation has a special form

(5.12)  ρ_{F_{g}} = [\begin{matrix}ρ_{g}|_{G_{K}} \\ & (ρ_{g} ⊗ ϵ)|_{G_{K}} \end{matrix} ] ,

where ρ_{g} is the Galois representation attached to g (cf. section 2.3) and ϵ is the ℓ-adic cyclotomic character. Let f be as in Proposition 5.14. Set R′ := ∏_{F ∈N_{F_{f}}^{M}} O

and let R be the O-subalgebra of R′ generated by the tuples (λ_{F} (T))_{F ∈N_{F_{f}}^{M}} for all

T ∈ T_{O}^{M}. Note that the expression λ_{F} (T) makes sense since S_{k}^{M}(Γ_{Z}) is Hecke stable. Then R is a complete Noetherian local O-algebra with residue field F = O/λ. It is a standard argument to show that R ≅ T_{O,𝔪_{F_{f}}^{M}}^{M} .

Proof of Proposition 5.14. Let I_{ℓ} denote the inertia group at ℓ. For every g ∈ N, ordinary at ℓ, we have by (5.12) and Theorem 3.26 (2) in [27] that

ρ_{F_{g}}|_{I_{ℓ}} ≅ [\begin{matrix}ϵ^{k−2} & ∗ \\ & 1 \\ & & ϵ^{k−1} & ∗ \\ & & & ϵ \end{matrix}] .

If ℓ ∤ (k − 1)(k − 2)(k − 3) it is easy to see that there exists σ ∈ I_{ℓ} such that the elements β_{1} := ϵ^{k−2}(σ), β_{2} := 1, β_{3} := ϵ^{k−1}(σ), β_{4} := ϵ(σ) are all distinct mod λ. For every g as above, we choose a basis of the space of ρ_{g} so that ρ_{g} is O-valued and ρ_{F_{g}}(σ) = diag(β_{1}, β_{2}, β_{3}, β_{4}). Let S be the set consisting of the places of K lying over ℓ and the place (i + 1). Note that we can treat ρ_{F_{g}} as a representation of G_{K,S}, the Galois group of the maximal Galois extension of K unramified away from S. Moreover, tr ρ_{F_{g}}(G_{K,S}) ⊂ R, since G_{K,S} is generated by conjugates of Frob_{𝔭}, 𝔭 ∉ S and for such a 𝔭, tr ρ_{F_{g}}(Frob_{𝔭}) ∈ R by Theorem 9.2 (i) and the fact that the coefficients of the characteristic polynomial of ρ_{F_{g}}(Frob_{𝔭}) belong to T_{O}^{h}. Set

e_{j} = ∏_{l≠j} \frac{σ − β_{l}}{β_{j} − β_{l}} ∈ O[G_{K,S}] ↪ R[G_{K,S}]

and e := e_{1} + e_{2}. Let

ρ := ∏_{F_{g}∈N_{F_{f}}^{M}} ρ_{F_{g}} : G_{K,S} → ∏_{F_{g}∈N_{F_{f}}^{M}} GL_{4}(O).

We extend ρ to an R-algebra map ρ′ : R[G_{K,S}] → M_{4}(R′). Note that

ρ′(Frob_{π} e) = ∏_{F_{g}∈N_{F_{f}}^{M}} ρ_{F_{g}}(Frob_{π})ρ′_{F_{g}}(e) = ∏_{F_{g}∈N_{F_{f}}^{M}} ρ_{g}(Frob_{π})

and thus

tr ρ′(Frob_{π} e) = (a_{g}(p))_{F_{g}∈N_{F_{f}}^{M}} ∈ R,

where g = ∑_{n=1}^{∞} a_{g}(n)q^{n}. Define T^{M}(p) to be the image of tr ρ′(Frob_{π} e) under the O-algebra isomorphism R ⟶^{∼} T_{O,𝔪_{F_{f}}^{M}}^{M} . □

Corollary 5.15. If f ∈ N, f ≠ f^{ρ} is ordinary at ℓ, and ℓ ∤ (k − 1)(k − 2)(k − 3), then for every split prime p = π\overline{π}, p ∤ ℓ, there exists T^{h}(p) ∈ T_{O,𝔪_{F_{f}}^{(2)}}^{h,(2)} such that

Desc(T^{h}(p)) ∈ T′_{O,𝔪′_{f}} equals the image of T_{p} ∈ T′_{O} under the canonical projection T′_{O} ↠ T′_{O,𝔪′_{f}}.

6. The standard L-function of a Maass lift

Let F_{f} be the Maass lift of f ∈ N. The goal of this section is to study the numerator of the coefficient C_{F_{f}} in formula (1.1). To do so we need to define the cusp form Ξ in (1.1). This will be done in subsection 7.3 (formula (7.15)). In this section we define an Eisenstein series E(Z, s, m, Γ^{h}) and a theta series θ_{χ} such that their product is closely related to Ξ. We then express the inner product ⟨F_{f}, E(Z, s, m, Γ^{h})θ_{χ}⟩ by an L-function associated to f.

We begin by defining the appropriate theta series which will be used in the inner product. Let 𝔣 be an ideal of O_{K} and χ a Hecke character of K with conductor 𝔣. We assume that the infinity component of χ has the form

χ_{∞}(x_{∞}) = \frac{|x_{∞}|^{t}}{x_{∞}^{t}} ,

for some integer −k ≤ t < −6. Following [50] we fix a Hecke character ϕ of K such that

ϕ_{∞}(y_{∞}) = \frac{|y_{∞}|}{y_{∞}} and ϕ|_{A^{×}} = (\frac{−4}{·}) .

Such a character always exists, but is not unique (cf. [51], lemma A.5.1). Put ψ′ = χ^{−1}ϕ^{−2}. Let l = t + k + 2 and μ = l − 2. Let τ ∈ S be such that the Fourier coefficient c_{F_{f}} (τ) is non-zero. Let b ∈ Q be such that g^{∗} τ g ∈ bZ for all g ∈ O_{K}^{2}, and let c′ ∈ Z be such that g^{∗} τ^{−1} g ∈ (c′)^{−1}Z for all g ∈ O_{K}^{2}. Let c ∈ Z be such that bc generates the Z-fractional ideal (4c′)N_{K/Q}(𝔣) ∩ (b)𝔣. Note that when b = 1,

(c) = (4c′N_{K/Q}(𝔣)).

Define a Schwartz function λ : M_{2}(A_{K,f}) → C by setting λ(x) = χ_{𝔣}(det x) if x ∈ ∏_{𝔭∤∞} M_{2}(O_{K,𝔭}) and λ(x) = 0 otherwise. Then the theta series of our interest is defined by:

θ_{χ}(Z) = ∑_{ξ∈M_{2}(K)} λ(ξ) (\overline{det ξ})^{μ} e(tr (ξ^{∗} τξZ)).

We have θ_{χ} ∈ M_{l}(Γ_{0}^{h}(b, c), ψ′) by [50], appendix, Proposition 7.16 and [51], page 278. In fact, since μ ≠ 0, θ_{χ} is cusp form ([50], appendix, page 277). In this section we will denote by Γ_{1}^{h} a congruence subgroup of Γ_{Z} such that θ_{χ} ∈ M_{l}(Γ_{1}^{h}) and Γ_{1}^{h} ∩ K^{×} = {1}. We set Γ^{h} := Γ_{1}^{h} ∩ G_{1}(Q). Note that we have F_{f} ∈ M_{k}(Γ^{h}). We also define an Eisenstein series of weight m = k − l and level Γ^{h} by putting:

E(Z, s, m, Γ^{h}) = ∑_{γ∈Γ^{h}∩P (Q)\Γ^{h}}(det Im Z)^{s− \frac{m}{2}} |_{m}γ.

The Petersson inner product of F_{f} against E(·, s, m, Γ^{h}) θ_{χ} has the form

⟨F_{f}, E(·, s, m, Γ^{h}) θ_{χ})⟩_{Γ^{h}} = ∫_{Γ^{h}\H} F_{f}(Z) \overline{E(Z, s, m, Γ^{h})} \overline{θ_{χ}(Z)}(det Im Z)^{k−4}dXdY.

Note that we use a volume form, which is 4 times the volume form used in [51]. By combining formulas (22.9), (22.18b) and (20.19) from [51] we arrive at the following:

⟨F_{f}, E(·, \overline{s}, m, Γ^{h}) θ_{χ})⟩_{Γ^{h}} = 64[Γ_{0}^{h}(c) : Γ_{1}^{h}(c)]b^{−4}Γ((s − 2))(det τ)^{−s− \frac{1}{2} (k+l)+2}×

× \frac{c_{F_{f}} (τ)L_{st}(F_{f}, s + 1, χ)}{B(s) L_{c}(2s, χ_{Q}) L_{c} (2s − 1, χ_{Q} (\frac{−4}{·}))}.

(6.1)

The meaning of the various factors in the product is explained below. We start with the L-function

L_{st}(F_{f}, s, χ) = ∏_{p∤∞} L_{st}(F_{f}, s, χ)_{p}.

This is the standard L-function of F_{f} twisted by the Hecke character χ: (6.2)

L_{st}(F_{f}, s, χ)_{p} = \begin{cases}∏_{j=1}^{4}\{(1 − N(𝔭)^{4}λ_{p,j}^{−1}χ^{∗}(𝔭)N(𝔭)^{−s})(1 − N(\overline{𝔭})^{4}λ_{p,j}χ^{∗}(\overline{𝔭})N(\overline{𝔭})^{−s})\} \\ ∏_{j=1}^{2}\{(1 − N(𝔭)^{2}λ_{p,j}^{−1}χ^{∗}(𝔭)N(𝔭)^{−s})(1 − N(𝔭)λ_{p,j}χ^{∗}(𝔭)N(𝔭)^{−s})\}^{−1}, \end{cases}

for (p) = 𝔭\overline{𝔭} and (p) = 𝔭^{e}, respectively. Here λ_{p,i} denote the p-Satake parameters of F_{f}. (For the definition of p-Satake parameters when p inerts or ramifies in K, see [29], and for the case when p splits in K, see [24].) The L-function in the denominator of (6.1) is the Dirichlet L-function with Euler factors at all p | c removed (cf. Definition 7.2). Furthermore,

Γ((s)) = (4π)^{−2s−k−l+1} Γ (s + \frac{1}{2} (k + l)) Γ (s + \frac{1}{2} (k + l) − 1) ,

and B(s) = ∏_{v∈b} g_{p}(χ^{∗}(pO_{K})p^{−2s}), where b denotes the set of primes at which b^{−1}τ is not regular in the sense of ([51], 16.1) and g_{p} is a polynomial with coefficients in Z and constant term 1.

For a prime 𝔭 of O_{K} of residue characteristic p, with p odd, set α_{𝔭,j} := α_{p,j}^{d}, where α_{p,j}, j = 1, 2 denote the p-Satake parameters of f (cf. section 4.2), and d is the degree over F_{p} of the field O_{K}/𝔭. For the prime 𝔭 = (i + 1) of O_{K}, set α_{𝔭,1} := a(2) and α_{𝔭,2} := \overline{a(2)}.

Definition 6.1. For a Hecke character ψ of K, set

L(BC(f), s, ψ) := ∏_{𝔭∤∞} ∏_{j=1}^{2}(1 − ψ^{∗}(𝔭)α_{𝔭,j}(N𝔭)^{−s}).

Remark 6.2. If π_{f} denotes the automorphic representation of GL_{2}(A) associated with f, then L(BC(f), s, ψ) is the classical analogue of the L-function attached to the base change of π_{f} to K twisted by ψ.

Remark 6.3. Let g_{ψ} be the modular form associated with the character ψ (cf.

[31], section 12.3) and suppose that ψ_{∞}(x_{∞}) = (\frac{x_{∞}}{|x_{∞}|})^{u}. Then

L(BC(f), s, ψ) = (1 − ψ^{∗}(𝔭)\overline{a(2)}2^{−s})^{−1}D(s + u/2, f^{ρ}, g_{ψ}),

where D(s, ·, ·) denotes the convolution L-function defined in [26], where it is denoted by L(λ_{f^{ρ}} ⊗ λ_{g_{ψ}}, s). Here 𝔭 denotes the prime of O_{K} lying over (2).

Proposition 6.4. Let χ be as before. The following identity holds

(6.3)  L_{st}(F_{f}, s, χ) = L(BC(f), s − 2 + k/2, ωχ)L(BC(f), s − 3 + k/2, ωχ).

Here ω is the unique Hecke character of K unramified at all finite places with infinity type ω_{∞}(z) = (\frac{z}{\overline{z}} )^{−k/2}.

Proof. This is a straightforward calculation on the Satake parameters of f and of F_{f}. □

7. Congruence

In this section we define a hermitian modular form Ξ as in (1.1) and formulate the main congruence result (Theorem 7.12). The form Ξ will be constructed (in section 7.3) as a combination of a certain Eisenstein series and a theta series, whose arithmetic properties are studied below.

7.1. Fourier coefficients of Eisenstein series. We keep the notation from section 6 and assume b = 1. Consider the set X_{m,c} of Hecke characters χ′ of K, such that

(7.1)  χ′_{∞}(x_{∞}) = \frac{x_{∞}^{m}}{|x_{∞}|^{m}} ,
(7.2)  χ′_{p}(x_{p}) = 1 if p ∤ ∞, x_{p} ∈ O_{K,p}^{×} and x_{p} − 1 ∈ cO_{K,p}.

Here m = k − l = −t − 2 > 0 (since t < −6) denotes the weight of the Eisenstein series E(Z, s, m, Γ^{h}) defined in section 6. For g ∈ G(A), let E(g, s, c, m, χ′) denote the Siegel Eisenstein series defined in section 3.2. We put, as before,

E(Z, s, m, χ′, c) = j(g_{∞}, i)^{m}E(g, s, c, m, χ′),

where Z = g_{∞}i and g = (g_{∞}, 1). Recall that in section 6 we made use of a congruence subgroup Γ_{1}^{h} of G(Q) such that θ_{χ} ∈ M_{l}(Γ_{1}^{h}) and Γ_{1}^{h} ∩ K^{×} = {1}. In this section we fix a particular choice of Γ_{1}^{h}, namely, we set Γ_{1}^{h} := Γ_{1}^{h}(c). Note that as long as c ∤ 2, we have Γ_{1}^{h}(c) ∩ K^{×} = {1} and since (cond ψ′) | c, where ψ′ is the character of θ_{χ}, we have θ_{χ} ∈ M_{l}(Γ_{1}^{h}(c)). The following lemma provides a connection between E(Z, s, m, χ′, c) and E(Z, s, m, Γ_{1}^{h}(c)). Here E(Z, s, m, Γ_{1}^{h}(c)) is defined in the same way as E(Z, s, m, Γ^{h}) in section 6. Recall that Γ^{h} := Γ_{1}^{h}∩G_{1}(Q).

Lemma 7.1. The set X_{m,c} is non-empty and

(#X_{m,c}) E(Z, s, m, Γ_{1}^{h}(c)) = ∑_{χ′∈X} E(Z, s, m, χ′, c).

Proof. This is identical to the proof of Lemma 17.2 in [51]. Note that Γ_{1}^{h}(c) ⊃ Γ^{h}(c). □

Definition 7.2. Let M be a non-zero integer. For a Hecke character ψ of Q set

L_{M}(s, ψ) := L(s, ψ) ∏_{p|M}(1 − ψ^{∗}(p)p^{−s}),

where L(s, ψ) denotes the Dirichlet L-function.

Recall that for any Hecke character ψ : K^{×} \ A_{K}^{×} → C^{×} we denote by ψ_{Q} its restriction to A^{×}. Moreover, if ψ satisfies (7.1) and (7.2) for c ∈ Z, set ψ^{c}(x) = ψ(\overline{x}). Let

(7.3) D(Z, s, m, χ′, c) = L_{c}(2s, χ′_{Q})L_{c} (2s − 1, χ′_{Q} (\frac{−4}{·})) E(Z, s, m, χ′, c).

It has been shown in [51] (Theorem 17.12(iii)) that D(Z, s, m, χ′, c) is holomorphic in the variable Z for s = 2 − \frac{m}{2} as long as m ≥ 2. In our case m = −t − 2 > 4 as t < −6. It follows from formula (18.6.2) in [50] that

(7.4) D(Z, s, m, χ′, c)|_{m}γ = (χ′_{c})(det d_{γ})D(Z, s, m, χ′, c) =

= ((χ′)^{c})_{c}^{−1}(det a_{γ})D(Z, s, m, χ′, c).

Instead of looking at D(Z, s, m, χ′, c) we will study the Fourier expansion of a transform D^{∗}(Z, s, m, χ′, c) defined by

(7.5)  D^{∗}(Z, s, m, χ′, c) = D(Z, s, m, χ′, c)|_{m}J.

First note that since D is holomorphic at s = 2 − \frac{m}{2}, so is D^{∗}. Write

D^{∗}(Z, 2 − m/2, m, χ′, c) = ∑_{h∈S} c_{h}^{χ′}e(tr hZ)

for the Fourier expansion of D^{∗}. Here S := {h ∈ M_{2}(K) | h^{∗} = h}.

Lemma 7.3.

(7.6) c_{h}^{χ′} = i^{−2m}2^{2m+1}π^{3}c^{2}×

× ∏_{j=0}^{1−rank(h)} L_{c} (2 − m − j, χ′ (\frac{−4}{·})^{j−1}) ∏_{p∈c} f_{h,Y^{1/2},p}(χ′(p)p^{m−4}),

where f_{h,Y^{1/2},p} is a polynomial with coefficients in Z and constant term 1, and c is a certain finite set of primes. If n < 1 we set ∏_{j=0}^{n} = 1.

Proof. The lemma follows from Propositions 18.14 and 19.2 in [50], combined with Lemma 18.7 of [50] and formulas (4.34K) and (4.35K) in [49]. It is a straightforward calculation. □

Proposition 7.4. Fix a prime ℓ ∤ 2c, and assume that −k ≤ t < −6. Set χ_{Q,c} := ∏_{p|c} χ_{Q,p}. Let E′ be a finite extension of Q_{ℓ} containing K(χ_{Q,c}), the finite extension of K generated by the values of χ_{Q,c}. Denote by O′ the valuation ring of E′. For every h ∈ S, we have π^{−3}c_{h}^{χ_{Q}} ∈ O′.

Proof. The proposition follows from Lemma 7.3 upon noting that for every Dirichlet character ψ of conductor dividing c and every n ∈ Z_{<0}, one has L(n, ψ) ∈ Z_{ℓ}[ψ] (by a simple argument using [59], Corollary 5.13) and (1 − ψ(p)p^{−n}) ∈ Z_{ℓ}[ψ] for every p | c. □

Let θ_{χ} be as above. Set θ_{χ}^{∗} := θ_{χ}|_{l}J.

Corollary 7.5. Fix a prime ℓ ∤ 2c, and assume that −k ≤ t < −6. Let E′ be a finite extension of Q_{ℓ} containing K(χ_{Q,c}, μ_{c}), where μ_{c} denotes the set of c-th roots of 1. Denote by O′ the valuation ring of E′. Then the Fourier coefficients of

π^{−3}D^{∗}(Z, 2 − m/2, m, (ψ′)^{c}, c) θ_{χ}^{∗}(Z)

all lie in O′.

Proof. Note that it follows from the definition of θ_{χ} and Theorem 2.2 that the Fourier coefficients of θ_{χ}^{∗}(Z) lie in O. Thus the Corollary is a consequence of Proposition 7.4. □

7.2. Some formulae. We keep notation from the previous section. Note that since θ_{χ} ∈ M_{l}(c, ψ′), we have D(Z, 2 − m/2, m, (ψ′)^{c}, c) θ_{χ}(Z) ∈ M_{k}(c) by (7.4). For f ∈ N we can write (7.7)

D(Z, 2 − m/2, m, (ψ′)^{c}, c) θ_{χ}(Z) = \begin{matrix}⟨D(·, 2 − m/2, m, (ψ′)^{c}, c) θ_{χ}, F_{f}⟩ & Γ_{0}^{h}(c) \\ ⟨F_{f}, F_{f}⟩ & Γ_{0}^{h}(c) \end{matrix} F_{f} + F′,

where F′ ∈ M_{k}(c) and ⟨F_{f}, F′⟩ = 0. Our goal now is to express

(7.8)  ⟨D(·, 2 − m/2, m, (ψ′)^{c}, c) θ_{χ}, F_{f}⟩_{Γ_{0}^{h}(c)}

in terms of L-functions of f. In section 6 we already carried out this task for the inner product ⟨F_{f}, E(·, s, m, Γ^{h}) θ_{χ}⟩_{Γ^{h}} with Γ^{h} = Γ_{1}^{h}(c) ∩ G_{1}(Q), so we will now relate the two inner products to each other. We first relate the inner product (7.8) to ⟨F_{f}, E(·, s, m, Γ_{1}^{h}(c)) θ_{χ}⟩_{Γ_{1}^{h}(c)}.

We have

(7.9) ⟨E(·, s, m, Γ_{1}^{h}(c)) θ_{χ}, F_{f}⟩_{Γ_{1}^{h}(c)} =

∫_{Γ_{1}^{h}(c)\H} E(Z, s, m, Γ_{1}^{h}(c)) θ_{χ}(Z) \overline{F_{f}(Z)} (det Y)^{k−4} dX dY =

= ∫_{Γ_{0}^{h}(c)\H} θ_{χ}(Z) ( ∑_{γ∈Γ_{1}^{h}(c)\Γ_{0}^{h}(c)} ψ′_{c}(det a_{γ}) E(Z, s, m, Γ_{1}^{h}(c))|_{m}γ) \overline{F_{f}(Z)} dX dY.

Using Lemma 7.1 (note that (ψ′)^{c} ∈ X_{m,c}) we get

(7.10) ∑_{γ∈Γ_{1}^{h}(c)\Γ_{0}^{h}(c)} ψ′_{c}(det a_{γ}) E(Z, s, m, Γ_{1}^{h}(c))|_{m}γ =

= (#X_{m,c})^{−1} ∑_{χ′∈X} ∑_{γ∈Γ_{1}^{h}(c)\Γ_{0}^{h}(c)} ψ′_{c}(det a_{γ}) E(Z, s, m, χ′, c)|_{m}γ =

= (#X_{m,c})^{−1} ∑_{χ′∈X} E(Z, s, m, χ′, c) ∑_{γ∈Γ_{1}^{h}(c)\Γ_{0}^{h}(c)}(ψ′((χ′)^{c})^{−1})_{c}(det a_{γ}) =

= (#X_{m,c})^{−1}[Γ_{0}^{h}(c) : Γ_{1}^{h}(c)]E(Z, s, m, (ψ′)^{c}, c),

where the last equality follows from the orthogonality relation for characters upon noting that both ψ′ and χ′ are trivial on Γ_{1}^{h}(c). Thus (7.3), (7.9) and (7.10) imply that

(7.11) ⟨D(·, s, m, (ψ′)^{c}, c)θ_{χ}, F_{f}⟩_{Γ_{0}^{h}(c)} = [Γ_{0}^{h}(c) : Γ_{1}^{h}(c)]^{−1} #X_{m,c} · L_{c}(2s, ψ′_{Q})×

× L_{c} (2s − 1, ψ′_{Q} (\frac{−4}{·})) ⟨E(·, s, m, Γ_{1}^{h}(c))θ_{χ}, F_{f}⟩_{Γ_{1}^{h}(c)} .

Moreover, by [51], formula (17.5) and Remark 17.12(2), we have

E(Z, s, m, Γ_{1}^{h}(c)) = \frac{1}{[Γ_{1}^{h}(c) : Γ^{h}]} ∑_{α∈Γ^{h}\Γ_{1}^{h}(c)} E(Z, s, m, Γ^{h})|_{m}α.

Hence we get

(7.12) ⟨D(·, s, m, (ψ′)^{c}, c)θ_{χ}, F_{f}⟩_{Γ_{0}^{h}(c)} = [Γ_{0}^{h}(c) : Γ^{h}]^{−1} #X_{m,c} · L_{c}(2s, ψ′_{Q})×

× L_{c} (2s − 1, ψ′_{Q} (\frac{−4}{·})) ⟨E(·, s, m, Γ^{h})θ_{χ}, F_{f}⟩_{Γ^{h}} .

Using (6.1) we obtain

⟨D(·, s, m, (ψ′)^{c}, c)θ_{χ}, F_{f}⟩_{Γ_{0}^{h}(c)} = 16π\overline{((4π)^{−2s′})}(det τ)^{−s′}B(\overline{s})^{−1} · #X_{m,c}×

× L_{c}(2s, ψ′_{Q})L_{c} (2s − 1, ψ′_{Q} (\frac{−4}{·})) ×

× \overline{Γ(s′)} \overline{Γ(s′ − 1)} \begin{matrix}\overline{c_{F_{f}} (τ)} & \overline{L_{st}(F_{f}, \overline{s} + 1, χ)} \\ \overline{L_{c}(2\overline{s}, χ_{Q})} & \overline{L_{c} (2\overline{s} − 1, χ_{Q} (\frac{−4}{·}))} \end{matrix},

(7.13)

where s′ := \overline{s} + k − 1 + t/2, and finally

\begin{aligned}⟨D(·, 2 − m/2, m, (ψ′)^{c}, c)θ_{χ}, F_{f}⟩_{Γ_{0}^{h}(c)} & = Rπ^{−2t−2k−3}Γ(t + k + 2)Γ(t + k + 1)× \\ & × \overline{L_{st}(F, 3 − m/2, χ)}, \end{aligned} (7.14)

where R := #X_{m,c} · 2^{−4(t+k+1)}\overline{c_{F_{f}} (τ)}B(2 − m/2)^{−1} (det τ)^{−t−k−2}.

7.3. Main congruence result. We will now prove the first main result of this paper. We will show that λ^{n}-divisibility of the algebraic part of L(Symm^{2} f, k) implies the existence of a non-Maass cusp form congruent to F_{f} modulo λ^{n}. We keep notation from previous sections.

7.3.1. Algebraicity of C_{F_{f}}. Let ℓ ∤ 2c be a rational prime, and let E be a finite extension of Q_{ℓ} with valuation ring O. We will always assume that E is “sufficiently large” in the sense that it contains certain algebraic numbers/number fields, which will be specified later. In particular, we assume that E contains the field E′ of Corollary 7.5. Fix a uniformizer λ ∈ O. We denote the λ-adic valuation by ord_{λ}. To shorten notation in this section we set D(Z) := D(Z, 2 − m/2, m, (ψ′)^{c}, c), and D^{∗}(Z) = D^{∗}(Z, 2 − m/2, m, (ψ′)^{c}, c). Applying the operator |_{k}J to both sides of (7.7), we get

D^{∗}θ_{χ}^{∗} = \frac{⟨Dθ_{χ}, F_{f}⟩}{⟨F_{f}, F_{f}⟩}F_{f} + G′ ∈ M_{k}(J^{−1}Γ_{0}^{h}(c)J)

where G′ := F′|_{k}J and we have ⟨F_{f}, G′⟩ = 0. By Corollary 7.5, the Fourier coefficients of π^{−3}D^{∗}θ_{χ}^{∗} lie in O. Define a trace operator

tr : M_{k}(J^{−1}Γ_{0}^{h}(c)J) → M_{k}(Γ_{Z})

by

F′ ↦ ∑_{γ∈J^{−1}Γ_{0}^{h}(c)J\Γ_{Z}} F′|_{k}γ

and set

(7.15)  Ξ := π^{−3}tr (D^{∗}θ_{χ}^{∗}) = [Γ_{Z} : Γ_{0}^{h}(c)]π^{−3} \frac{⟨Dθ_{χ}, F_{f}⟩}{⟨F_{f}, F_{f}⟩} F_{f} + G′′,

where G′′ = π^{−3}tr G′ ∈ M_{k}(Γ_{Z}) and we have ⟨F_{f}, G′′⟩ = 0. By the q-expansion principle (Theorem 2.2), the Fourier coefficients of Ξ lie in O. Set

C_{F_{f}} := [Γ_{Z} : Γ_{0}^{h}(c)]π^{−3} \frac{⟨Dθ_{χ}, F_{f}⟩}{⟨F_{f}, F_{f}⟩}.

By Proposition 4.5, the Fourier coefficients of F_{f} lie in the ring of integers \overline{Z}_{ℓ} of \overline{Q}_{ℓ} and generate a finite extension of Q_{ℓ}. We assume E contains all the Fourier coefficients of F_{f}. The numerator and denominator of \frac{⟨Dθ_{χ},F_{f} ⟩}{⟨F_{f} ,F_{f} ⟩} were studied in sections 7.2 and 4.2 respectively.

Lemma 7.6.

\frac{⟨Dθ_{χ}, F_{f}⟩}{⟨F_{f}, F_{f}⟩} = (∗) α \frac{L^{alg}(BC(f), 1 + \frac{t+k}{2} , \overline{χ}\overline{ω})L^{alg}(BC(f), 2 + \frac{t+k}{2} , \overline{χ}\overline{ω})}{L^{alg}(Symm^{2} f, k)},

where

α := #X_{m,c} · B(2 − m/2)^{−1}(det τ)^{−k−t−2}π^{3}\overline{c_{F_{f}} (τ)},

L^{alg}(BC(f), j + (t + k)/2, \overline{χ}\overline{ω}) := \frac{Γ(t + k + j)L(BC(f), j + \frac{t+k}{2} , \overline{χ}\overline{ω})}{π^{t+k+2j} ⟨f, f⟩},

L^{alg}(Symm^{2} f, n) := \frac{Γ(n)L(Symm^{2} f, n)}{π^{n+2} ⟨f, f⟩}

for any integer n, and (∗) ∈ \overline{Q} ∩ E is a λ-adic unit.

Proof. This is a straightforward calculation using (7.14), Proposition 6.4 and Theorem 4.8. □

It follows from Remark 6.3 and from Theorem 1 on page 325 in [26] that

(7.16)  L^{alg}(BC(f), 1 + (t + k)/2, \overline{χ}\overline{ω}) ∈ \overline{Q}

and

(7.17)  L^{alg}(BC(f), 2 + (t + k)/2, \overline{χ}\overline{ω}) ∈ \overline{Q}

and from a result of Sturm [53] that

(7.18)  L^{alg}(Symm^{2} f, k) ∈ \overline{Q}.

We note here that [53] uses a definition of the Petersson norm of f which differs from ours by a factor of \frac{3}{π}, the volume of the fundamental domain for the action of SL_{2}(Z) on the complex upper half-plane. We assume that E contains values (7.16), (7.17), and (7.18).

Corollary 7.7. C_{F_{f}} ∈ \overline{Q} ∩ E.

As we are ultimately interested in (mod λ) congruences between hermitian modular forms, we will use “integral periods” Ω_{f}^{+}, Ω_{f}^{−} instead of ⟨f, f⟩ (cf. section 8.3). It follows from Proposition 8.15 in section 8.3 that we have:

(7.19)  ⟨f, f⟩ = (∗) η Ω_{f}^{+}Ω_{f}^{−},

where η ∈ \overline{Z}_{ℓ} is defined in section 8.3 and (∗) is a λ-adic unit as long as f is ordinary at ℓ and ℓ > k, which we assume in what follows. We also assume that E contains η and that ℓ ∤ #X_{m,c}.

Corollary 7.8. (7.20)

C_{F_{f}} = (∗) \overline{c_{F_{f}} (τ)} η^{−1} \frac{L^{int}(BC(f), 2 + (t + k)/2, \overline{χ}\overline{ω})L^{int}(BC(f), 1 + (t + k)/2, \overline{χ}\overline{ω})}{L^{int}(Symm^{2} f, k)},

where

L^{int}(BC(f), j + (t + k)/2, \overline{χ}\overline{ω}) := \frac{Γ(t + k + j)L(BC(f), j + (t + k)/2, \overline{χ}\overline{ω})}{π^{t+k+2j} Ω_{f}^{+}Ω_{f}^{−}},

L^{int}(Symm^{2} f, k) := \frac{Γ(k)L(Symm^{2} f, k)}{π^{k+2} Ω_{f}^{+}Ω_{f}^{−}},

and (∗) ∈ E with ord_{λ}((∗)) ≤ 0.

Proof. This follows directly from Lemma 7.6 upon noting that ord_{λ}(B(2−m/2)) ≥ 0 and ord_{λ}(det τ) ≥ 0. □

We are now going to show that we can choose τ to make \overline{c_{F_{f}} (τ)} in (7.20) a λ-adic unit. Since we have derived our formulas with the assumption b = 1, where b is defined in section 6, we need to choose τ appropriately so this assumption remains valid. If τ is ℓ-ordinary in the sense of the following definition, then we can take b = 1.

Definition 7.9. For a rational prime ℓ, we will say that τ ∈ S is ℓ-ordinary if the following two conditions are simultaneously satisfied:

• {g^{∗}τg}_{g∈O_{K}^{2}} = Z

• there exists c′ ∈ Z with (c′, ℓ) = 1 such that {g^{∗}τ^{−1}g}_{g∈O_{K}^{2}} ⊂ (c′)^{−1}Z.

Lemma 7.10. If f ∈ N is such that the Galois representation \overline{ρ}_{f}|_{G_{K}} is absolutely irreducible, then there exists an ℓ-ordinary τ ∈ S such that ord_{λ}(\overline{c_{F_{f}} (τ)}) = 0.

Proof. Write the Fourier expansion of f as f = ∑_{n=1}^{∞} b(n)q^{n}. Since F_{f} is a Maass form, we have c_{F_{f}} (τ) = ∑_{d∈Z_{>0}, d|ϵ(τ)} d^{k−1}c_{F_{f}}^{∗} (4 det τ/d^{2}), where c_{F_{f}}^{∗} and ϵ(·) were defined in Definition 4.2. Note that τ = [ \begin{matrix}n \\ & 1 \end{matrix} ] is ℓ-ordinary (with c′ = n) for any positive integer n with (n, ℓ) = 1. Using Proposition 4.5 and Fact 2.1 we get

\overline{c_{F_{f}} ([ \begin{matrix}1 \\ & 1 \end{matrix} ])} = −2i(b(2)^{2} − \overline{b(2)}^{2})

and

\overline{c_{F_{f}} ([ \begin{matrix}p \\ & 1 \end{matrix} ])} = 2ib(p)(b(2)^{2} + \overline{b(2)}^{2})

if p ≠ ℓ is inert in K. As will be shown in Proposition 8.13, absolute irreducibility of \overline{ρ}_{f}|_{G_{K}} implies that there exists an inert prime p_{0}, distinct from ℓ, such

that b(p_{0}) is a λ-adic unit. Suppose now that both ord_{λ} (\overline{c_{F_{f}} ([ \begin{matrix}1 \\ & 1 \end{matrix} ])}) > 0 and

ord_{λ} (\overline{c_{F_{f}} ([ \begin{matrix}p_{0} \\ & 1 \end{matrix} ])}) > 0. Then we must have ord_{λ}(b(2)) > 0, which is impossible as

ℓ is odd and |b(2)| = 2^{(k−2)/2} (cf. [31], formula (6.90)). □

Definition 7.11. For f ∈ N such that the Galois representation \overline{ρ}_{f}|_{G_{K}} is absolutely irreducible, let S_{f,ℓ} denote the set of positive integer n with (n, ℓ) = 1 such that ord_{λ} (\overline{c_{F_{f}} ([ \begin{matrix}n \\ & 1 \end{matrix} ])}) = 0. By the proof of Lemma 7.10 the set S_{f,ℓ} is non-empty.

7.3.2. Congruence between F_{f} and a non-Maass form. Our goal is to prove that F_{f} is congruent to a non-Maass form. Note that if C_{F_{f}} = aλ^{−n}, with a ∈ O^{×} and n > 0, then F_{f} is congruent to −a^{−1}λ^{n}G′′ mod λ^{n}. However, G′′ need not a priori be orthogonal to the Maass space. We overcome this obstacle by introducing a certain Hecke operator T̃^{h} which will kill the “Maass part” of G′′. For g ∈ N and F ∈ N^{h}, the set

Σ := {λ_{g,C}(T) | g ∈ N , T ∈ T_{Z}} ∪ {λ_{F,C}(T) | F ∈ N^{h}, T ∈ T_{Z}^{h}}

is contained in the ring of integers of a finite extension of Q (cf. section 5.1 and Theorem 5.9). We assume that E contains Σ. From now on assume that the Galois representation \overline{ρ}_{f}|_{G_{K}} is absolutely irreducible. Without loss of generality we also assume that F_{f} ∈ N^{h} (cf. Remark 5.13). For any F ∈ N^{h}, let 𝔪_{F} ⊂ T_{O}^{h} be the maximal ideal corresponding to F. It follows from (5.8) that there exists T^{h} ∈ T_{O}^{h} such that T^{h}F_{f} = F_{f} and T^{h}F = 0 for all F ∈ N^{h} such that 𝔪_{F} ≠ 𝔪_{F_{f}}. We apply T^{h} to both sides of

Ξ = C_{F_{f}} F_{f} + G′′.

As the Fourier coefficients of F_{f} and Ξ lie in O, so do the Fourier coefficients of T^{h}Ξ by Lemma 5.8. Moreover, since θ_{χ} is a cusp form, so are Ξ and T^{h}Ξ. Let S_{k,F_{f}}^{(2)} ⊂ S_{k}(Γ_{Z}) denote the subspace spanned by

N_{F_{f}}^{h,(2)} := {F ∈ N^{h} | 𝔪_{F}^{(2)} = 𝔪_{F_{f}}^{(2)}},

where 𝔪_{F}^{(2)} and 𝔪_{F_{f}}^{(2)} are the maximal ideals of T_{O}^{h,(2)} corresponding to F and F_{f},

respectively (cf. section 5.5). Then T^{h}Ξ, T^{h}F_{f} = F_{f}, T^{h}G′′ ∈ S_{k,F_{f}}^{(2)} . The image

of T_{O}^{h,(2)} inside End_{C}(S_{k,F_{f}}^{(2)} ) can be naturally identified with T_{O,𝔪_{F_{f}}^{(2)}}^{h,(2)}. By the commutativity of diagram (5.10) and the discussion following the diagram, the Oalgebra map Desc : T_{O}^{h,(2)} ↠ T′_{O} factors through T_{O,𝔪_{F_{f}}^{(2)}}^{h,(2)} ↠ T′_{O,𝔪′_{f}}. The algebra

T′_{O,𝔪′_{f}} can be identified with the image of T′_{O} inside End_{C}(S_{k−1,f}), where S_{k−1,f} ⊂

S_{k−1} (4, (\frac{−4}{·})) is the subspace spanned by N′_{f} := \{g ∈ N | 𝔪′_{f} = 𝔪′_{g}\}. Here 𝔪′_{f} and 𝔪′_{g} denote the maximal ideals of T′_{O} corresponding to f and g, respectively. Denote by ϕ_{f} the natural projection T′_{O} ↠ T′_{O,𝔪′_{f}}, and by Φ_{f} the natural projection

T_{O}^{h,(2)} ↠ T_{O,𝔪_{F_{f}}^{(2)}}^{h,(2)}. Assume ℓ > k, hence in particular ℓ ∤ (k − 1)(k − 2)(k − 3). By

Corollary 5.15, for every split prime p = π\overline{π}, p ∤ ℓ, there exists T^{h}(p) ∈ T_{O,𝔪_{F_{f}}^{(2)}}^{h,(2)}

such that Desc(T^{h}(p)) = ϕ_{f}(T_{p}) ∈ T′_{O,𝔪′_{f}}. As will be proven in section 8 (cf. Proposition 8.14) there exists a Hecke operator T ∈ T′_{O,𝔪′_{f}} such that T f = ηf, T f^{ρ} = ηf^{ρ}, and T g = 0 for all g ∈ N′_{f}, g ≠ f, f^{ρ}. The operator T is a polynomial P_{T} in the elements of ϕ_{f}(Σ′) with coefficients in O (here Σ′ is as in Definition 5.1). Let T̃^{h} ∈ T_{O,𝔪_{F_{f}}^{(2)}}^{h,(2)} be the Hecke operator given by the polynomial PT̃^{h} obtained

from P_{T} by substituting • Φ_{f}(T_{p}^{h} − p^{k−1} − p^{k−2} − p^{k−3}) for ϕ_{f}(T_{p^{2}}) if p inert in K,

• T^{h}(p) for ϕ_{f}(T_{p}) if p ∤ ℓ splits in K,
• Φ_{f}(λ_{0}^{k}ℓ^{2−k}(ℓ + 1)^{−1}T_{λ_{0}}^{h}) for ϕ(T_{ℓ}) if ℓ = λ_{0}\overline{λ}_{0} splits in K.

Note that λ_{0}^{k}ℓ^{2−k}(ℓ + 1)^{−1}T_{λ_{0}}^{h} is indeed an element of T_{O,𝔪_{F_{f}}^{(2)}}^{h} as ℓ + 1 is invertible

in O. It follows from (5.9) that Desc(T̃^{h}) = T. Apply T̃^{h} to both sides of

T^{h}Ξ = C_{F_{f}} F_{f} + T^{h}G′′.

Note that T̃^{h}T^{h}Ξ is again a cusp form. The operator T̃^{h} preserves the Maass space and its orthogonal complement by Theorem 5.10. Another application of Lemma 5.8 shows that the Fourier coefficients of T̃^{h}T^{h}Ξ lie in O. Moreover, since Desc is a C-algebra map, it is clear from the definition of T̃^{h} that T̃^{h}F_{f} = ηF_{f} and T̃^{h}F = 0 for any F inside the Maass space of S_{k}(Γ_{Z}) which is orthogonal to F_{f}. We thus get

(7.21)  T̃^{h}T^{h}Ξ = ηC_{F_{f}} F_{f} + T̃^{h}T^{h}G′′

with T̃^{h}T^{h}G′′ orthogonal to the Maass space.

As C_{F_{f}} ∈ \overline{Q} ∩ E ⊂ C by Corollary 7.7, it makes sense to talk about its λadic valuation. Suppose ord_{λ}(η C_{F_{f}}) = −n ∈ Z_{<0}. We write F ≡ F′ (mod λ^{n}) to mean that ord_{λ}(c_{F} (h) − c_{F′}(h)) ≥ n for every h ∈ S. Note that since the Fourier coefficients of T̃^{h}T^{h}Ξ and of F_{f} lie in O, but η C_{F_{f}} ∉ O, we must have that either T̃^{h}T^{h}G′′ ≠ 0 or F_{f} ≡ 0 mod λ. However, by Proposition 4.5, the latter is only possible if f ≡ f^{ρ} mod λ and this contradicts absolute irreducibility of \overline{ρ}_{f}|_{G_{K}} by Proposition 8.13 in section 8.2. Hence we must have T̃^{h}T^{h}G′′ ≠ 0. Write η C_{F_{f}} = aλ^{−n} with a ∈ O^{×}. Then the Fourier coefficients of λ^{n}T̃^{h}T^{h}G′′ lie in O and one has

F_{f} ≡ −a^{−1}λ^{n}T̃^{h}T^{h}G′′ (mod λ^{n}).

As explained above, −a^{−1}λ^{n}T̃^{h}T^{h}G′′ is a hermitian modular form orthogonal to the Maass space.

We have proven the following theorem:

Theorem 7.12. Let k a positive integer divisible by 4 and ℓ > k a rational prime. Let f ∈ N be ordinary at ℓ and such that \overline{ρ}_{f}|_{G_{K}} is absolutely irreducible. Fix a positive integer m ∈ S_{f,ℓ} and a Hecke character χ of K such that ord_{ℓ}(cond χ) = 0, χ_{∞}(z) = (\frac{z}{|z|})^{−t} with −k ≤ t < −6, and ℓ ∤ #X_{−t−2,4mN_{K/Q}(cond χ)}. Let E be a sufficiently large finite extension of Q_{ℓ} with uniformizer λ. If

−n := ord_{λ} (∏_{j=1}^{2} L^{int}(BC(f), j + (t + k)/2, \overline{χ}\overline{ω})) − ord_{λ}(L^{int}(Symm^{2} f, k)) < 0

where ω is the unique Hecke character of K which is unramified at all finite places

and such that ω_{∞}(z) = (\frac{z}{|z|})^{−k}, then there exists F′ ∈ S_{k}(Γ_{Z}), orthogonal to the Maass space, such that F′ ≡ F_{f} (mod λ^{n}).

Remark 7.13. For χ and m as in Theorem 7.12, set c = 4mN_{K/Q}(cond χ). In Theorem 7.12, we say that E is sufficiently large if it contains the field K(χ_{Q,c}, μ_{c}), the set Σ, the elements (7.16), (7.17), (7.18), the Fourier coefficients of F_{f} and the number η.

Corollary 7.14. Suppose that χ in Theorem 7.12 can be chosen so that

ord_{λ} (∏_{j=1}^{2} L^{int}(BC(f), j + (t + k)/2, \overline{χ}\overline{ω})) = 0,

then n in Theorem 7.12 can be taken to be ord_{λ}(L^{int}(Symm^{2} f, k)).

Remark 7.15. The existence of character χ as in Corollary 7.14 is not known in general. Some results in this direction (although not applicable to the case considered here) have been obtained by Vatsal in [57]. The problem in our case is that one would need to control the λ-adic valuation of two L-values at the same time.

Remark 7.16. The ordinarity assumption on f in Theorem 7.12 is crucial to our method and is used in section 8 to construct the Hecke operator T annihilating the Maass part of G′′ as above as well as to ensure that (∗) in (7.19) is a λ-adic unit. One expects that the set of primes ℓ of Q such that a given (non-CM) form f is ordinary at ℓ has Dirichlet density one, but for now no proof of this fact is known. An analogous statement for elliptic curves was proved by Serre [48].

7.4. Congruence between F_{f} and a non-CAP eigenform.

Corollary 7.17. Under the assumptions of Theorem 7.12 there exists a non-CAP cuspidal Hecke eigenform F such that ord_{λ}(λ_{F_{f}} (T^{h}) − λ_{F} (T^{h})) > 0 for all Hecke operators T^{h} ∈ T_{O}^{h}.

Proof. Let F′ be as in Theorem 7.12. Using the decomposition (5.8), we see that there exists a Hecke operator T_{0}^{h} ∈ T_{O}^{h} such that T_{0}^{h}F_{f} = F_{f} and T_{0}^{h}F = 0 for each F ∈ S_{k}(Γ_{Z}) which is orthogonal to all Hecke eigenforms whose eigenvalues are congruent to those of F_{f} (mod λ). Suppose all the elements of N^{h} whose eigenvalues are congruent to those of F_{f} (mod λ) are CAP forms. Then applying T_{0}^{h} to the congruence F_{f} ≡ F′, we get F_{f} ≡ 0 (mod λ). By Proposition 4.5 this is only possible if f ≡ f^{ρ} (mod λ). This however leads to a contradiction by Proposition 8.13. □

7.5. The CAP ideal. Recall that we have a Hecke-stable decomposition

S_{k}(Γ_{Z}) = S_{k}^{M}(Γ_{Z}) ⊕ S_{k}^{NM}(Γ_{Z}),

where S_{k}^{NM}(Γ_{Z}) denotes the orthogonal complement of S_{k}^{M}(Γ_{Z}) inside S_{k}(Γ_{Z}). Denote by T_{O}^{NM} the image of T_{O}^{h} inside End_{C}(S_{k}^{NM}(Γ_{Z})) and let ϕ : T_{O}^{h} ↠ T_{O}^{NM} be the canonical O-algebra epimorphism. Let Ann(F_{f}) ⊂ T_{O}^{h} denote the annihilator of F_{f}. It is a prime ideal of T_{O}^{h} and λ_{F_{f}} : T_{O}^{h} ↠ O induces an O-algebra isomorphism T_{O}^{h}/Ann(F_{f}) ⟶^{∼} O.

Definition 7.18. As ϕ is surjective, ϕ(Ann(F_{f})) is an ideal of T_{O}^{NM}. We call it the CAP ideal associated to F_{f}.

There exists a non-negative integer r for which the diagram

(7.22)  \xymatrix{T_{O}^{h} \ar[r]^{ϕ} \ar[d] & T_{O}^{NM} \ar[d] \\ T_{O}^{h}/Ann(F_{f}) \ar[r]^{ϕ} \ar[d]^{≀}_{λ_{F_{f}}} & T_{O}^{NM}/ϕ(Ann(F_{f})) \ar[d]^{≀} \\ O \ar[r] & O/λ^{r}O}

all of whose arrows are O-algebra epimorphisms, commutes.

Corollary 7.19. If r is the integer from diagram (7.22), and n is as in Theorem 7.12, then r ≥ n.

Proof. Set N^{NM} := {F ∈ N^{h} | F ∈ S_{k}^{NM}(Γ_{Z})}. Choose any T^{h} ∈ ϕ^{−1}(λ^{r}) ⊂ T_{O}^{h}. Suppose that r < n, and let F′ be as in Theorem 7.12. We have

(7.23)  F_{f} ≡ F′ (mod λ^{n}).

and T^{h}F′ = λ^{r}F′. Hence applying T^{h} to both sides of (7.23), we obtain 0 ≡ λ^{r}F′ (mod λ^{n}), which leads to

(7.24)  F′ ≡ 0 (mod λ^{n−r}).

Since r < n, (7.23) and (7.24) imply that F_{f} ≡ 0 (mod λ), which is impossible as shown in the proof of Corollary 7.17. □

Remark 7.20. The CAP ideal can be regarded as an analogue of the Eisenstein ideal in the case of classical modular forms (see e.g. [39]). It measures congruences between F_{f} and non-CAP modular forms. We will show in section 9.2 that ord_{ℓ}(#T_{O}^{NM}/ϕ(Ann(F_{f}))) provides a lower bound for the ℓ-adic valuation of the order of the Selmer group we study in section 9.1.

8. Hecke algebras and deformation rings

The goal of this section is to prove Proposition 8.14 which was used in section 7.3 to prove Theorem 7.12, as well as some auxiliary results.

8.1. Congruences and weak congruences. Let E denote a finite extension of Q_{ℓ} containing all Hecke eigenvalues of all the elements of N. Let O be the valuation ring of E with uniformizer λ, and put F = O/λ. Whenever we refer to a prime p being split or inert we will always mean split in K or inert in K. Let T_{Z}, T′_{Z} be as in Definition 5.1. To ease notation in this section we set T := T_{O} and T′ := T′_{O}. Moreover, if a, b ∈ O, we write a ≡ b if λ | (a − b).

Let λ_{f} : T → O be as in section 5.1 and as before set 𝔪_{f} = ker \overline{λ}_{f}. Moreover, set λ′_{f} := λ_{f}|_{T′} and denote by \overline{λ}′_{f} the reduction of λ′_{f} modulo λ. Put 𝔪′_{f} := ker \overline{λ}′_{f}.

From now on let f = ∑_{n=1}^{∞} a(n)q^{n} and g = ∑_{n=1}^{∞} b(n)q^{n} denote two elements of N. We denote by ρ_{f}, ρ_{g} : G_{Q} → GL_{2}(E) the ℓ-adic Galois representations attached to f and g, respectively and by \overline{ρ}_{f} and \overline{ρ}_{g} their mod λ reductions with respect to some lattice in E^{2}. We write \overline{ρ}_{f}^{ss} for the semi-simplification of \overline{ρ}_{f}. The isomorphism class of \overline{ρ}_{f}^{ss} is independent of the choice of the lattice. (cf. section 2.3).

Definition 8.1. We will say that f and g are congruent (resp. weakly congruent), denoted by f ≡ g (resp. f ≡_{w} g) if 𝔪_{f} = 𝔪_{g} (resp. 𝔪′_{f} = 𝔪′_{g}). We will say that f and g are congruent at p if a(p) ≡ b(p). Let A be a set of finite primes of Z of density zero. We will say that f and g are A-congruent, denoted by f ≡_{A} g if f and g are congruent at p for all primes p ∉ A.

We note that decompositions analogous to (5.1) and (5.2) hold for T′ and that the localizations T_{𝔪} and T′_{𝔪′} are Noetherian, local, complete O-algebras. For a maximal ideal 𝔪′ ⊂ T′, we denote by M(𝔪′) the set of maximal ideals of T which contract to 𝔪′. Note that the inclusion T′ ↪ T factors into a direct product (over all maximal ideals 𝔪′ of T′) of injections T′_{𝔪′} ↪ ∏_{𝔪∈M(𝔪′)} T_{𝔪}. We will now examine the sets M(𝔪′) a little closer.

Lemma 8.2. Let f, g ∈ N and let A be a density zero set of finite primes of Z not containing ℓ. Then f ≡ g if and only if f ≡_{A} g.

Proof. One direction is a tautology, so assume f ≡_{A} g. We have tr \overline{ρ}_{f}(Frob_{p}) =

a(p) (mod λ), tr \overline{ρ}_{g}(Frob_{p}) = b(p) (mod λ) and det \overline{ρ}_{f}(Frob_{p}) = (\frac{−4}{p}) p^{k−2} = det \overline{ρ}_{g}(Frob_{p}) for p ≠ 2, ℓ. Hence by the Tchebotarev Density Theorem together with the Brauer-Nesbitt Theorem we get \overline{ρ}_{f}^{ss} ≅ \overline{ρ}_{g}^{ss}, and thus, a(p) ≡ b(p) for all

p ∈ A, p ≠ 2. Moreover, we have ρ_{f}|_{D_{2}} ≅ [\begin{matrix}μ_{f}^{1}χ \\ & μ_{f}^{2} \end{matrix}], where D_{2} denotes the

decomposition group at 2, μ_{f}^{1} and μ_{f}^{2} are unramified characters, with μ_{f}^{2}(Frob_{2}) = a(2), and χ is the Galois character associated with the Dirichlet character (\frac{−4}{·}) (cf. [27], Theorem 3.26 (3)). An analogous result holds for ρ_{g}. Let σ ∈ D_{2} be any lift of Frob_{2}, and let τ ∈ I_{2} be such that χ(τ) = −1, where I_{2} denotes the inertia group at 2. We want to show that μ_{f}^{2}(σ) ≡ μ_{g}^{2}(σ) (mod λ). We have tr ρ_{f}(σ) = μ_{f}^{1}(σ)χ(σ) + μ_{f}^{2}(σ) and tr ρ_{f}(τσ) = μ_{f}^{1}(σ)χ(τ)χ(σ) + μ_{f}^{2}(σ). Then as χ(τ) = −1, we get μ_{f}^{2}(σ) = \frac{1}{2}(tr ρ_{f}(σ) + tr ρ_{f}(τσ)). Similarly we get μ_{g}^{2}(σ) = \frac{1}{2}(tr ρ_{g}(σ) + tr ρ_{g}(τσ)). Since \overline{ρ}_{f}^{ss} ≅ \overline{ρ}_{g}^{ss} implies the equality of traces of \overline{ρ}_{f} and \overline{ρ}_{g}, μ_{f}^{2}(σ) ≡ μ_{g}^{2}(σ) and the lemma is proved. □

Proposition 8.3. If f ≡_{w} g, then either f ≡ g or f ≡ g^{ρ}.

Proof. Assume f ≡_{w} g. Using the Tchebotarev density Theorem and the Brauer-Nesbitt Theorem, we see that \overline{ρ}_{f}^{ss}|_{G_{K}} ≅ \overline{ρ}_{g}^{ss}|_{G_{K}}. By possibly changing a basis of, say, \overline{ρ}_{g}, we may assume that \overline{ρ}_{f}^{ss}|_{G_{K}} = \overline{ρ}_{g}^{ss}|_{G_{K}}. This implies that \overline{ρ}_{f}^{ss} = χ\overline{ρ}_{g}^{ss}, where χ

is either as in the proof of Lemma 8.2 or trivial. Hence a(p) ≡ (\frac{−4}{·})^{i} b(p) for some i and all p ≠ 2, ℓ. Thus by Lemma 8.2 we are done if we show that a(ℓ) ≡ (\frac{−4}{ℓ})^{i} b(ℓ). If ℓ is split, then (since f ≡_{w} g) we have a(ℓ) ≡ b(ℓ), so assume ℓ is inert. In that case, a(ℓ)^{2} ≡ b(ℓ)^{2} hence if a(ℓ) ≡ 0, we are done. Otherwise, f and g are

ℓ-ordinary, and in such case ρ_{f}|_{D_{ℓ}} ≅ \left[{μ_{f}^{1} ∗ \atop μ_{f}^{2}}\right] with μ_{f}^{2} unramified and μ_{f}^{2}(Frob_{ℓ}) is

the unit root α_{f} of X^{2} −a(ℓ)X +(\frac{−4}{ℓ}) ℓ^{k−2} (cf. [27], Theorem 3.26 (2)). Analogous statements hold for ρ_{g}. Now, since \overline{ρ}_{f} ≅ \overline{ρ}_{g} ⊗ χ^{i}, we must have α_{f} ≡ (\frac{−4}{ℓ})^{i} α_{g}. As α_{f} is the unique unit root of the polynomial X^{2} −a(ℓ)X +(\frac{−4}{ℓ}) ℓ^{k−2}, we must have a(ℓ) ≡ α_{f}, and similarly b(ℓ) ≡ α_{g}, hence the proposition is proved. □

Corollary 8.4. If f ≡ f^{ρ}, then M(𝔪′_{f}) = {𝔪_{f}}. If f ≢ f^{ρ}, then M(𝔪′_{f}) = {𝔪_{f}, 𝔪_{f^{ρ}}}. Hence, if f ≡ f^{ρ}, we have an injection T′_{𝔪′_{f}} ↪ T_{𝔪_{f}}, while if f ≢ f^{ρ}, we have T′_{𝔪′_{f}} ↪ T_{𝔪_{f}} × T_{𝔪_{f^{ρ}}}.

Proposition 8.5. If f ∈ N, then the canonical O-algebra map ϕ_{0} : T′_{𝔪′_{f}} → T_{𝔪_{f}} is injective.

Proof. If f ≡ f^{ρ}, then T′_{𝔪′_{f}} injects into T_{𝔪_{f}} by Corollary 8.4. Assume that f ≢ f^{ρ}. Note that in that case g ≡_{w} f implies g ≢ g^{ρ}. By Proposition 8.3, g ≡ f or g ≡ f^{ρ}. Without loss of generality assume that f ≡ g. Consider T_{𝔪_{f}} as a subalgebra of ∏_{g∈N ,g≡f} O via T ↦ (λ_{g}(T))_{g}, and T_{𝔪_{f^{ρ}}} as a subalgebra of ∏_{g∈N ,g≡f^{ρ}} O via T ↦ (λ_{g^{ρ}}(T))_{g}. By Corollary 8.4 we have T′_{𝔪′_{f}} ↪ T_{𝔪_{f}} ×T_{𝔪_{f}}^{ρ}, so we just need to prove that the composite T′_{𝔪′_{f}} ↪ T_{𝔪_{f}} × T_{𝔪_{f^{ρ}}} ↠ T_{𝔪_{f}} is injective, where the last arrow is projection. Identifying T_{𝔪_{f}} × T_{𝔪_{f^{ρ}}} with a subalgebra of R := ∏_{g∈N ,g≡f} O × ∏_{g∈N ,g≡f^{ρ}} O by the embeddings specified above, we see that T ∈ T′_{𝔪′_{f}} maps to an element of R, whose g-entry in the first product is the same as the corresponding g^{ρ}-entry in the second product for every g ∈ N, g ≡ f (this is so, because T g = ag implies T g^{ρ} = ag^{ρ} for T ∈ T′_{𝔪′_{f}}). Hence if T maps to zero under the composite T′_{𝔪′_{f}} ↪ T_{𝔪_{f}} × T_{𝔪_{f^{ρ}}} → T_{𝔪_{f}}, it must be zero in T_{𝔪_{f}} × T_{𝔪_{f^{ρ}}}. □

8.2. Deformations of Galois representations. The goal of this section is to prove surjectivity of ϕ_{0} : T′_{𝔪′_{f}} → T_{𝔪_{f}}. We will use the theory of deformations of Galois representations. For an introduction to the subject see e.g. [40].

8.2.1. Universal deformation ring. Let C denote the category of local, complete Oalgebras with residue field F. A morphism between two objects in C is a continuous O-algebra homomorphism which induces the identity on the residue fields. For an object R of C we denote by 𝔪_{R} its maximal ideal. Let G be a profinite group. Two continuous representations ρ : G → GL_{2}(R) and ρ′ : G → GL_{2}(R) are called strictly equivalent if ρ(g) = xρ′(g)x^{−1} for every g ∈ G with x ∈ 1 + M_{2}(𝔪_{R}) independent of g. We will write ρ ≈ ρ′ if ρ and ρ′ are strictly equivalent. Consider a continuous representation \overline{ρ} : G → GL_{2}(F). If R is an object of C, a continuous representation ρ : G → GL_{2}(R) or, more precisely, a strict equivalence of such, is called a deformation of \overline{ρ} if \overline{ρ} = ρ mod 𝔪_{R}. A pair (R^{univ}, ρ^{univ}) consisting of an object R^{univ} of C and a deformation ρ^{univ} : G → GL_{2}(R^{univ}) is called a universal couple if for every deformation ρ : G → GL_{2}(R), where R is an object in C, there exists a unique O-algebra homomorphism ϕ : R^{univ} → R such that ϕ ∘ ρ^{univ} ≈ ρ in GL_{2}(R). The ring R^{univ} is called the universal deformation ring of \overline{ρ}. By the universal property stated above, it is unique if it exists. Note that any O-algebra homomorphism between objects in C is automatically local, since all objects of C have the same residue fields.

Theorem 8.6 (Mazur). Suppose that \overline{ρ} : G → GL_{n}(F) is absolutely irreducible. Then there exists a universal deformation ring R^{univ} in C and a universal deformation ρ^{univ} : G → GL_{n}(R^{univ}).

Proof. [27], Theorem 2.26. □

8.2.2. Hecke algebras as quotients of deformation rings. Consider f ∈ N and let ρ_{f} : G_{Q} → GL_{2}(O) be the associated Galois representation (after fixing a lattice in E^{2}). Let \overline{ρ}_{f} : G_{Q} → GL_{2}(F) be its reduction modulo λ. Since ρ_{f} is unramified away from S = {2, ℓ}, it factors through G_{Q,S}, the Galois group of the maximal Galois extension of Q unramified away from S. Let G_{K,S} be the image of G_{K} under the map G_{K} ↪ G_{Q} ↠ G_{Q,S}. We will be considering deformations of the representation \overline{ρ}_{f} : G_{Q,S} → GL_{2}(F) and of \overline{ρ}_{f,K} := \overline{ρ}_{f}|_{G_{K,S}}. From now on we assume that \overline{ρ}_{f,K} is absolutely irreducible. Let (R_{Q}, ρ_{Q}) and (R_{K}, ρ_{K}) denote the universal couples of \overline{ρ}_{f} and \overline{ρ}_{f,K}, respectively, which exist by Theorem 8.6. We will denote 𝔪_{R_{Q}} and 𝔪_{R_{K}} by 𝔪_{Q} and 𝔪_{K}, respectively. Let A be a density zero set of primes of Q and g ∈ N, g ≡_{A} f. Then after possibly changing the basis of ρ_{g} we may assume (by the Tchebotarev Density Theorem together with the Brauer-Nesbitt Theorem) that \overline{ρ}_{f} = \overline{ρ}_{g}. Hence ρ_{g} : G_{Q,S} → GL_{2}(O) is a deformation of \overline{ρ}_{f}, and ρ_{g}|_{G_{K}} is a deformation of \overline{ρ}_{f,K}. As in the proof of Proposition 8.5 we identify T_{𝔪_{f}} and T′_{𝔪′_{f}} with appropriate subalgebras of ∏_{g∈N , g≡f} O and of ∏_{g∈N , g≡_{w}f} O, respectively. Let T̃ denote the O-subalgebra of T generated by the operators T_{p} for p ≠ 2, ℓ and let T̃′ denote the O-subalgebra of T′ generated by the set Σ′, where Σ′ is as in Definition 5.1. We put 𝔪̃_{f} := T̃∩𝔪_{f} and 𝔪̃′_{f} := T̃′∩𝔪_{f}. Let Σ_{f} denote the subset of N consisting of those eigenforms which are congruent to f except possibly at 2 or ℓ. Similarly let Σ′_{f} be the subset of N consisting of those eigenforms which are weakly congruent to f except possibly at 2 or ℓ. We have Σ_{f} ⊂ Σ′_{f}. We again identify T̃𝔪̃_{f} (resp. T̃′ ; 𝔪̃′_{f}) with a subalgebra of ∏_{g∈Σ_{f}} O (resp. ∏_{g∈Σ′_{f}} O) in an obvious

way. Consider the representations ρ := ∏_{g∈Σ_{f}} ρ_{g} : G_{Q,S} → GL_{2} (∏_{g∈Σ_{f}} O), and ρ′ := ρ|_{G_{K,S}}. Choose bases for each ρ_{g} so that \overline{ρ}_{g} = \overline{ρ}_{g′} for all g, g′ ∈ Σ_{f}, and so that ρ_{g}(c) = [ \begin{matrix}1 \\ & −1 \end{matrix} ] for all g ∈ Σ_{f}, where c is the complex conjugation. We allow ourselves to enlarge E, O and F if necessary.

Lemma 8.7. The image of the representation ρ is contained in GL_{2}(T̃𝔪̃_{f}).

Proof. [11], Lemma 3.27. □

We claim that ρ′(G_{K,S}) is contained in the image of GL_{2}(T̃′ ; 𝔪̃′_{f}) inside GL_{2}(T̃𝔪̃_{f}).

To prove it, let ϕ denote the map T̃′ ; 𝔪̃′_{f} → T̃𝔪̃_{f} induced by T̃′ ↪ T̃. It is easy

to see that ϕ(T̃′ ; 𝔪̃′_{f}) is an object of C. Consider ρ̃′ : G_{K,S} → GL_{2} (∏_{g∈Σ′_{f}} O), ρ̃′(σ) = (ρ_{g}(σ))_{g∈Σ′_{f}}. We have ϕ ∘ ρ̃′ = ρ′. For τ ∈ G_{K,S} we denote by [τ] the conjugacy class of τ in G_{K,S}. Note that G_{K,S} is topologically generated by the set ⋃_{𝔭∈Spec O_{K},𝔭∩Z∉S}[Frob_{𝔭}]. For a split p = 𝔭\overline{𝔭}, we have tr ρ′(Frob_{𝔭}) = tr ρ′(Frob\overline{𝔭}) = tr ρ′(Frob_{p}) = T_{p} ∈ T̃′ ; 𝔪̃′_{f} while for p inert, tr ρ′(Frob_{p}^{2}) = T_{p}^{2} − p^{k−2} ∈ T̃′ ; 𝔪̃′_{f}.

Thus tr ρ̃′(G_{K,S}) ⊂ T̃′ ; 𝔪̃′_{f}, and hence tr ρ′(G_{K,S}) ⊂ ϕ(T̃′ ; 𝔪̃′_{f}). Since we know that

ρ′(G_{K,S}) ⊂ GL_{2}(T̃𝔪̃_{f}), a theorem of Mazur ([40], Corollary 6, page 256)) implies that (after possibly changing the basis of ρ′), we have ρ′(G_{K,S}) ⊂ GL_{2}(ϕ(T̃′ ; 𝔪̃′_{f})).

Then ρ is a deformation of \overline{ρ}_{f} and ρ′ : G_{K,S} → GL_{2}(ϕ(T̃′ ; 𝔪̃′_{f})) is a deformation

of \overline{ρ}_{f,K}. Hence there are unique O-algebra homomorphisms ϕ_{Q} : R_{Q} → T̃𝔪̃_{f} and

ϕ_{K} : R_{K} → ϕ(T̃′ ; 𝔪̃′_{f}), such that ϕ_{Q} ∘ ρ_{Q} ≈ ρ, and ϕ_{K} ∘ ρ_{K} ≈ ρ′. In fact as ρ_{Q}|_{G_{K}} is

a deformation of \overline{ρ}_{f,K}, there is a unique O-algebra homomorphism ψ : R_{K} → R_{Q}, such that ψ ∘ ρ_{K} ≈ ρ_{Q}|_{G_{K}}. Hence we get the following diagram

(8.1)  \xymatrix{R_{K} \ar[r]^{ψ} \ar[d]_{ϕ_{K}} & R_{Q} \ar[d]^{ϕ_{Q}} \\ ϕ(T̃′ ; 𝔪̃′_{f}) \ar[r]^{ι} & T̃𝔪̃_{f}}

where ι denotes the embedding ϕ(T̃′ ; 𝔪̃′_{f}) ⊂ T̃𝔪̃_{f}. Note that diagram (8.1) commutes. [Indeed, as ι∘ρ′ is a deformation of \overline{ρ}_{f,K}, there is a unique O-algebra homomorphism α : R_{K} → T̃𝔪̃_{f}, such that α∘ρ_{K} ≈ ι∘ρ′. Since ϕ_{K} ∘ρ_{K} ≈ ρ′ we get ι∘ϕ_{K} ∘ρ_{K} ≈ ι∘ρ′, and hence ι ∘ ϕ_{K} = α by uniqueness of α. On the other hand as stated in the paragraph before diagram (8.1), ψ ∘ ρ_{K} ≈ ρ_{Q}|_{G_{K}}, thus ϕ_{Q} ∘ ψ ∘ ρ_{K} ≈ ϕ_{Q} ∘ ρ_{Q}|_{G_{K}}. Since ϕ_{Q} ∘ ρ_{Q} ≈ ρ, we have ϕ_{Q} ∘ ρ_{Q}|_{G_{K}} ≈ ρ|_{G_{K}} = ι ∘ ρ′. Hence ϕ_{Q} ∘ ψ ∘ ρ_{K} ≈ ι ∘ ρ′, which implies as before that ϕ_{Q} ∘ ψ = α. So, ι ∘ ϕ_{K} = ϕ_{Q} ∘ ψ.] Furthermore, note that ϕ_{Q} and ϕ_{K} are surjective. Our goal is to prove surjectivity of ψ which will imply surjectivity of ι. From this we will deduce surjectivity of ϕ_{0}.

The map ψ : R_{K} → R_{Q} is local, hence induces an F-linear homomorphism on the cotangent spaces 𝔪_{K}/(𝔪_{K}^{2}, λR_{K}) → 𝔪_{Q}/(𝔪_{Q}^{2}, λR_{Q}), which we will call ψ_{ct}. We will show that ψ_{ct}^{∗} : α ↦ α ∘ ψ_{ct} in the exact sequence of dual maps

0 → Hom_{F}(C, F) → Hom_{F}(𝔪_{Q}/(𝔪_{Q}^{2}, λR_{Q}), F) ⟶^{ψ_{ct}^{∗}} Hom_{F}(𝔪_{K}/(𝔪_{K}^{2}, λR_{K}), F)

is injective, which will imply C := coker ψ_{ct} = 0.

Let G be a profinite group and (R^{univ}, ρ^{univ}) the universal couple of an absolutely irreducible representation \overline{ρ} : G → GL_{2}(F).

Lemma 8.8. One has Hom_{F}(𝔪_{R^{univ}}/(𝔪_{R^{univ}}^{2}, λR^{univ}), F) ≅ H^{1}(G, ad(\overline{ρ})), where H^{1} stands for continuous group cohomology and ad(\overline{ρ}) denotes the discrete G-module M_{2}(F) with the G-action given by g · M := \overline{ρ}(g)M\overline{ρ}(g)^{−1}.

Proof. [27], Lemma 2.29. □

When G = G_{Q,S} (or G = G_{K,S}) and R^{univ} = R_{Q} (or R^{univ} = R_{K}), we will denote the isomorphism from Lemma 8.8 by t_{Q} (or t_{K}, respectively).

Proposition 8.9. The following diagram is commutative:

(8.2)  \xymatrix{Hom_{F}(𝔪_{Q}/(𝔪_{Q}^{2}, λR_{Q}), F) \ar[r]^{ψ_{ct}^{∗}} \ar[d]_{t_{Q}}^{≀} & Hom_{F}(𝔪_{K}/(𝔪_{K}^{2}, λR_{K}), F) \ar[d]^{t_{K}}_{≀} \\ H^{1}(G_{Q,S}, ad(\overline{ρ})) \ar[r]^{res} & H^{1}(G_{K,S}, ad(\overline{ρ}))}

Proof. This follows from unraveling the definitions of the maps in diagram (8.2). We omit the details. □

8.2.3. Isomorphism between T′_{𝔪′_{f}} and T_{𝔪_{f}}. Note that since # ad(\overline{ρ}_{f}) is a power of ℓ, and Gal(K/Q) has order 2, the first cohomology group in the inflation-restriction exact sequence

0 → H^{1}(Gal(K/Q), ad(\overline{ρ}_{f})^{G_{K,S}}) → H^{1}(G_{Q,S}, ad(\overline{ρ}_{f})) → H^{1}(G_{K,S}, ad(\overline{ρ}_{f}))

is zero, hence the restriction map in diagram (8.2) is injective, and thus so is ψ_{ct}^{∗}. Hence C = 0 and thus ψ_{ct} is surjective. An application of the complete version of Nakayama’s Lemma (cf. [18], exercise 7.2) now implies that ψ is surjective.

Corollary 8.10. Let f ∈ N and suppose that \overline{ρ}_{f}|_{G_{K}} is absolutely irreducible. Then ϕ : T̃′ ; 𝔪̃′_{f} → T̃𝔪̃_{f} is surjective.

Proof. This is essentially a summary of the arguments we have carried out so far. □

Proposition 8.11. Assume that f ∈ N is ordinary at ℓ and that \overline{ρ}_{f}|_{G_{K}} is absolutely irreducible. Then ϕ_{0} : T′_{𝔪′_{f}} → T_{𝔪_{f}} is surjective.

Proof. Consider the commutative diagram

(8.3)  \xymatrix{T̃′ ; 𝔪̃′_{f} \ar[r]^{ϕ} \ar[d] & T̃𝔪̃_{f} \ar[d] \\ T′_{𝔪′_{f}} \ar[r]^{ϕ_{0}} & T_{𝔪_{f}}}

where 𝔪̃_{f}, 𝔪′_{f} and 𝔪̃′_{f} are contractions of 𝔪_{f} to T̃, T′ and T̃′ respectively. Since f satisfies the assumptions of Corollary 8.10, ϕ is surjective. For p ≠ 2, ℓ, it is clear that T_{p} ∈ T_{𝔪_{f}} is inside the image of ϕ_{0}. If ℓ is split, then T_{𝔪_{f}} contains T_{ℓ} by definition, so assume ℓ is inert. Then T_{ℓ}^{2} ∈ T_{𝔪_{f}}. Since f = ∑_{n=1}^{∞} a(n)q^{n} is ordinary at ℓ, we must have a(ℓ) ∉ λ, hence the image of T_{ℓ} in F is not zero, i.e., T_{ℓ} ∉ 𝔪_{f}. Thus the equation X^{2} − T_{ℓ}^{2} splits in T_{𝔪_{f}} /𝔪_{f} into relatively prime factors X − T_{ℓ} and X + T_{ℓ}. Since T′_{𝔪′_{f}} /𝔪′_{f} ≅ T_{𝔪_{f}} /𝔪_{f}, X^{2} − T_{ℓ}^{2} splits in T′_{𝔪′_{f}} /𝔪′_{f}, and then by Hensel’s lemma it splits in T′_{𝔪′_{f}}. This shows that T_{ℓ} is in the image of ϕ_{0}. It remains to show that T_{2} is in the image of ϕ_{0}.

Let ρ_{g} : G_{Q,S} → GL_{2}(O) denote the Galois representation associated to g =

∑_{n=1}^{∞} b(n)q^{n}, g ≡ f. Arguing as in Lemma 8.2, we get ρ_{g}|_{D_{2}} ≅ [\begin{matrix}μ_{g}^{1}χ \\ & μ_{g}^{2} \end{matrix}] with

μ_{g}^{2}(σ) = \frac{1}{2}(tr ρ_{g}(σ) + tr ρ_{g}(τσ)) (for notation see the proof of Lemma 8.2). Let L be the fixed field of G_{Q,S}, and L′ ⊂ L always denote a finite Galois extension of Q. Using the Tchebotarev Density Theorem we can write

σ = \underleftarrow{lim}_{Q⊂L′⊂L} ξ(L′) Frob_{p(L′)} ξ(L′)^{−1},

where p(L′) is a choice of p ∈ S and ξ(L′) ∈ G_{Q,S} is such that

σ|_{L′} = ξ(L′)|_{L′} Frob_{p(L′)} |_{L′}ξ(L′)^{−1}|_{L′}.

Hence (tr ρ_{g}(σ))_{g} = \underleftarrow{lim}_{Q⊂L′⊂L} (tr (Frob_{p(L′)}))_{g} = \underleftarrow{lim}_{Q⊂L′⊂L} T_{p(L′)}, where each T_{p} is considered as an element of ∏_{g∈N ,g≡f} O. Since every T_{p(L′)} ∈ Im(ϕ_{0}), and Im(ϕ_{0}) being the image of T′_{𝔪′_{f}} is complete, (tr ρ_{g}(σ))_{g} ∈ Im(ϕ_{0}). Similarly one shows that (tr ρ_{g}(τσ))_{g} ∈ Im(ϕ_{0}), and hence T_{2} ∈ Im(ϕ_{0}). □

Corollary 8.12. Assume f ∈ N is ordinary at ℓ. If \overline{ρ}_{f}|_{G_{K}} is absolutely irreducible then the canonical O-algebra map T′_{𝔪′_{f}} → T_{𝔪_{f}} is an isomorphism.

Proposition 8.13. If \overline{ρ}_{f}|_{G_{K}} is absolutely irreducible, then f ≢ f^{ρ}.

Proof. Assume that \overline{ρ}_{f} : G_{Q} → GL_{2}(F) is absolutely irreducible when restricted to G_{K}. Suppose f = ∑_{n=1}^{∞} a(n)q^{n} ≡ f^{ρ} = ∑_{n=1}^{∞} \overline{a(n)}q^{n}. Let p be a prime inert in K. By Fact 2.1, a(p) = −\overline{a(p)}, hence a(p) ≡ −a(p), and thus tr \overline{ρ}_{f}(Frob_{p}) ≡ a(p) ≡ 0. Let L be the splitting field of \overline{ρ}_{f} and denote by c ∈ Gal(L/Q) the complex conjugation. By possibly replacing F with a finite extension, we can choose a basis of the space of \overline{ρ}_{f} such that with respect to that basis \overline{ρ}_{f}(c) = [ \begin{matrix}1 \\ & −1 \end{matrix} ]. Let σ ∈ Gal(L/K), and suppose that \overline{ρ}_{f}(σ) = [ \begin{matrix}a & b \\ c & d \end{matrix} ]. By Tchebotarev Density Theorem there exists a prime p and an element τ ∈ Gal(L/Q) such that cσ = τ Frob_{p} τ^{−1}. Since σ ∈ Gal(L/K), we must have Frob_{p} ∉ Gal(L/K), and thus p is inert in K. Hence tr \overline{ρ}_{f}(Frob_{p}) = a − d = 0. Let σ′ ∈ Gal(L/K) and write \overline{ρ}_{f}(σ′) = [ \begin{matrix}a′ & b′ \\ c′ & d′ \end{matrix} ].

Then \overline{ρ}_{f}(σσ′) = [ \begin{matrix}aa′+bc′ & ab′+bd′ \\ ca′+dc′ & cb′+dd′ \end{matrix} ]. Since the argument carried out for σ may also be applied to σ′ and σσ′ ∈ Gal(L/K), we have a′ = d′ and bc′ = cb′, and this condition implies that σσ′ = σ′σ. Hence Gal(L/K) is abelian, which contradicts the absolute irreducibility of \overline{ρ}_{f}|_{G_{K}}. The proposition follows. □

8.3. Hida’s congruence modules. Fix f ∈ N and set N_{f} := {g ∈ N | 𝔪_{g} = 𝔪_{f}}. Write T_{𝔪_{f}} ⊗ E = E × B_{E}, where B_{E} = ∏_{g∈N_{f} \{f}} E and let B denote the image

of T_{𝔪} under the composite T_{𝔪} ↪ T_{𝔪} ⊗ E ⟶^{π_{f}} B_{E}, where π_{f} is projection. Denote by δ : T_{𝔪_{f}} ↪ O × B the map T ↦ (λ_{f}(T), π_{f}(T)). If E is sufficiently large, there exists η ∈ O such that coker δ ≅ O/ηO. This cokernel is usually called the congruence module of f. Set N′_{f} := {g ∈ N | 𝔪′_{g} = 𝔪′_{f}}.

Proposition 8.14. Assume f ∈ N is ordinary at ℓ and the associated Galois representation ρ_{f} is such that \overline{ρ}_{f}|_{G_{K}} is absolutely irreducible. Then there exists T ∈ T′_{𝔪′_{f}} such that T f = ηf, T f^{ρ} = ηf^{ρ} and T g = 0 for all g ∈ N′_{f} \ {f, f^{ρ}}.

Proof. First note that T′_{𝔪′_{f}} can be identified with the image of T′ inside End_{C}(S_{k−1,f}),

where S_{k−1,f} ⊂ S_{k−1} (4, (\frac{−4}{·})) is the subspace spanned by N′_{f}. By Corollary 8.12, the natural O-algebra map T′_{𝔪′_{f}} → T_{𝔪_{f}} is an isomorphism. So, it is enough to find T ∈ T_{𝔪_{f}} such that T f = ηf and T g = 0 for every g ∈ N_{f} \ {f}. (Note that by Proposition 8.13, f^{ρ} ∉ N_{f}.) It follows from the exactness of the sequence 0 → T_{𝔪_{f}} ⟶^{δ} O × B → O/ηO → 0, that (η, 0) ∈ O × B is in the image of T_{𝔪_{f}} ↪ O × B. Let T be the preimage of (η, 0) under this injection. Then T has the desired property. □

Proposition 8.15 ([25], Theorem 2.5). Suppose ℓ > k. If f ∈ N is ordinary at ℓ, then

η = (∗)\frac{⟨f, f⟩}{Ω_{f}^{+}Ω_{f}^{−}} ,

where Ω_{f}^{+}, Ω_{f}^{−} denote the “integral” periods defined in [56] and (∗) is a λ-adic unit.

9. Galois representations and Selmer groups

In this section we will give a lower bound on the order of (the Pontryagin dual of) the Selmer group ad^{0} ρ_{f}|_{G_{K}} (−1) in terms of the CAP ideal (Theorem 9.10) as well as in terms of the special L-value L^{int}(Symm^{2} f, k) (Corollary 9.11). We will also discuss the relationship between Corollary 9.11 and the Bloch-Kato conjecture for the “motives” ad^{0} M_{0}(−1) and ad^{0} M_{0}(2), where M_{0} is the motif (over Q) associated to f (section 9.3).

9.1. Galois representations. It is well-known that one can attach an ℓ-adic Galois representation to every f ∈ N (cf. section 2.3). In this section we gather some basic facts concerning Galois representations attached to hermitian modular forms.

Let F ∈ S_{k}(Γ_{Z}) be an eigenform. For every rational prime p, let λ_{p,j}(F), j = 1, …, 4, denote the p-Satake parameters of F. (For the definition of p-Satake parameters when p inerts or ramifies in K, see [29], and for the case when p splits in K, see [24].) Let 𝔭 be a prime of O_{K} lying over p. Set

λ̃_{𝔭,j}(F) := (N𝔭)^{−2+k/2}ω^{∗}(𝔭)λ_{p,j}(F),

where ω is the unique Hecke character of K unramified at all finite places with infinity type ω_{∞}(x_{∞}) = (\frac{z}{\overline{z}} )^{−k/2}.

Definition 9.1. The elements λ̃_{𝔭,j}(F) will be called the Galois-Satake parameters of F at 𝔭.

By Theorem 5.9 there exists a finite extension L_{F} of Q containing the Hecke eigenvalues of F. In what follows for a number field L and a prime 𝔭 of L we denote by Frob_{𝔭} the arithmetic Frobenius at 𝔭.

Theorem 9.2. There exists a finite extension E_{F} of Q_{ℓ} containing L_{F} and a 4dimensional semisimple Galois representation ρ_{F} : G_{K} → GL_{E_{F}} (V) unramified away from the primes of K dividing 2ℓ and such that

(i) For any prime 𝔭 of K such that 𝔭 ∤ 2ℓ, the set of eigenvalues of ρ_{F} (Frob_{𝔭}) coincides with the set of the Galois-Satake parameters of F at 𝔭 (cf. Definition 9.1);

(ii) If 𝔭 is a place of K over ℓ, the representation ρ_{F} |_{D_{𝔭}} is crystalline (cf.

section 9.2). (iii) If ℓ > m, and 𝔭 is a place of K over ℓ, the representation ρ_{F} |_{D_{𝔭}} is short.

(For a definition of short we refer the reader to [13], section 1.1.2.)

Remark 9.3. We know of no reference in the existing literature for the proof of this theorem, although it is widely regarded as a known result. For some discussion regarding Galois representations attached to hermitian modular forms, see [4] or [2]. We assume Theorem 9.2 in what follows.

Remark 9.4. It is not known if the representation ρ_{F} is also unramified at the prime i + 1. See [1] for a discussion of this issue.

As before, we assume that E is a sufficiently large finite extension of Q_{ℓ} with valuation ring O, uniformizer λ and residue field F = O/λ. Let f = ∑_{n=1}^{∞} a(n)q^{n} ∈ N be such that \overline{ρ}_{f}|_{G_{K}} is absolutely irreducible. Then by Proposition 8.13, F_{f} ≠ 0. From now on we also assume that ad^{0} \overline{ρ}_{f}|_{G_{K}}, the trace-0-endomorphisms of the representation space of \overline{ρ}_{f}|_{G_{K}} with the usual G_{K}-action, is absolutely irreducible. Let ϵ denote the ℓ-adic cyclotomic character. It follows from Proposition 6.4 that the Galois representation ρ_{F_{f}} ≅ ρ_{f,K} ⊕ (ρ_{f,K} ⊗ ϵ). From now on we assume in addition that 2^{k} ≢ a(2) ≢ 2^{k−4} (mod λ).

9.2. Selmer group. Set N^{NM} := {F ∈ N^{h} | F ∈ S_{k}^{NM}(Γ_{Z})}. Let M^{h} denote the set of maximal ideals of T_{O}^{h} and M^{NM} the set of maximal ideals of T_{O}^{NM}. We have T_{O}^{NM} = ∏_{𝔪∈M^{NM}} T_{𝔪}^{NM}, where T_{𝔪}^{NM} denotes the localization of T_{O}^{NM} at 𝔪. Let ϕ : T_{O}^{h} → T_{O}^{NM} be the natural projection. We have M^{h} = M^{c} ⊔ M^{nc}, where M^{c} consists of those 𝔪 ∈ M^{h} which are preimages (under ϕ) of elements of M^{NM} and M^{nc} := M^{h}\M^{c}. Note that ϕ factors into a product ϕ = ∏_{𝔪∈M^{c}} ϕ_{𝔪}×∏_{𝔪∈M^{nc}} 0_{𝔪}, where ϕ_{𝔪} : T_{𝔪}^{h} → T_{𝔪′}^{h} is the projection, with 𝔪′ ∈ M^{NM} being the unique maximal ideal such that ϕ^{−1}(𝔪′) = 𝔪 and 0_{𝔪} is the zero map. For F ∈ N^{h} we denote by 𝔪_{F} (respectively 𝔪_{F}^{NM}) the element of M^{h} (resp. of M^{NM}) corresponding to F. In particular, 𝔪_{F_{f}}^{NM} ∈ M^{NM} is such that ϕ^{−1}(𝔪_{F_{f}}^{NM}) = 𝔪_{F_{f}}.

We now define the Selmer group relevant for our purposes. For a profinite group G and a G-module M (where we assume the action of G on M to be continuous) we will consider the group H_{cont}^{1}(G, M) of cohomology classes of continuous cocycles G → M. To shorten notation we will suppress the subscript ‘cont’ and simply write H^{1}(G, M). For a field L, and a Gal(\overline{L}/L)-module M (with a continuous action of Gal(\overline{L}/L)) we sometimes write H^{1}(L, M) instead of H_{cont}^{1}(Gal(\overline{L}/L), M). We also write H^{0}(L, M) for the submodule M^{Gal(\overline{L}/L)} consisting of the elements of M fixed by Gal(\overline{L}/L).

Let L be a number field. For a rational prime p denote by Σ_{p} the set of primes of L lying over p. Let Σ ⊃ Σ_{ℓ} be a finite set of primes of L and denote by G_{Σ} the Galois group of the maximal Galois extension L_{Σ} of L unramified outside of Σ. Let V be a finite dimensional E-vector space with a continuous G_{Σ}-action. Let T ⊂ V be a G_{Σ}-stable O-lattice. Set W := V/T.

We begin by defining local Selmer groups. For every 𝔭 ∈ Σ set

H_{un}^{1}(L_{𝔭}, M) := ker{H^{1}(L_{𝔭}, M) ⟶^{res} H^{1}(I_{𝔭}, M)}.

Define the local 𝔭-Selmer group (for V) by

H_{f}^{1}(L_{𝔭}, V) := \begin{cases}H_{un}^{1}(L_{𝔭}, V) & 𝔭 ∈ Σ \ Σ_{ℓ} \\ ker\{H^{1}(L_{𝔭}, V) → H^{1}(L_{𝔭}, V ⊗ B_{crys})\} & 𝔭 ∈ Σ_{ℓ}. \end{cases}

Here B_{crys} denotes Fontaine’s ring of ℓ-adic periods (cf. [19]).

For 𝔭 ∈ Σ_{ℓ}, we call the D_{𝔭}-module V crystalline (or the G_{L}-module V crystalline at 𝔭) if dim_{Q_{ℓ}} V = dim_{Q_{ℓ}} H^{0}(L_{𝔭}, V ⊗ B_{crys}). When we refer to a Galois representation ρ : G_{L} → GL(V) as being crystalline at 𝔭, we mean that V with the G_{L}-module structure defined by ρ is crystalline at 𝔭.

For every 𝔭, define H_{f}^{1}(L_{𝔭}, W) to be the image of H_{f}^{1}(L_{𝔭}, V) under the natural map H^{1}(L_{𝔭}, V) → H^{1}(L_{𝔭}, W). Using the fact that Gal(\overline{κ}_{𝔭} : κ_{𝔭}) = Ẑ has cohomological dimension 1, one easily sees that if W is unramified at 𝔭 and 𝔭 ∉ Σ_{ℓ}, then H_{f}^{1}(L_{𝔭}, W) = H_{un}^{1}(L_{𝔭}, W). Here κ_{𝔭} denotes the residue field of L_{𝔭}.

For a Z_{ℓ}-module M, we write M^{∨} for its Pontryagin dual defined as

M^{∨} = Hom_{cont}(M, Q_{ℓ}/Z_{ℓ}).

Moreover, if M is a Galois module, we denote by M(n) := M ⊗ ϵ^{n} its n-th Tate twist.

Definition 9.5. For each finite set Σ′ ⊂ Σ \ Σ_{ℓ}, the group

Sel_{Σ}(Σ′, W) := ker \{H^{1}(G_{Σ}, W) ⟶^{res} ⨁_{𝔭∈Σ′∪Σ_{ℓ}} \frac{H^{1}(L_{𝔭}, W)}{H_{f}^{1}(L_{𝔭}, W)}\}

is called the (global) Selmer group of the triple (Σ, Σ′, W). We also set S_{Σ}(Σ′, W) := Sel_{Σ}(Σ′, W)^{∨}, Sel_{Σ}(W) := Sel_{Σ}(∅, W) and S_{Σ}(W) = S_{Σ}(∅, W). Define Sel_{Σ}(Σ′, V) in the same way with V instead of W.

For L = Q, the group Sel_{Σ}(Σ \ Σ_{ℓ}, W) is the standard Selmer group H_{f}^{1}(Q, W) defined by Bloch and Kato [5], section 5.

Let Σ, Σ′ be as above. Let ρ : G_{Σ} → GL_{E}(V) denote the representation giving the action of G_{Σ} on V. The following two lemmas are easy (cf. [46], Lemma 1.5.7 and [52]).

Lemma 9.6. S_{Σ}(Σ′, W) is a finitely generated O-module.

Lemma 9.7. If the mod λ reduction \overline{ρ} of ρ is absolutely irreducible, then the length of S_{Σ}(Σ′, W) as an O-module is independent of the choice of the lattice T.

Remark 9.8. For an O-module M, ord_{ℓ}(#M) = [O/λ : F_{ℓ}] length_{O}(M).

Example 9.9. Let L = K, Σ = Σ_{ℓ}, ρ_{f,K} := ρ_{f}|_{G_{K}} and let V denote the representation space of

ad^{0} ρ_{f,K}(−1) = ad^{0} ρ_{f,K} ⊗ ϵ^{−1} ⊂ Hom_{E}(ρ_{f,K} ⊗ ϵ, ρ_{f,K})

of G_{K}. Let T ⊂ V be some choice of a G_{K}-stable lattice. Set W = V/T. Note that the action of G_{K} on V factors through G_{Σ}. Since the mod λ reduction of ad^{0} ρ_{f,K} ⊗ϵ^{−1} is absolutely irreducible by assumption, ord_{ℓ}(S_{Σ}(W)) is independent of the choice of T.

Our goal is to prove the following theorem.

Theorem 9.10. Let L, Σ and W be as in Example 9.9. Suppose that for each F ∈ N_{F_{f}}^{NM}, the representation ρ_{F} : G_{K} → GL_{4}(E) is absolutely irreducible. Then

ord_{ℓ}(#S_{Σ}(W)) ≥ ord_{ℓ}(#T_{𝔪_{F_{f}}}^{NM} /ϕ_{𝔪_{F_{f}}} (Ann(F_{f}))).

Corollary 9.11. With the same assumptions and notation as in Theorem 7.12 and Theorem 9.10 we have

ord_{ℓ}(#S_{Σ}(W)) ≥ n.

If in addition the character χ in Theorem 7.12 can be taken as in Corollary 7.14, then

ord_{ℓ}(#S_{Σ}(W)) ≥ ord_{ℓ}(#O/L^{int}(Symm^{2} f, k)).

Proof. The corollary follows immediately from Theorem 9.10 and Corollary 7.19. □

9.3. Relations to the Bloch-Kato Conjecture. In this section we discuss how our results (Theorem 9.10 and Corollary 9.11) are related to the Bloch-Kato conjecture. We begin by recalling the statement of the conjecture in our particular case. We follow closely the exposition in [13]. For more details as well as precise definitions the reader is encouraged to consult [13], section 2.4 and [20].

Let E_{0} be a number field, which we will assume to be “sufficiently large” (in particular we assume that E_{0} contains all the Hecke eigenvalues of f) and write E for its completion at a prime λ lying over ℓ determined by our choice of the embedding \overline{Q} ↪ \overline{Q}_{ℓ}. This is consistent with our previous definition of E as a sufficiently large finite extension of Q_{ℓ}. Let M_{0} be the “premotivic structure” attached to f over Q with coefficients in E_{0}. Write

M := {M_{B}, M_{dR}, {M_{v}}_{v}, I^{∞}, {I_{B}^{v}}_{v}, {I^{v}}_{v}, {W^{i}}_{i}},

for the premotivic structure ad^{0} M_{0}(−1). Here v runs over the set of finite places of E_{0}, M_{B} (resp. M_{dR}; M_{v}) is a finite dimensional vector space over E_{0} (resp. E_{0}; E_{0,v}), with an action of Gal(C/R) (resp. with a decreasing filtration Fil^{i}; with a pseudo-geometric action of G_{Q}), I^{∞} : C ⊗ M_{dR} → C ⊗ M_{B} (resp. I_{B}^{v} : E_{0,v} ⊗_{E_{0}} M_{B} → M_{v}; I^{v} : B_{dR,p} ⊗_{Q_{p}} E_{0,v} ⊗_{E_{0}} M_{dR} → B_{dR,p} ⊗_{Q_{p}} M_{v}) is a C ⊗ E_{0}- linear (resp. E_{0,v}-linear; B_{dR,p} ⊗_{Q_{p}} E_{0,v}-linear with v | p) isomorphism respecting the Gal(C/R)-action (resp. the Gal(C/R)-action; the G_{Q_{p}}-action and filtrations), where B_{dR,p} is the ring defined by Fontaine, and W^{i} are the so called weight filtrations, whose definition we omit. Similarly one defines the premotivic structure ad^{0} M_{0}(2), which we denote by M^{∗}. We have M_{λ} = V_{Q} and M_{λ}^{∗} = V_{Q}^{∗} := V_{Q}(3), where V_{Q} is the E[G_{Q}]-module ad^{0} ρ_{f}(−1).

From now on let M ∈ {M, M^{∗}}. We adopt similar notation for other symbols, e.g., V_{Q} ∈ {V_{Q}, V_{Q}^{∗}}, where the choice of M determines the choice of V and other symbols related to M in an obvious way.

Let

Δ_{f}(M) = Hom_{E_{0}}(det_{E_{0}} M_{B}^{+}, det_{E_{0}}(M_{dR}/Fil^{0}M_{dR}))

be the fundamental line for M and write δ_{f}(M) for the O-lattice in E ⊗_{E_{0}} Δ_{f}(M) defined by Fontaine and Perrin-Riou (see [20], section II.4 or [13], p. 700 or [33] for details). Here ^{+} indicates the subspace fixed by Gal(C/R).

The Bloch-Kato conjecture relates the lattice δ_{f}(M) to the value at 0 of an L-function of M normalized by a certain period, both of which we now define.

The isomorphism I^{∞} gives rise to an R ⊗ E_{0}-linear isomorphism

R ⊗ M_{B}^{+} → (C ⊗ M_{B})^{+} ⟶^{(I^{∞})^{−1}} R ⊗ M_{dR} → R ⊗ M_{dR}/Fil^{0}M_{dR},

whose determinant over R ⊗ E_{0} is c^{+}(M) ∈ R ⊗ Δ_{f}(M) which we will refer to as the Deligne period of M. Note that c^{+}(M) is canonically defined, i.e., not just up to a multiplication by an element of E_{0}^{×} as in [12].

Let α_{p,1} and α_{p,2} be the p-Satake parameters of f as defined in section 4.2. For an odd prime p, set

L_{p}(ad^{0} M_{0}, s) := (1 − α_{p,1}α_{p,2}^{−1}p^{−s})^{−1}(1 − p^{−s})^{−1}(1 − α_{p,1}^{−1}α_{p,2}p^{−s})^{−1}

and put

L_{2}(ad^{0} M_{0}, s) := (1 − 2^{−s})^{−1}.

Then the L-function of ad^{0} M_{0}(n) is defined as

(9.1)  L(ad^{0} M_{0}(n), s) := ∏_{p} L_{p}(ad^{0} M_{0}, s + n).

In particular we have

L(M, s) = L(ad^{0} M_{0}, s − 1)

and

L(M^{∗}, s) = L(ad^{0} M_{0}, s + 2).

The properties of L(ad^{0} M_{0}, s) are summarized in [13], p. 686. In particular L(ad^{0} M_{0}, s) is entire as a function of s and satisfies a functional equation with respect to s ↦ 1 − s.

Remark 9.12 (Geometric vs. arithmetic Frobenius). In general one defines the L-function of a motive N as ∏_{p} det(1 − f_{p}p^{−s}), where f_{p} denotes the action of the Frobenius element at p on the inertia invariants of the space of N_{λ} (or on Crys(N_{λ}) if p = ℓ) - see [5], p. 361 for more details. However, this definition depends in general on whether one uses the geometric or arithmetic Frobenius. In [5] Bloch and Kato use the geometric Frobenius and then their conjecture relates L(N , 0) to the Selmer group of N. Moreover, in that case one has L(N (n), s) = L(N , s + n). Also note that L(ad^{0} M_{0}, s) is independent of the choice of geometric or arithmetic Frobenius, since the set of eigenvalues of ad^{0} ρ_{f} is of the form {α, 1, α^{−1}} hence is invariant under taking the inverse. To be able to keep with the spirit of the original paper of Bloch and Kato, we defined our L-function in (9.1) so that it agrees with the L-function defined in [5], i.e., L(ad^{0} M_{0}(n), s) := ∏_{p}(1 − f_{p}p^{−s}), where f_{p} denotes the action of Frob_{p}^{−1} on the appropriate space (see above) and Frob_{p} is the arithmetic Frobenius at p as before.

It follows from a result of Sturm [53] that there exists a basis b(M) of Δ_{f}(M) such that

L(M, 0)(1 ⊗ b(M)) = c^{+}(M).

The first version of (the λ-part of) the Bloch-Kato conjecture can be formulated as follows.

Conjecture 9.13 (Bloch-Kato, cf. [13], Conjecture 2.14). One has

(9.2)  δ_{f}(M) = (1 ⊗ b(M))O

as lattices in E ⊗ Δ_{f}(M).

In [20], Diamond, Flach and Guo give an alternative description of the lattice δ_{f}(M) in terms of Tate-Shafarevich groups. This description will allow us to state a different version of Conjecture 9.13 and relate it to Theorem 9.10 and Corollary 9.11.

Set Σ := Σ_{ℓ} ∪ Σ_{2}. Note that the representation V_{Q} is unramified outside Σ. To formulate the second version of the Bloch-Kato conjecture one needs to choose “integral structures” on the one-dimensional E-vector spaces E ⊗ det_{E_{0}} M_{B}^{+} and E ⊗ det_{E_{0}}(M_{dR}/ Fil^{0} M_{dR}). One does this by choosing a free rank one O-module ω(M) ∈ E ⊗ det_{E_{0}}(M_{dR}/Fil^{0}M_{dR})), which in the following we abbreviate as ω, and a Galois stable O-lattice T_{Q} ⊂ V_{Q} which gives rise to a free rank one O-module in E ⊗ det_{E_{0}} M_{B}^{+} via the isomorphism I_{B}^{λ}. Set W_{Q} := V_{Q}/T_{Q}. Let

� (T_{Q}) := \frac{Sel_{Σ}(Σ_{2}, W_{Q})}{Sel_{Σ}(Σ_{2}, V_{Q}) ⊗ (E/O)}

be the Tate-Shafarevich group of T_{Q}. The group � (T_{Q}) is finite ([20], Proposition II.5.3.5). Put T_{Q}^{D} := Hom_{O}(T_{Q}, O(1)) and set V_{Q}^{D} := T_{Q}^{D} ⊗_{O} E and W_{Q}^{D} := V_{Q}^{D}/T_{Q}^{D}. Note that V_{Q}^{D} ≅ V_{Q}^{∗} and (V_{Q}^{∗})^{D} ≅ V_{Q}.

Assume that

(9.3)  Sel_{Σ}(Σ_{2}, V_{Q}) = Sel_{Σ}(Σ_{2}, V_{Q}^{D}) = 0.

This follows from a conjecture on the order of vanishing of L(M, 0) (cf. [20], section III.4.2.2) and the fact that in our case

(9.4)  H^{0}(Q, V_{Q}) = H^{0}(Q, V_{Q}^{D}) = 0.

For a commutative ring R and a finitely generated R-module N, denote by Fitt_{R}(N) the Fitting ideal of N in R. For the definition and basic properties of Fitting ideals see for example the Appendix of [41]. Using Theorem II.5.3.6 in [20], Diamond, Flach and Guo show that

(9.5)  δ_{f}(M) = \frac{Fitt_{O} H^{0}(Q, W_{Q}) · Fitt_{O} H^{0}(Q, W_{Q}^{D})}{Fitt_{O} � (T_{Q}^{D}) · Tam_{ω}^{0}(T_{Q})}L_{ω}(T_{Q}),

where L_{ω}(T_{Q}) is a lattice in E ⊗ Δ_{f}(M) depending on the choice of the “integral structures” T_{Q} and ω = ω(M), and

Tam_{ω}^{0}(T_{Q}) = Tam_{ℓ,ω}^{0}(T_{Q}) · Tam_{∞}^{0}(T_{Q}) · ∏_{p≠ℓ,∞} Tam_{p}^{0}(T_{Q})

is the Tamagawa ideal of T_{Q} relative to ω (cf. [20], section II.5.3). It follows from Proposition II.4.2.2 in [20] that Tam_{p}^{0}(T_{Q}) = Tam_{∞}^{0}(T_{Q}) = O for all p ≠ ℓ.

The integral structures T_{Q} and ω give an identification of E ⊗ Δ_{f}(M) with E. Then the quotient L_{ω}(T_{Q})/ Tam_{ℓ,ω}^{0}(T_{Q}) is identified with a fractional ideal of E whose inverse we denote by Tam_{ω}(T_{Q}). Similarly, (1 ⊗ b(M))O is identified with a fractional ideal (Ω_{ω}(T_{Q})/L(M, 0)) · O of E for some Ω_{ω}(T_{Q}) ∈ E/O^{×}. Using our assumption (9.3), we get � (T_{Q}) = Sel_{Σ}(Σ_{2}, W_{Q}). Moreover, as explained in [13], p. 708, one also has an O-linear isomorphism � (T_{Q}^{D}) ≅ Hom_{Z_{ℓ}}(� (T_{Q}), Q_{ℓ}/Z_{ℓ}) and the latter group is just S_{Σ}(Σ_{2}, W_{Q}).

Using the above arguments and (9.4), Conjecture 9.13 can be rephrased in the following way.

Conjecture 9.14 (Bloch-Kato conjecture, second version). For M ∈ {M, M^{∗}} one has

(9.6)  #S_{Σ}(Σ_{2}, W_{Q}) · Tam_{ω}(T_{Q}) = \frac{L(M, 0)}{Ω_{ω}(T_{Q})}O

as fractional ideals of E.

Let T_{K} be T_{Q} considered as an O[G_{K}]-module. Set V_{K} := T_{K} ⊗ E and W_{K} := V_{K}/T_{K}. Then V_{K} is unramified away from primes of K lying over ℓ.

The following is just a restatement of Corollary 9.11.

Theorem 9.15. With the same assumptions as in Theorem 7.12 and in Corollary 9.11 we have the following containment of fractional ideals of E:

(9.7)  #S_{Σ_{ℓ}}(W_{K}) · O ⊃ L^{int}(Symm^{2} f, k) · O.

We will now discuss the relation between Theorem 9.15 and Conjecture 9.14. As before, let M ∈ {M, M^{∗}}. Write s(M) = k − 3 and s(M^{∗}) = k.

Theorem 9.15 falls short of proving that the left-hand side of (9.6) contains the right-hand side of (9.6), but gives some evidence for this containment. First note that

L(M, 0) = L(ad^{0} M_{0}, s(M) − (k − 2)) = α(M)L(Symm^{2} f, s(M)),

where α(M) comes from the discrepancy in the definitions of the Euler factors at 2 of L(M, s) and L(Symm^{2} f, s) and α(M) ∈ O since λ ∤ 2. Moreover, using the functional equation for L(Symm^{2} f, s) (cf. [47]) one concludes that

ord_{ℓ}(L^{int}(Symm^{2} f, k − 3)) = ord_{ℓ}(L^{int}(Symm^{2} f, k)).

Recall that the order of the Selmer group is independent of the choice of the lattice T_{Q}, hence we can fix T_{Q} as in [13], section 1.6.2 and ω(M) as in [13], p. 709. It has been shown by Dummigan [17], p.11 using Proposition 7.7 in [12] and some arguments in [13] that with these choices of T_{Q} and ω(M), one has Ω_{ω}(T_{Q}) = uπ^{−s(M)−2}Ω_{f}^{+}Ω_{f}^{−} for u an ℓ-adic unit. Thus the right-hand side of (9.6) is the same for M and M^{∗} and is contained in the right-hand side of (9.7). This containment is an equality if α(M) is a λ-adic unit.

Similarly, using the fact that � (T_{Q}^{D}) ≅ � (T_{Q})^{∨} one sees that #S_{Σ}(Σ_{2}, W_{Q}) = #S_{Σ}(Σ_{2}, W_{Q}^{∗}). On the other hand Theorem 9.15 concerns the group S_{Σ_{ℓ}}(W_{K}) = S_{Σ}(Σ_{2}, W_{K}), which can be potentially larger than S_{Σ}(Σ_{2}, W_{Q}) (this follows from the inflation-restriction sequence), so (9.7) does not imply an analogous containment for S_{Σ}(W_{Q}) or S_{Σ}(W_{Q}^{∗}) - see also Remark 9.16 below. Finally, we are not able to show that with the choice of T_{Q} and ω as above, one has Tam_{ω}(T_{Q}) ⊃ O. Dummigan in [15], section 7 and [16], section 6 showed that Tam_{ω}(T_{Q}) = O if f is a modular form of level 1. See [15], section 10 for a discussion of the difficulties involved, when the level of f is larger than one. Diamond, Flach and Guo in [13] have computed the Tamagawa ideal for the motives ad^{0} M_{0} and ad^{0} M_{0}(1) (cf. the proof of Theorem 2.15 and Proposition 2.16 in [13]). However, their calculations cannot be extended to our case.

To summarize, if f is such that #S_{Σ}(Σ_{2}, W_{K}) = #S_{Σ}(Σ_{2}, W_{Q}) and Tam_{ω}(T_{Q}) = O, Theorem 9.15 implies that the right-hand side of (9.6) is contained in the lefthand side of (9.6).

Remark 9.16. One can state a conjecture similar to Conjecture 9.14 for the restriction M|_{K} of the premotivic structure M to G_{K}. Then the L-function L(M|_{K}, 0) factors as

L(M, 0)L (M, 0, (\frac{−4}{·})) = α(M)L(Symm^{2} f, k − 3)L (Symm^{2} f, k − 3, (\frac{−4}{·}))

and the Selmer group on the left-hand-side of (9.6) is replaced by S_{Σ_{ℓ}}(W_{K}). So, this version of the Bloch-Kato conjecture gives us the same Selmer group as the one in Theorem 9.15, but an extra L-value L (Symm^{2} f, k − 3, (\frac{−4}{·})). Unfortunately we are unaware of any rationality results for that special value. One needs a statement that would involve the period c^{+}(M|_{K}) used in the formulation of the Bloch-Kato conjecture. For the value L(Symm^{2} f, k) we have used a rationality result due to Sturm [53], who uses a period related to the Petersson inner product ⟨f, f⟩, but his theorem (cf. [53], p. 220) specifically excludes the value L (Symm^{2} f, k − 3, (\frac{−4}{·})).

Remark 9.17. In [13] Diamond, Flach and Guo proved the λ-part of the Bloch-Kato conjecture for the motives ad^{0} M_{0} and ad^{0} M_{0}(1) using an extension of the methods of Taylor and Wiles [61, 54]. The latter two motives are in duality and the two L-values L(ad^{0} M_{0}, 0) and L(ad^{0} M_{0}(1), 0) are related by the functional equation. Hence our result provides evidence for an extension of their theorem to the motives ad^{0} M_{0}(−1) and ad^{0} M_{0}(2).

9.4. Degree n Selmer groups. In this section we collect some technical results regarding Selmer groups which will be used in the proof of Theorem 9.10. Let G be a group, R a commutative ring with identity, M a finitely generated R-module with an R-linear action of G given by a homomorphism ρ : G → Aut_{R}(M). For any two such pairs (M′, ρ′), (M′′, ρ′′), the R-module Hom_{R}(M′′, M′) is naturally a G-module with the G-action given by

(g · ϕ)(m′′) = ρ′(g)ϕ(ρ′′(g^{−1})m′′).

Suppose there exists (M, ρ) which fits into an exact sequence of R[G]-modules

X : 0 → M′ → M → M′′ → 0,

that splits as a sequence of R-modules. Choose s_{X} : M′′ → M, an R-section of X. Define ϕ_{X} : G → Hom_{R}(M′′, M′) to be the map sending g to the homomorphism m′′ ↦ ρ(g)s_{X}(ρ′′(g)^{−1}m′′) − s_{X}(m′′).

Lemma 9.18. Let Ext_{R[G]}(M′′, M′) denote the set of equivalence classes of R[G]- extensions of M′′ by M′ which split as extensions of R-modules. The map X ↦ ϕ_{X} defines a bijection between Ext_{R[G]}(M′′, M′) and H^{1}(G, Hom_{R}(M′′, M′)).

Proof. The proof is a simple modification of the proof of Proposition 4 in [58]. □

Let E, O and λ be as before. Let L be a number field and Σ a finite set of places of L containing Σ_{ℓ}. Let ρ′ : G_{Σ} → GL_{E}(V′), ρ′′ : G_{Σ} → GL_{E}(V′′) be two Galois representations. Choose G_{Σ}-stable O-lattices T′ ⊂ V′, T′′ ⊂ V′′, and denote the corresponding representations by (T′, ρ′_{T′}) and (T′′, ρ′′_{T′′}) respectively. Define W′ := V′/T′, and W′′ := V′′/T′′. Set V = Hom_{E}(V′′, V′). Let T ⊂ V be a G_{Σ}-stable O-lattice, and set W = V/T. For an O-module M, let M[n] denote the submodule consisting of elements killed by λ^{n}. For 𝔭 ∈ Σ, Lemma 9.18 provides a natural bijection between H^{1}(L_{𝔭}, W [n]) and Ext_{O/λ^{n}[D_{𝔭}]}(W′′[n], W′[n]). We now define degree n local Selmer groups. If 𝔭 ∈ Σ \ Σ_{ℓ}, set

H_{f}^{1}(L_{𝔭}, W [n]) := H_{un}^{1}(L_{𝔭}, W [n]), where W is as above.

If 𝔭 ∈ Σ_{ℓ}, define H_{f}^{1}(L_{𝔭}, W [n]) ⊂ H^{1}(L_{𝔭}, W [n]) to be the subset consisting of those cohomology classes which correspond to extensions

0 → W′[n] → W̃ [n] → W′′[n] → 0 ∈ Ext_{O/λ^{n}[D_{𝔭}]}(W′′[n], W′[n])

such that W̃ [n] is in the essential image of the functor V defined in [13], section 1.1.2. We will not need the precise definition of V. It is shown in [13] that H_{f}^{1}(L_{𝔭}, W [n]) is an O-submodule of H^{1}(L_{𝔭}, W [n]) and that H_{f}^{1}(L_{𝔭}, W [n]) is the preimage of H_{f}^{1}(L_{𝔭}, W [n + 1]) under the natural map H^{1}(L_{𝔭}, W [n]) → H^{1}(L_{𝔭}, W [n + 1]) (cf. Section 2.1, loc. cit.).

Lemma 9.19. Fix 𝔭 ∈ Σ_{ℓ}. Let ρ̃: G_{L} → GL_{E}(Ṽ) be a Galois representation short at 𝔭, T̃ ⊂ Ṽ an O[D_{𝔭}]-stable lattice and W̃ := Ṽ /T̃. If W̃ [n] fits into an exact sequence

0 → W′[n] → W̃ [n] → W′′[n] → 0 ∈ Ext_{O/λ^{n}[D_{𝔭}]}(W′′[n], W′[n]),

then such an extension gives rise to an element of H_{f}^{1}(L_{𝔭}, W [n]).

Proof. See [13], Section 1.1.2. □

Proposition 9.20. The natural isomorphism

\underrightarrow{lim}_{n} H^{1}(L_{𝔭}, W [n]) ≅ H^{1}(L_{𝔭}, W)

induces a natural isomorphism

\underrightarrow{lim}_{n} H_{f}^{1}(L_{𝔭}, W [n]) ≅ H_{f}^{1}(L_{𝔭}, W).

Proof. See [13], Proposition 2.2. □

9.5. Proof of Theorem 9.10. The key ingredient in the proof of Theorem 9.10 is Lemma 9.21 below. Before we state it, we need some notation. Let L be any number field, Σ ⊃ Σ_{ℓ} a finite set of primes of L. Let n′, n′′ ∈ Z_{≥0} and n := n′ + n′′. Let V′ (respectively V′′) be an E-vector space of dimension n′ (resp. n′′), affording a continuous absolutely irreducible representation ρ′ : G_{Σ} → Aut_{E}(V′) (resp. ρ′′ : G_{Σ} → Aut_{E}(V′′)). Assume that the residual representations \overline{ρ}′ and \overline{ρ}′′ are also absolutely irreducible (hence well-defined) and non-isomorphic. Let V_{1}, …, V_{m} be n-dimensional E-vector spaces each of them affording an absolutely irreducible continuous representation ρ_{i} : G_{Σ} → Aut_{E}(V_{i}), i = 1, …, m. Moreover assume that the mod λ reductions \overline{ρ}_{i} (with respect to some G_{Σ}-stable lattice in V_{i} and hence with respect to all such lattices) satisfy

\overline{ρ}_{i}^{ss} ≅ \overline{ρ}′ ⊕ \overline{ρ}′′.

For σ ∈ G_{Σ}, let ∑_{j=0}^{n} a_{j}(σ)X^{j} ∈ O[X] be the characteristic polynomial of (ρ′ ⊕ ρ′′)(σ) and let ∑_{j=0}^{n} c_{j}(i, σ)X^{j} ∈ O[X] be the characteristic polynomial of

ρ_{i}(σ). Put c_{j}(σ) := [\begin{matrix}c_{j}(1, σ) \\ … \\ c_{j}(m, σ) \end{matrix}] ∈ O^{m} for j = 0, 1, …, n − 1. Let T ⊂ O^{m} be

the O-subalgebra generated by the set {c_{j}(σ) | 0 ≤ j ≤ n − 1, σ ∈ G_{Σ}}. By continuity of the ρ_{i} this is the same as the O-subalgebra of O^{m} generated by {c_{j}(Frob_{𝔭}) | 0 ≤ j ≤ n − 1, 𝔭 ∉ Σ}. Note that T is a finite O-algebra. Let I ⊂ T be the ideal generated by the set {c_{j}(Frob_{𝔭}) − a_{j}(Frob_{𝔭}) | 0 ≤ j ≤ n − 1, 𝔭 ∉ Σ}. From the definition of I it follows that the O-algebra structure map O → T/I is surjective. Let J be the kernel of this map, so we have O/J = T/I. The following lemma is due to Urban.

Lemma 9.21. Suppose F^{×} contains n distinct elements. Then there exists a G_{Σ}- stable T-submodule L ⊂ ⨁_{i=1}^{m} V_{i}, T-submodules L′, L′′ ⊂ L (not necessarily G_{Σ}- stable) and a finitely generated T-module T such that

(1) as T-modules we have L = L′ ⊕ L′′ and L′′ ≅ T^{n′′};

(2) L has no T[G_{Σ}]-quotient isomorphic to \overline{ρ}′;

(3) L′/IL′ is G_{Σ}-stable and there exists a T[G_{Σ}]-isomorphism

L/(IL + L′) ≅ M′′ ⊗_{O} T/I

for any G_{Σ}-stable O-lattice M′′ ⊂ V′′.

(4) Fitt_{T}(T ) = 0 and there exists a T[G_{Σ}]-isomorphism

L′/IL′ ≅ M′ ⊗_{O} T /IT

for any G_{Σ}-stable O-lattice M′ ⊂ V′.

Proof. Lemma 9.21 follows from Theorem 1.1 of [55]. We only indicate how one proves that Fitt_{T}(T ) = 0, which is not directly stated in [55]. By Lemma 1.5 (i) in [loc. cit.], L′ ≅ T^{n′}, hence it is enough to show that 𝔞 := Fitt_{T}(L′) = 0. Since 𝔞 ⊂ Ann_{T}(L′), if 𝔞 ≠ 0, there exists a non-zero t ∈ T such that tL′ = 0. Let 1 ≤ i ≤ m be such that the projection t_{i} of t onto the ith component of T ⊂ O^{m} is non-zero. Then t_{i} annihilates the image of L′ under the projection of ⨁_{j=1}^{m} V_{j} ↠ V_{i}. Since 0 ≠ t_{i} ∈ O and O is a domain, we must have that the image of L′ in V_{i} is zero. Thus the composite L ↪ ⨁_{j=1}^{m} V_{j} ↠ V_{i} factors through L/L′ ≅ L′′ ≅ T^{n′′} by part (1) of the Lemma. Hence the image of L in V_{i} is a G_{Σ}-stable, rank n′′ O-module which contradicts the assumption that ρ_{i} is absolutely irreducible. We conclude that Fitt_{T}(L′) = 0. □

We will now show how Lemma 9.21 implies Theorem 9.10. For this we set • n′ = n′′ = 2;

• L = K, Σ = Σ_{ℓ} ∪ {(i + 1)}, Σ′ := {(i + 1)};

• ρ′ = ρ_{f,K}, ρ′′ = ρ_{f,K} ⊗ ϵ, V′, V′′ = representation spaces of ρ′, ρ′′ respectively;

• T = T_{𝔪_{F_{f}}}^{NM} ;

• N_{F_{f}}^{NM} = {F ∈ N^{NM} | ϕ^{−1}(𝔪_{F}^{NM}) = 𝔪_{F_{f}} } (we denote the elements of N_{F_{f}}^{NM} by F_{1}, …, F_{m});

• I = the ideal of T generated by ϕ_{𝔪_{F_{f}}} (Ann F_{f})

• (V_{i}, ρ_{i}) = the representation ρ_{F_{i}}, i = 1, …, m.

Remark 9.22. As mentioned in section 9.2, ρ′ and ρ′′ factor not only through G_{Σ}, but also through G_{Σ_{ℓ}}, however, the ρ_{i} do not necessarily factor through G_{Σ_{ℓ}} (cf. Theorem 9.2), and hence we have to work with Σ as defined above. Nevertheless, for any G_{Σ} module M which is unramified at (i + 1) we have an exact sequence (cf. [58], Proposition 6)

0 → H^{1}(G_{Σ_{ℓ}}, M) → H^{1}(G_{Σ}, M) → H^{1}(I_{(i+1)}, M).

Hence in particular the group S_{Σ}(W) from Theorem 9.10 is isomorphic to S_{Σ}({(i + 1)}, W), which we study below.

Lemma 9.21 guarantees the existence of L, L′, L′′ and T with properties (1)-(4) as in the statement of the lemma. Let M′ (resp. M′′) be a G_{Σ}-stable O-lattice inside V′ (resp. V′′). The split short exact sequence of T-modules (cf. Lemma 9.21, (1))

(9.8)  0 → L′ → L → L/L′ → 0

gives rise to a short exact sequence of (T/I)[G_{Σ}]-modules, which splits as a sequence of T/I-modules (cf. Lemma 9.21, (3) and (4))

(9.9)  0 → M′ ⊗_{O} T /IT → L/IL → M′′ ⊗_{O} T/I → 0.

(Note that L/IL ≅ L ⊗_{T} T/I ≅ L ⊗_{O} T/I, hence (9.9) recovers the sequence from Theorem 1.1 of [55].) Let s : M′′ ⊗_{O} T/I → L/IL be a section of T/I-modules. Define a class c ∈ H^{1}(G_{Σ}, Hom_{T/I}(M′′ ⊗_{O} T/I, M′ ⊗_{O} T /IT )) by

g ↦ (m′′ ⊗ t ↦ s(m′′ ⊗ t) − g · s(g^{−1} · m′′ ⊗ t)).

The following lemma will be used in the proof of Lemma 9.25.

Lemma 9.23. Let I_{(i+1)} denote the inertia group of the prime ideal (i + 1). We have c|_{I_{(i+1)}} = 0.

Proof. For simplicity set J := I_{(i+1)} and D := D_{(i+1)}. We identify D with Gal(\overline{K}_{(i+1)}/K_{(i+1)}). It is enough to show that J acts trivially on L/IL. Let ϕ : D → Aut_{T/I}(L/IL) be the homomorphism giving the action of D on L/IL and denote by K_{s} the splitting field of ϕ. Set G := Gal(K_{s}/K_{(i+1)}). Note that for

g ∈ D we can write ϕ(g) = \left[{ϕ_{11}(g) ϕ_{12}(g) \atop ϕ_{22}(g)}\right], where ϕ_{11}(g) ∈ Aut_{T/I}(M′ ⊗_{O} T /IT ), ϕ_{22}(g) ∈ Aut_{T/I}(M′′ ⊗_{O} T/I) and ϕ_{12}(g) ∈ Hom_{T/I}(M′′ ⊗_{O} T/I, M′ ⊗_{O} T /IT ). Also note that ϕ_{11} and ϕ_{22} are group homomorphisms from G into the appropriate groups of automorphisms. Let K^{un}/K_{(i+1)} be the maximal unramified subextension of K_{s}/K_{(i+1)} and let σ ∈ Gal(K^{un}/K_{(i+1)}) be the Frobenius generator. Let τ be a topological generator of the totally tamely ramified extension K_{s}/K^{un}. On the one hand ϕ(τ) = \left[{1 ϕ_{12}(τ) \atop 1}\right] since M′ and M′′ are unramified at (i + 1), and on the other hand, ϕ(στσ^{−1}) = ϵ(σ)ϕ(τ). This implies that

(9.10)  ϕ_{11}(σ)ϕ_{12}(τ)ϕ_{22}(σ)^{−1} = ϵ(σ)ϕ_{12}(τ).

Writing f = ∑_{n=1}^{∞} a(n)q^{n} and using Theorem 3.26(ii) from [27], we get ρ_{f,K}|_{D} ≅ [ \begin{matrix}μ_{1} \\ & μ_{2} \end{matrix} ], where μ_{j} are unramified characters with μ_{2}(σ) = a(2). Assume M′ = M′′ as O-submodules of V′ (= V′′ as an E-vector space) and choose an O-basis {e_{1}, e_{2}} of M′ so that in that basis ρ_{f,K}|_{D} = [ \begin{matrix}μ_{1} \\ & μ_{2} \end{matrix} ]. Since M′ ⊗_{O} T /IT ≅ (T /IT )^{2}, it follows that every element x ∈ M′ ⊗_{O} T /IT can be written as e_{1} ⊗ t_{1} + e_{2} ⊗ t_{2}, where t_{1}, t_{2} ∈ T /IT are uniquely determined by x. Hence ρ′(σ)(e_{j}) = μ_{j}(σ)e_{j} and ρ′′(σ)(e_{j}) = μ_{j}(σ)ϵ(σ)e_{j}. Write ϕ_{12}(τ)(e_{j} ⊗ 1) = e_{1} ⊗ t_{j1} + e_{2} ⊗ t_{j2}. Then (9.10) implies that t_{11} = t_{22} = 0. Moreover, if t_{12} ≠ 0, we must have μ_{1}(σ)μ_{2}(σ)^{−1} ≡ ϵ(σ)^{−2} (mod λ), while if t_{21} ≠ 0, we must have μ_{1}(σ)μ_{2}(σ)^{−1} ≡ ϵ(σ)^{2} (mod λ). Since det ρ′(σ) ≡ μ_{1}(σ)μ_{2}(σ) ≡ ϵ^{k−2}(σ) (mod λ) by the Tchebotarev Density Theorem, we get μ_{2}(σ) ≡ ϵ(σ)^{k} ≡ 2^{k} (mod λ) if t_{12} ≠ 0 and μ_{2}(σ) ≡ ϵ(σ)^{k−4} ≡ 2^{k−4} (mod λ) if t_{21} ≠ 0. Since none of these congruences can hold due to our assumption on f, we get ϕ_{12}(τ) = 0 and the lemma follows. □

Note that Hom_{T/I}(M′′ ⊗_{O} T/I, M′ ⊗_{O} T /IT ) ≅ Hom_{O}(M′′, M′) ⊗_{O} T /IT , so c can be regarded as an element of

H^{1}(G_{Σ}, Hom_{O}(M′′, M′) ⊗_{O} T /IT ).

Define a map

\begin{matrix}ι : Hom_{O}(T /IT , E/O) → & H^{1}(G_{Σ}, Hom_{O}(M′′, M′) ⊗_{O} E/O) \\ f ↦ & (1 ⊗ f)(c). \end{matrix} ; (9.11)

Note that T̃ := Hom_{O}(M′′, M′) is a G_{Σ}-stable O-lattice inside Ṽ = ad ρ_{f,K} ⊗ϵ^{−1} = Hom_{E}(V′′, V′). Then W̃ = Hom_{O}(M′′, M′) ⊗_{O} E/O = W ⊕ E/O(−1), where W is as in Theorem 9.10.

Lemma 9.24. We have S_{Σ}(W) = S_{Σ}(W̃).

Proof. Let O_{χ} be a free rank-one O-module on which G_{Σ} operates by a (non-trivial) character χ, and set W_{χ} = E/O ⊗O_{χ}. Since every element in Sel_{Σ}(W_{χ}) is killed by a power of ℓ, we have Sel_{Σ}(W_{χ}) = 0 if and only if the λ-torsion part Sel_{Σ}(W_{χ})[1] of Sel_{Σ}(W_{χ}) is zero. Hence it is enough to show that Sel_{Σ}(W_{ϵ^{−1}})[1] = 0. Note that the natural map H^{1}(G_{Σ}, W_{χ}[1]) → H^{1}(G_{Σ}, W_{χ}) is an injection since H^{0}(G_{Σ}, W_{χ}) = 0 for a non-trivial χ. Hence Sel_{Σ}(W_{χ})[1] = Sel_{Σ}(W_{χ})∩H^{1}(G_{Σ}, W_{χ}[1]). Thus, we have Sel_{Σ}(W_{ϵ^{−1}})[1] = Sel_{Σ}(W_{ϵ^{−1}}) ∩ H^{1}(G_{Σ}, W_{ϵ^{−1}}[1]). Since W_{ϵ^{−1}}[1] = W_{ω^{−1}}[1], where ω : G_{Σ} → Z_{ℓ}^{×} is the Teichmuller lift of the mod ℓ cyclotomic character, we conclude that Sel_{Σ}(W_{ϵ^{−1}})[1] = Sel_{Σ}(W_{ω^{−1}})[1]. So it suffices to show that Sel_{Σ}(W_{ω^{−1}})[1] = 0. Its Pontryagin dual S_{Σ}(W_{ω^{−1}}) is isomorphic to Cl_{K(ζ_{ℓ})}^{ω^{−1}}, the ω^{−1}-isotypical part of

the ℓ-primary part of the class group of K(ζ_{ℓ}). This in turn is isomorphic to Cl_{Q(ζ_{ℓ})}^{ω^{−1}}, since ℓ is odd ([41], Remark (3), p. 216). By [41], Theorem 2, p. 216, the ℓ-adic valuation of the order of Cl_{Q(ζ_{ℓ})}^{ω^{−1}} is equal to the ℓ-adic valuation of B_{1}(ω)^{[E:Q_{ℓ}]}, where B_{1}(χ) is the first generalized Bernoulli number of χ. Since B_{1}(ω) ≡ \frac{1}{6} (mod ℓ), and ℓ > 3, we obtain our claim. □

By Lemma 9.24 it is enough to work with S_{Σ}(W̃) instead of S_{Σ}(W). Since the mod λ reduction of the representation ad^{0}(ρ_{f,K}) ⊗ ϵ^{−1} is absolutely irreducible, Lemma 9.7 implies that our conclusion is independent of the choice of T. Hence we can work with T̃ chosen as above.

Lemma 9.25. The image of ι is contained inside Sel_{Σ}({(i + 1)}, W).

Lemma 9.26. ker(ι)^{∨} = 0.

We first prove that Lemma 9.25 and Lemma 9.26 imply Theorem 9.10.

Proof of Theorem 9.10. By Remark 9.22, S_{Σ_{ℓ}}(W̃) ≅ S_{Σ}({(i + 1)}, W̃), so it is enough to bound the size of the latter group. It follows from Lemma 9.25 that

ord_{ℓ}(#S_{Σ}(W̃)) ≥ ord_{ℓ}(# Im(ι)^{∨}),

and from Lemma 9.26 that

(9.12)  ord_{ℓ}(# Im(ι)^{∨}) = ord_{ℓ}(# Hom_{O}(T /IT , E/O)^{∨}).

Since Hom_{O}(T /IT , E/O)^{∨} ≅ (T /IT )^{∨∨} = T /IT (cf. [27], page 98), we have

ord_{ℓ}(# Im(ι)^{∨}) = ord_{ℓ}(#T /IT ).

So, it remains to show that ord_{ℓ}(#T /IT ) ≥ ord_{ℓ}(#T/I). Since Fitt_{T}(T ) = 0 (Lemma 9.21 (4)), we have Fitt_{T}(T ⊗_{T} T/I) ⊂ I and thus ord_{ℓ}(#(T ⊗_{T} T/I)) ≥ ord_{ℓ}(#T/I). As ord_{ℓ}(#T /IT ) = ord_{ℓ}(#(T ⊗_{T} T/I)), the claim follows. □

Proof of Lemma 9.25. Consider f ∈ Hom_{O}(T /IT , E/O). Since c|_{I_{(i+1)}} = 0 by Lemma 9.23, we only need to show that (1 ⊗ f)(c)|_{D_{𝔭}} ∈ H_{f}^{1}(L_{𝔭}, W̃) for 𝔭 ∈ Σ_{ℓ}. Fix such a 𝔭. Note that since T /IT is a finitely generated T-module, it is also a finitely generated O-module (since T/I = O/J). In fact it is even of finite cardinality for the same reason. In any case, there exists a positive integer n such that Hom_{O}(T /IT , E/O) = Hom_{O}(T /IT , E/O[n]). Thus

Im(ι) ⊂ H^{1}(G_{Σ}, Hom_{O}(M′′, M′) ⊗_{O} E/O[n]) = H^{1}(G_{Σ}, W̃ [n]).

By Lemma 9.20, we have \underrightarrow{lim}_{j} H_{f}^{1}(L_{𝔭}, W̃_{j}) ≅ H_{f}^{1}(L_{𝔭}, W̃), hence it is enough to show

that Im(ι) ⊂ H_{f}^{1}(L_{𝔭}, W̃ [n]). However, this is clear by Lemma 9.19 since by Theorem 9.2, each ρ_{i} is short at 𝔭 (note that we are assuming that ℓ > k). □

Proof of Lemma 9.26. We follow [52], but see also [55], Fact 1 on page 520. First note that if f ∈ Hom_{O}(T /IT , E/O), then ker f has finite index in T /IT . Suppose that f ∈ ker ι. We will show that the image of c under the map

ϕ : H^{1}(G_{Σ}, Hom_{O}(M′′, M′) ⊗_{O} T /IT ) → H^{1}(G_{Σ}, Hom_{O}(M′′, M′) ⊗_{O} K_{f})

is zero. Here K_{f} := (T /IT )/ ker f. Assuming f ≠ 0, we will use this fact to produce a T[G_{Σ}]-quotient of L isomorphic to \overline{ρ}′ and thus arrive at a contradiction. Set I_{f} := (E/O)/ Im f and T̃ := Hom_{O}(M′′, M′). Tensoring the short exact sequence of O[G_{Σ}]-modules

0 → K_{f} ⟶^{f} E/O → I_{f} → 0,

with ⊗_{O}T̃ and considering a piece of the long exact sequence in cohomology together with the map ϕ we obtain commutative diagram with the bottom row being exact (9.13)

\xymatrix{ & H^{1}(G_{Σ}, T̃ ⊗_{O} T /IT ) \ar[d]^{ϕ} \ar@{-->}[dr]^{H^{1}(1⊗f)} \\ H^{0}(G_{Σ}, T̃ ⊗_{O} I_{f}) \ar[r] & H^{1}(G_{Σ}, T̃ ⊗_{O} K_{f}) \ar[r]^{H^{1}(1⊗f)} & H^{1}(G_{Σ}, T̃ ⊗_{O} E/O).}

Since f ∈ ker ι, we get H^{1}(1 ⊗ f) ∘ ϕ(c) = 0. As the action of G_{Σ} on M′ and M′′ respectively gives rise to absolutely irreducible non-isomorphic representations, H^{0}(G_{Σ}, T̃ ⊗_{O} I_{f}) = 0. So, exactness of the bottom row of (9.13) implies that ϕ(c) = 0. From now on assume that 0 ≠ f ∈ ker ι. Since ker f ≠ 0, there exists an O-module A with ker f ⊂ A ⊂ T /IT such that (T /IT )/A ≅ O/λ = F. Since the image of c in H^{1}(G_{Σ}, T̃ ⊗_{O} ((T /IT )/A)) under the composite

(9.14) H^{1}(G_{Σ}, T̃ ⊗_{O} T /IT ) ⟶^{ϕ} H^{1}(G_{Σ}, T̃ ⊗_{O} ((T /IT )/ ker f)) →

→ H^{1}(G_{Σ}, T̃ ⊗_{O} ((T /IT )/A)).

is zero, the sequence

(9.15)  0 → M′ ⊗_{O} F → (L/IL)/(λL + M′ ⊗_{O} A) → M′′ ⊗_{O} F → 0

splits a sequence of T[G_{Σ}]-modules. As G_{Σ} acts on M′ ⊗_{O} F via \overline{ρ}′, this contradicts the fact that L has no quotient isomorphic \overline{ρ}′. Hence ker ι = 0 and thus (ker ι)^{∨} = 0 as well. □

References

1. J. Bellaïche and G. Chenevier, Formes non tempérées pour U(3) et conjectures de Bloch-Kato, Ann. Sci. École Norm. Sup. (4) 37 (2004), no. 4, 611–662.

2. , p-adic families of galois representations and higher rank Selmer groups, Astérisque (to appear).

3. T. Berger, An Eisenstein ideal for imaginary quadratic fields and the Bloch-Kato conjecture for Hecke characters, Preprint (2007), arXiv:math.NT/0701177.

4. D. Blasius and J. D. Rogawski, Zeta functions of Shimura varieties, Motives (Seattle, WA, 1991), Proc. Sympos. Pure Math., vol. 55, Amer. Math. Soc., Providence, RI, 1994, pp. 525– 571.

5. S. Bloch and K. Kato, L-functions and Tamagawa numbers of motives, The Grothendieck Festschrift, Vol. I, Progr. Math., vol. 86, Birkhäuser Boston, Boston, MA, 1990, pp. 333–400.

6. H. Braun, Hermitian modular functions, Ann. of Math. (2) 50 (1949), 827–855.

7. , Hermitian modular functions. II. Genus invariants of Hermitian forms, Ann. of Math. (2) 51 (1950), 92–104.

8. , Hermitian modular functions. III, Ann. of Math. (2) 53 (1951), 143–160.

9. J. Brown, Saito-Kurokawa lifts and applications to the Bloch-Kato conjecture, Compos. Math. 143 (2007), no. 2, 290–322.

10. D. Bump, Automorphic forms and representations, Cambridge Studies in Advanced Mathematics, vol. 55, Cambridge University Press, Cambridge, 1997.

11. H. Darmon, F. Diamond, and R. Taylor, Fermat’s last theorem, Elliptic curves, modular forms & Fermat’s last theorem (Hong Kong, 1993), Internat. Press, Cambridge, MA, 1997, pp. 2–140.

12. P. Deligne, Valeurs de fonctions L et périodes d’intégrales, Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, With an appendix by N. Koblitz and A. Ogus, pp. 313–346.

13. F. Diamond, M. Flach, and L. Guo, The Tamagawa number conjecture of adjoint motives of modular forms, Ann. Sci. École Norm. Sup. (4) 37 (2004).

14. F. Diamond and J. Im, Modular forms and modular curves, Seminar on Fermat’s Last Theorem (Toronto, ON, 1993–1994), CMS Conf. Proc., vol. 17, Amer. Math. Soc., Providence, RI, 1995, pp. 39–133.

15. N. Dummigan, Period ratios of modular forms, Math. Ann. 318 (2000), no. 3, 621–636.

16. , Symmetric square L-functions and Shafarevich-Tate groups, Experiment. Math. 10 (2001), no. 3, 383–400.

17. , Symmetric square L-functions and Shafarevich-Tate groups, II, Preprint (2008).

18. D. Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995, With a view toward algebraic geometry.

19. J.-M. Fontaine, Sur certains types de représentations p-adiques du groupe de Galois d’un corps local; construction d’un anneau de Barsotti-Tate, Ann. of Math. (2) 115 (1982), no. 3, 529–577.

20. J.-M. Fontaine and B. Perrin-Riou, Autour des conjectures de Bloch et Kato: cohomologie galoisienne et valeurs de fonctions L, Motives (Seattle, WA, 1991), Proc. Sympos. Pure Math., vol. 55, Amer. Math. Soc., Providence, RI, 1994, pp. 599–706.

21. E. Freitag, Siegelsche Modulfunktionen, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 254, Springer-Verlag, Berlin, 1983.

22. S. Gelbart, Automorphic forms on adèle groups, Princeton University Press, Princeton, N.J., 1975, Annals of Mathematics Studies, No. 83.

23. V. A. Gritsenko, The Maass space for SU(2, 2). The Hecke ring, and zeta functions, Trudy Mat. Inst. Steklov. 183 (1990), 68–78, 223–225, Translated in Proc. Steklov Inst. Math. 1991, no. 4, 75–86, Galois theory, rings, algebraic groups and their applications (Russian).

24. , Parabolic extensions of the Hecke ring of the general linear group. II, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 183 (1990), no. Modul. Funktsii i Kvadrat. Formy. 1, 56–76, 165, 167.

25. H. Hida, Galois representations and the theory of p-adic Hecke algebras, Sūgaku 39 (1987), no. 2, 124–139, Sugaku Expositions 2 (1989), no. 1, 75–102.

26. , Elementary theory of L-functions and Eisenstein series, London Mathematical Society Student Texts, vol. 26, Cambridge University Press, Cambridge, 1993.

27. , Modular forms and Galois cohomology, Cambridge Studies in Advanced Mathematics, vol. 69, Cambridge University Press, Cambridge, 2000.

28. , p-adic automorphic forms on Shimura varieties, Springer Monographs in Mathematics, Springer-Verlag, New York, 2004.

29. T. Hina and T. Sugano, On the local Hecke series of some classical groups over 𝔭-adic fields, J. Math. Soc. Japan 35 (1983), no. 1, 133–152.

30. T. Ikeda, On the lifting of hermitian modular forms, Preprint (2007).

31. H. Iwaniec, Topics in classical automorphic forms, Graduate Studies in Mathematics, vol. 17, American Mathematical Society, Providence, RI, 1997.

32. H. Kim, Automorphic L-functions, Lectures on automorphic L-functions, Fields Inst. Monogr., vol. 20, Amer. Math. Soc., Providence, RI, 2004, pp. 97–201.

33. G. Kings, The Bloch-Kato conjecture on special values of L-functions. A survey of known results, J. Théor. Nombres Bordeaux 15 (2003), no. 1, 179–198, Les XXIIèmes Journées Arithmetiques (Lille, 2001).

34. K. Klosin, Congruences among automorphic forms on the unitary group U(2, 2), Thesis, University of Michigan, Ann Arbor, 2006.

35. , Adelic Maass spaces on U(2, 2), preprint (2007), arXiv:math.NT/0706.2828.

36. H. Kojima, An arithmetic of Hermitian modular forms of degree two, Invent. Math. 69 (1982), no. 2, 217–227.

37. A. Krieg, The Maaß spaces on the Hermitian half-space of degree 2, Math. Ann. 289 (1991), no. 4, 663–681.

38. R. P. Langlands, On the functional equations satisfied by Eisenstein series, Springer-Verlag, Berlin, 1976, Lecture Notes in Mathematics, Vol. 544.

39. B. Mazur, Modular curves and the Eisenstein ideal, Inst. Hautes Études Sci. Publ. Math. (1977), no. 47, 33–186 (1978).

40. , An introduction to the deformation theory of Galois representations, Modular forms and Fermat’s last theorem (Boston, MA, 1995), Springer, New York, 1997, pp. 243–311.

41. B. Mazur and A. Wiles, Class fields of abelian extensions of Q, Invent. Math. 76 (1984), no. 2, 179–330.

42. T. Miyake, Modular forms, Springer-Verlag, Berlin, 1989, Translated from the Japanese by Yoshitaka Maeda.

43. C. Mœglin and J.-L. Waldspurger, Le spectre résiduel de GL(n), Ann. Sci. École Norm. Sup. (4) 22 (1989), no. 4, 605–674.

44. S. Raghavan and J. Sengupta, A Dirichlet series for Hermitian modular forms of degree 2, Acta Arith. 58 (1991), no. 2, 181–201.

45. K. A. Ribet, A modular construction of unramified p-extensions of Q(μ_{p}), Invent. Math. 34 (1976), no. 3, 151–162.

46. K. Rubin, Euler systems, Annals of Mathematics Studies, vol. 147, Princeton University Press, Princeton, NJ, 2000, Hermann Weyl Lectures. The Institute for Advanced Study.

47. C.-G. Schmidt, p-adic measures attached to automorphic representations of GL(3), Invent. Math. 92 (1988), no. 3, 597–631.

48. J.-P. Serre, Groupes de Lie l-adiques attachés aux courbes elliptiques, Les Tendances Géom. en Algébre et Théorie des Nombres, Éditions du Centre National de la Recherche Scientifique, Paris, 1966, pp. 239–256.

49. G. Shimura, Confluent hypergeometric functions on tube domains, Math. Ann. 260 (1982), no. 3, 269–302.

50. , Euler products and Eisenstein series, CBMS Regional Conference Series in Mathematics, vol. 93, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1997.

51. , Arithmeticity in the theory of automorphic forms, Mathematical Surveys and Monographs, vol. 82, American Mathematical Society, Providence, RI, 2000.

52. C. M. Skinner, Selmer groups, Preprint (2004).

53. J. Sturm, Special values of zeta functions, and Eisenstein series of half integral weight, Amer. J. Math. 102 (1980), no. 2, 219–240.

54. R. Taylor and A. Wiles, Ring-theoretic properties of certain Hecke algebras, Ann. of Math. (2) 141 (1995), no. 3, 553–572.

55. E. Urban, Selmer groups and the Eisenstein-Klingen ideal, Duke Math. J. 106 (2001), no. 3, 485–525.

56. V. Vatsal, Canonical periods and congruence formulae, Duke Math. J. 98 (1999), no. 2, 397–419.

57. , Special values of anticyclotomic L-functions, Duke Math. J. 116 (2003), no. 2, 219– 261.

58. L. C. Washington, Galois cohomology, Modular forms and Fermat’s last theorem (Boston, MA, 1995), Springer, New York, 1997, pp. 101–120.

59. , Introduction to cyclotomic fields, second ed., Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1997.

60. A. Wiles, The Iwasawa conjecture for totally real fields, Ann. of Math. (2) 131 (1990), no. 3, 493–540.

61. , Modular elliptic curves and Fermat’s last theorem, Ann. of Math. (2) 141 (1995), no. 3, 443–551.