Anticyclotomic Main Conjectures

To John Coates

Haruzo Hida^{1}

Received: August 3, 2005 Revised: May 16, 2006

Abstract. In this paper, we prove many cases of the anticyclotomic main conjecture for general CM fields with p-ordinary CM type.

2000 Mathematics Subject Classification: 11F27, 11F30, 11F33, 11F41, 11F60, 11F80, 11G10, 11G15, 11G18, 11R23, 11R34, 11R42 Keywords and Phrases: Eisenstein series, Main conjecture, CM field, CM abelian variety, Shimura series, Basis problem

Contents

1. Introduction 466 2. Siegel’s theta series for GL(2) × GL(2) 470 2.1. Symmetric Domain of O(n, 2) 470 2.2. SL(2) × SL(2) as an orthogonal group 473 2.3. Growth of theta series 474 2.4. Partial Fourier transform 475 2.5. Fourier expansion of theta series 479 3. q–Expansion of Shimura series 481 3.1. Integral expression 482 3.2. Computation of q–expansion 484 4. Evaluation at CM points 487 4.1. CM points 487 4.2. Special values of Shimura series 488 4.3. An explicit formula of Petersson inner product 489 5. Jacquet-Langlands-Shimizu correspondence 491 5.1. Hilbert modular forms and Hecke algebras 492 5.2. q–Expansion of p–integral modular forms 495 5.3. Integral correspondence 497 6. Ordinary cohomology groups 501 6.1. Freeness as Hecke modules 502 6.2. Induced representations 506 6.3. Self-duality 512 7. Proof of the theorem 513 7.1. Integrality of values of modular forms 513 7.2. Error terms of integral decomposition 516 7.3. Proof 520 References 530

1The author is partially supported by the NSF grant: DMS 0244401 and DMS 0456252.

Documenta Mathematica · Extra Volume Coates (2006) 465–532

1. Introduction

Iwasawa’s theory for elliptic curves with complex multiplication was initiated by J. Coates in the 1970s in a series of papers (for example, [CW] and [CW1]), and it is now well developed (by the effort of a handful of number-theorists) into a solid theory for abelian varieties of CM type (or one may call it Iwasawa’s theory for CM fields). In this paper, we prove many cases of the anticyclotomic main conjecture for general CM fields with p-ordinary CM type.

Let M be a CM field with maximal real subfield F. The field F is totally real, and M is a totally imaginary quadratic extension of F (inside a fixed algebraic closure \overline{F} of F). We fix a prime p > 3 unramified in M/ℚ. We assume to have a p–ordinary CM type Σ of M. Thus, fixing an embedding i_{p} : \overline{ℚ} ↪ \overline{ℚ}_{p}, the embeddings i_{p} ∘ σ for σ ∈ Σ induce exactly a half Σ_{p} of the p–adic places of M. We identify Σ_{p} with a subset of prime factors of p in M. For the generator c of Gal(M/F), the disjoint union Σ_{p} ⊔ Σ_{p}^{c} gives the total set of prime factors of p in M. For a multi-index e = ∑_{𝔓|p} e(𝔓)𝔓 ∈ ℤ[Σ_{p} ⊔ Σ_{p}^{c}], we write 𝔓^{e} for ∏_{𝔓|p} 𝔓^{e(𝔓)}. We choose a complete discrete valuation ring W inside \overline{ℚ}_{p} finite flat and unramified over ℤ_{p}. A Hecke character ψ : M^{×}\M_{𝔸}^{×} → ℂ^{×} is called anticyclotomic if ψ(x^{c}) = ψ(x)^{−1}. We call ψ has split conductor if the conductor of ψ is divisible only by primes split in M/F. We fix a continuous anticyclotomic character ψ : Gal(\overline{F} /M) → W^{×} of finite order. It is an easy consequence of class field theory(see (7.18) and [HMI] Lemma 5.31) that we can always find another Hecke character φ : M_{𝔸}^{×}/M^{×}M_{∞}^{×} → ℂ^{×} such that ψ(x) = φ^{−}(x) = φ^{−1}(x)φ(x^{c}). Regarding φ and ψ as Galois characters, this is equivalent to ψ(σ) = φ^{−1}(σ)φ(cσc^{−1}) for any complex conjugation c in Gal(\overline{F} /F). We assume the following four conditions:

(1) The character ψ has order prime to p with exact conductor 𝔠𝔓^{e} for 𝔠 prime to p.

(2) The conductor 𝔠 is a product of primes split in M/F.

(3) The local character ψ_{𝔓} is non-trivial for all 𝔓 ∈ Σ_{p}.

(4) The restriction ψ^{∗} of ψ to Gal(\overline{F} /M[\sqrt{p^{∗}}]) for p^{∗} = (−1)^{(p−1)/2}p is non-trivial.

We study arithmetic of the unique ℤ_{p}^{[F:ℚ]}–extension M_{∞}^{−} of M (unramified outside p and ∞) on which cσc^{−1} = σ^{−1} for all σ ∈ Γ_{M}^{−} = Gal(M_{∞}^{−}/M). The extension M_{∞}^{−}/M is called the anticyclotomic tower over M. Let M(ψ)/M be the class field with ψ inducing the isomorphism Gal(M(ψ)/M) ≅ Im(ψ). Let L_{∞}/M_{∞}^{−}M(ψ) be the maximal p–abelian extension unramified outside Σ_{p}. Each γ ∈ Gal(L_{∞}/M) acts on the normal subgroup X = Gal(L_{∞}/M_{∞}^{−}M(ψ)) continuously by conjugation, and by the commutativity of X, this action factors through Gal(M(ψ)M_{∞}^{−}/M). We have a canonical splitting Gal(M(ψ)M_{∞}^{−}/M) = Γ_{M}^{−} × G_{tor}(ψ) for the maximal torsion subgroup G_{tor}(ψ) ≅ Im(ψ). Since ψ is of order prime to p, it factors through the maximal torsion subgroup G_{tor}(ψ). Then we look into the Γ_{M}^{−}–module: X[ψ] = X ⊗_{ℤ_{p}[G_{tor}(ψ)],ψ} W.

As is well known, X[ψ] is a W [[Γ_{M}^{−}]]–module of finite type, and it is a torsion module by a result of Fujiwara (cf. [H00] Corollary 5.4 and [HMI] Theorem 5.33) generalizing the fundamental work of Wiles [W] and Taylor-Wiles [TW]. Thus we can think of the characteristic element F^{−}(ψ) ∈ W [[Γ_{M}^{−}]] of the module X[ψ]. As we have seen in [HT1] and [HT2], we have the anticyclotomic p–adic Hecke L–function L_{p}^{−}(ψ) ∈ \overline{W} [[Γ_{M}^{−}]] (constructed by Katz), where \overline{W} is the completed p–adic integer ring of the maximal unramified extension of ℚ_{p} inside \overline{ℚ}_{p}. We regard W ⊂ \overline{W}. Then we prove

Theorem. We have the identity: F^{−}(ψ) = L_{p}^{−}(ψ) up to a unit in \overline{W} [[Γ_{M}^{−}]].

The condition p > 3 is necessary because at one point we need to choose a prime ideal 𝔮 of F with N_{F/ℚ}(𝔮) ≢ ±1 mod p. By implementing our idea more carefully, we might be able to include the prime p = 3, but there is no hope (without a new idea) of including p = 2. The condition (1) is probably inessential, and it could be avoided by using the nearly ordinary Galois deformation with fixed p–power order nearly ordinary characters instead of the minimal one we used, although some of our argument has to be done more carefully to incorporate p– power order characters. In such a generalization, we probably need to assume (2-4) replacing ψ by the Teichmüller lift of ψ mod 𝔪_{W} for the maximal ideal 𝔪_{W} of W. The condition (2) is imposed to guarantee the local representation at the prime 𝔩 given by Ind_{M_{𝔩}}^{F_{𝔩}} φ_{𝔩} is reducible; otherwise, we possibly need to work with quaternionic modular forms coming from a quaternion algebra ramifying at an inert or ramified prime 𝔩|𝔠, adding further technicality, though we hope that the obstacle is surmountable. The condition (3) is a rigidity condition for nearly ordinary Galois deformation of Ind_{M}^{F} φ, assuring the existence of the “universal” (not “versal”) deformation ring. To remove this, we need to somehow invent a reasonable requirement to rigidify the deformation problem. The condition (4) is a technical assumption in order to form a Taylor-Wiles system to identify the deformation ring with an appropriate Hecke algebra (see [TW], [Fu] and [HMI] Sections 3.2–3).

The type of the assertion (in the theorem) is called the anticyclotomic main conjecture for CM fields. The main conjecture for imaginary quadratic fields (including the cyclotomic ℤ_{p}–extension) and its anticyclotomic version for imaginary quadratic fields have been proved by K. Rubin [R] and [R1] refining Kolyvagin’s method of Euler systems, and basically at the same time, the anticyclotomic conjecture was treated by J. Tilouine (and B. Mazur) [Ti] and [MT] (for imaginary quadratic cases) by a method similar to the one exploited here combined with the class number formula of the ring class fields. A partial result towards the general conjecture was studied in [HT1], [HT2] and [H05d].

The present idea of the proof is a refinement of those exploited in [HT1], [HT2] and [H05d] Theorem 5.1, where we have proven L_{p}^{−}(ψ)|F^{−}(ψ) in \overline{W} [[Γ_{M}^{−}]]. One of the main ingredients of the proof is the congruence power series H(ψ) ∈ W [[Γ_{M}^{−}]] of the CM–component of the universal nearly ordinary Hecke algebra h for GL(2)_{/F}. In the joint works with Tilouine, we took h of (outside p) level N_{M/F} (ℭ)d(M/F) for the conductor ℭ of φ and the relative discriminant d(M/F) of M/F. In this paper, as in [H05d] Section 2.10, we take the Hecke algebra of level 𝔑(ψ) which is a product of 𝔠∩F and d(M/F) (introducing a new type of Neben character determined by φ with ψ = φ^{−}). Fujiwara formulated his results in [Fu] using such level groups. Another important ingredient is the divisibility proven in [H05d] Corollary 5.5:

(L)  (h(M)/h(F))L_{p}^{−}(ψ^{−})|H(ψ) in \overline{W} [[Γ_{M}^{−}]].

Here h(M) (resp. h(F)) is the class number of M (resp. F). On the other hand, Fujiwara’s result already quoted implies (see [Fu], [HT2], [H00] and [HMI] Sections 3.2–3 and 5.3):

(F)  H(ψ) = (h(M)/h(F))F^{−}(ψ^{−}) up to units in \overline{W} [[Γ_{M}^{−}]].

Thus we need to prove:

(R)  H(ψ)(κ)|(h(M)/h(F))L_{p}^{−}(ψ^{−})(κ) in \overline{W}

for a (single) weight κ specialization, where Φ(κ) is the value of a power series Φ ∈ \overline{W} [[Γ_{M}^{−}]] at κ ∈ Spec(W [[Γ_{M}^{−}]])(W). By (L) and Nakayama’s lemma, the reverse divisibility (R) (specialized at κ) implies the theorem. In the (finite dimensional) space S_{κ}^{n.ord}(𝔑(ψ)p^{∞}, ε_{λ}; W) of nearly p–ordinary cusp forms of weight κ with coefficients in W and with suitable Neben character ε_{λ}, we have a CM Hecke eigenform f(λ) of a Hecke character λ of weight κ (regarded as a Galois character) such that λ^{−} factors through Gal(M(ψ)M_{∞}^{−}/M) and λ^{−}|_{G_{tor}(ψ)} = ψ. We write 𝔑(λ) (𝔑(λ)|𝔑(ψ)p^{∞}) for the level of f(λ). This form studied in [H91] is of minimal level (possibly of level smaller than that of the primitive form). Since the CM local ring R of h is a Gorenstein ring (see [Fu], [H00] Corollary 5.3 (3) and [HMI] Proposition 1.53 and Theorem 3.59), the number H(ψ)(κ) is the maximal denominator of the numbers \frac{(f(λ),f)}{(f(λ),f(λ))} in W as f running through all elements of S_{κ}(𝔑(λ), ε_{λ}; W) (see again [H00] Corollary 5.3 (1) and [H86] Proposition 3.9), where (·, ·) is the Petersson inner product of level 𝔑(λ). As seen in [HT1] Theorem 7.1 and [H05d] Proposition 5.6, we have π^{κ_{1}−κ_{2}+Σ}(f(λ), f(λ)) = c_{1}(h(M)/h(F))L(1, λ^{−}) for an innocuous constant c_{1} ∈ W (for the constant c_{1}, see (7.17)). The quotient \frac{π^{2(κ_{1}−κ_{2})}W_{p}(λ^{−})(f(λ),f(λ))}{Ω^{2(κ_{1}−κ_{2})}} is then the value (h(M)/h(F))L_{p}^{−}(ψ^{−})(κ) ∈ \overline{W} (up to units in W). Here W_{p}(λ^{−}) is the local Gauss sum of λ^{−} at p, Ω is the Néron period of the abelian variety of CM type Σ (defined over \overline{ℚ} ∩ W), and the exponent κ_{1} − κ_{2} is determined by the weight κ. Since H(ψ)(κ) is the maximal denominator of \frac{(f(λ),f)}{(f(λ),f(λ))}, what we need to show (to prove (R))

is the W –integrality of \frac{π^{2(κ_{1}−κ_{2})}W_{p}(λ^{−})(f(λ),f)}{Ω^{2(κ_{1}−κ_{2})}} for all f ∈ S_{κ}(𝔑(λ), ε_{λ}; W). This we will show by a detailed analysis of the residue formulas of generalized Eisenstein series, which we call Shimura series, on orthogonal groups of signature (n, 2). The series have been introduced in [Sh1] and [Sh2], and we take those associated with a theta series of M and the determinant (quadratic form) of M_{2}(F). The validity of the q–expansion principle is very important to show the W –integrality, because we write the Petersson inner product as a value of a modular form (with integral q–expansion) at a CM point of (the product of two copies of) the Hilbert modular variety. This modular form is obtained as the residue of a Shimura series. However in the split case, the orthogonal similitude group of signature (2, 2) over F is isogenous to the product GL(2) × GL(2)_{/F}; so, basically we are dealing with Hilbert modular forms, and the q–expansion principle is known by a work of Ribet (see [PAF] Theorem 4.21).

Another important point is to write down every W –integral Hilbert cusp form as a W –integral linear combination of theta series of the definite quaternion algebra unramified at every finite (henselian) place. Such a problem over ℚ was first studied by Eichler (his basis problem) and then generalized to the Hilbert modular case by Shimizu and Jacquet-Langlands in different manners. We scrutinize the integrality of the Jacquet-Langlands-Shimizu correspondence (mainly using duality between Hecke algebras and their spaces of cusp forms; see [H05b]). At the last step of finalizing the W –integral correspondence, we again need a result of Fujiwara: Freeness theorem in [Fu] of quaternionic cohomology groups as Hecke modules, which is valid again under the assumptions (1-4) for cusp forms with complex multiplication (see [HMI] Corollary 3.42). The everywhere unramified definite quaternion algebra exists only when the degree [F : ℚ] is even; so, we will at the end reduce, by a base-change argument, the case of odd degree to the case of even degree.

The identity: (h(M)/h(F))L_{p}^{−}(ψ^{−}) = H(ψ) resulted from our proof of the theorem is the one (implicitly) conjectured at the end of [H86] (after Theorem 7.2) in the elliptic modular case. A similar conjecture made there for Eisenstein congruences has now also been proven by [O] under some mild assumptions.

2. Siegel’s theta series for GL(2) × GL(2)

Since the Shimura series has an integral presentation as a Rankin-Selberg convolution of Siegel’s theta series and a Hilbert modular form, we recall here the definition and some properties of the theta series we need later.

2.1. Symmetric Domain of O(n, 2). We describe the symmetric domain associated to an orthogonal group of signature (n, 2), following [Sh1] Section 2. Let V be a n + 2–dimensional space over ℝ. We consider a symmetric bilinear form S : V × V → ℝ of signature (n, 2) with n > 0. We define an orthogonal similitude group G by

(2.1) G(ℝ) = {α ∈ End_{ℝ}(V)|S(αx, αy) = ν(α)S(x, y) with ν(α) ∈ ℝ^{×}} .

We would like to make explicit the symmetric hermitian domain G(ℝ)^{+}/ℝ^{×}C for a maximal compact subgroup C ⊂ G(ℝ)^{+} for the identity connected component G(ℝ)^{+} of G(ℝ). We start with the following complex submanifold of V_{ℂ} = V ⊗ ℂ:

Y(S) = \{v ∈ V_{ℂ} = V ⊗_{ℝ} ℂ|S[v] = S(v, v) = 0, S(v, \overline{v}) < 0\} .

Since S is indefinite over ℂ, the space Y(S) is always non-empty. Obviously g ∈ G(ℝ) with ν(g) > 0 acts on Y(S) by v ↦ gv.

Take v ∈ Y(S), and write W for the subspace spanned over ℝ by v + \overline{v} and iv − i\overline{v} for i = \sqrt{−1}. Then we have

S(v + \overline{v}, v + \overline{v}) = 2S(v, \overline{v}) < 0
S(iv − i\overline{v}, iv − i\overline{v}) = 2S(v, \overline{v}) < 0
S(v + \overline{v}, iv − i\overline{v}) = −i · S(v, \overline{v}) + i · S(\overline{v}, v) = 0.

This shows that S|_{W} is negative definite. Let W^{⊥} = {w ∈ V |S(w, W) = 0}. Then we have an orthogonal decomposition: V = W ⊕W^{⊥} and S|_{W^{⊥}} is positive definite. We then define a positive definite bilinear form

P_{v}(x, y) = −S(x_{W} , y_{W}) + S(x_{W^{⊥}}, y_{W^{⊥}})

for the orthogonal projections x_{W} to W and x_{W^{⊥}} to W^{⊥} of x. The bilinear form P_{v} is called the positive majorant of S indexed by v ∈ Y(S). If g ∈ G(ℝ) fixes v ∈ Y(S), g fixes by definition the positive definite form P_{v}. Thus g has to be in the compact subgroup O(P_{v}) made up of orthogonal matrices preserving P_{v}. Thus G(ℝ)^{+}/O(P_{v}) ↪ Y(S). If we have two v, w ∈ Y(S), then by Sylvester’s theorem, we find g ∈ G(ℝ)^{+} such that gv = w, and hence G(ℝ)^{+}/O(P_{v}) ≅ Y(S).

Writing P_{v}[x] = P_{v}(x, x) for x = cv + \overline{c}\overline{v} + z with c ∈ ℂ and z ∈ W^{⊥}, we see

\begin{matrix}P_{v}[x] − S[x] = & P_{v}(cv + \overline{c}\overline{v} + z, cv + \overline{c}\overline{v} + z) − S(cv + \overline{c}\overline{v} + z, cv + \overline{c}\overline{v} + z) \\ = & − 2c^{2}S[v] − 2\overline{c}^{2}S[\overline{v}] − 4|c|^{2}S(v, \overline{v}) + S[z] − S[z] \\ = & 4|c|^{2}S(v, \overline{v}) = −4S(v, \overline{v})^{−1}|S(x, v)|^{2} ≥ 0. \end{matrix} (2.2)

We now make explicit the domain Y(S) as a hermitian bounded matrix domain.

Proposition 2.1. We have a ℂ–linear isomorphism A : V_{ℂ} ≅ ℂ^{n+2} such that

S(x, y) = ^{t}(Ax) · RAy, S(\overline{x}, y) = ^{t}\overline{(Ax)} · QAy,

where R and Q are real symmetric matrices given by

R = ( \begin{matrix}1_{n} & 0 & 0 \\ 0 & 0 & −1 \\ 0 & −1 & 0 \end{matrix} ) , Q = ( \begin{matrix}1_{n} & 0 \\ 0 & −1_{2} \end{matrix} ) .

Proof. Choose a base v_{1}, …, v_{n+2} of V over ℝ, identify V with ℝ^{n+2} by sending ∑_{i=1}^{n+2} x_{i}v_{i} ↦ ^{t}(x_{1}, …, x_{n+2}) ∈ ℝ^{n+2} and use the same symbol S for the symmetric matrix (S(v_{i}, v_{j}))_{i,j}. Then S(x, y) = ^{t}x · Sy for x, y ∈ V = ℝ^{n+2}. By a theorem of Sylvester, S is equivalent (in GL_{n+2}(ℝ)) to Q; so, we find an invertible matrix X ∈ GL_{n+2}(ℝ) with ^{t}X · SX = Q.

Choose B = diag[1_{n}, \sqrt{2}^{−1} ( \begin{matrix}1 & 1 \\ −i & i \end{matrix} )]. Then by computation ^{t}\overline{B} · QB = Q and tBQB = R. Then x ↦ Ax for A = (XB)^{−1} = B^{−1}X^{−1} does the desired job. □

By our choice of A, the map α ↦ AαA^{−1} gives an isomorphism of Lie groups:

(2.3) ι : G(ℝ) ≅ G(Q, R)

= \{α ∈ GL_{n+2}(ℂ)|^{t}α · Rα = ν(α)R, ^{t}\overline{α} · qα = ν(α)Q with α ∈ ℝ^{×}\} ,

and the map: v ↦ Av gives an isomorphism of complex manifolds:

(2.4)  j : Y(S) ≅ Y(Q, R) = \{u ∈ ℂ^{n+2}|^{t}u · Ru = 0, ^{t}\overline{u} · Qu < 0\} .

These two maps are equivariant:

ι(α)j(v) = j(αv).

We are going to show that Y(Q, R) has two connected components. Write u = ^{t}(u_{1}, …, u_{n+2}) ∈ Y(Q, R). Then we have

(∑_{i=1}^{n} u_{i}^{2}) − 2u_{n+1}u_{n+2} = ^{t}u · Ru = 0,

∑_{i=1}^{n} |u_{i}|^{2} < |u_{n+1}|^{2} + |u_{n+2}|^{2} ⇔ ^{t}\overline{u} · Qu < 0.

Assume |u_{n+1}| = |n_{n+2}| towards contradiction. Then we see

∑_{j=1}^{n} |u_{j}|^{2} ≥ | ∑_{j=1}^{n} u_{j}^{2}| = 2|u_{n+1}u_{n+2}| = |u_{n+1}|^{2} + |u_{n+2}|^{2},

a contradiction; hence we have either |u_{n+1}| > |u_{n+2}| or |u_{n+1}| < |u_{n+2}|. These two cases split the domain Y(Q, R) into two pieces of connected components. To see each component is connected, we may assume that |u_{n+2}| > |u_{n+1}| by interchanging indices if necessary; so, u_{n+2} ≠ 0. Put z_{j} = U_{j}/u_{n+2} for j ≤ n, and define a column vector z = ^{t}(z_{1}, z_{2}, …, z_{n}). Then w = u_{n+1}/u_{n+2} = tz · z/2, and defining

(2.5)  ℨ = ℨ_{n} = \{z ∈ ℂ^{n}|^{t}z · \overline{z} < 1 + \frac{1}{4}|^{t}z · z|^{2} < 2\} ,

ℂ^{×} ×ℨ is isomorphic to the connected component of Y(Q, R) given by |u_{n+2}| > |u_{n+1}| via (λ, z) ↦ λP(z), where

(2.6)  P(z) = ^{t}(z, (^{t}z · z)/2, 1).

From this expression, it is plain that Y(Q, R) has two connected components. We define the action of α ∈ G(ℝ) on ℨ and a factor of automorphy μ(α; z) (z ∈ ℨ) by

(2.7)  ι(α)P(z) = P(α(z))μ(α; z).

We look into spherical functions on V_{ℂ}. Choose a base v_{1}, …, v_{d} of V over ℝ. By means of this base, we identify V with ℝ^{d} (d = n+2); so, v ↦ (x_{1}, …, x_{d}) if v = ∑_{j} x_{j}v_{j}. We take the dual base v_{j}^{∗} so that S(v_{i}^{∗}, v_{j}) = δ_{ij} for the Kronecker symbol δ_{ij} and define a second-degree homogeneous differential operator Δ by

Δ = ∑_{i,j} S(v_{i}^{∗}, v_{j}^{∗})\frac{∂^{2}}{∂x_{i}∂x_{j}} .

A polynomial function η : V → ℂ is called a spherical function if Δη = 0. Writing S = (S(v_{i}, v_{j})), we see that this definition does not depend on the choice of the base v_{j}, because Δ = ^{t}∂S^{−1}∂ for ∂ = ^{t}(\frac{∂}{∂x_{1}} , …, \frac{∂}{∂x_{d}}). Since ∂(^{t}wSx) = Sw for a constant vector w = (w_{1}, …, w_{d}), we find that, for k ≥ 2

Δ(^{t}wSx)^{k} = k^{t}∂(S^{−1}Sw)(^{t}wSx)^{k−1} = k(k − 1)(^{t}wSw)(^{t}wSx)^{k−2}.

Thus the polynomial function x ↦ S(w, x)^{k} for k ≥ 2 is spherical if and only if S[w] := S(w, w) = 0. All homogeneous spherical functions of degree k ≥ 2 are linear combination of S(w, x)^{k} for a finite set of spherical vectors w with S[w] = 0. In particular, for v ∈ Y(S), the function x ↦ S(v, x)^{k} is a spherical function.

Note here that for v ∈ Y(S), S[v] = 0 and S(v, x) = −P_{v}(v, x), because P (v, x) = P (v, x_{W}) + P (v, x_{W^{⊥}}) = −S(v, x_{W}) = −S(v, x). Define ∂_{v} = ṽ · ∂, where ṽ = (λ_{1}, …, λ_{d}) when v = ∑_{j} λ_{j}v_{j}. Then we have, by computation,

(2.8)  ∂_{v}S[x] = 2S(v, x), ∂_{v}P_{v}[x] = 2P_{v}(v, x) = −S(v, x).

We define a Schwartz function Ψ on V for each τ = ξ + iη ∈ ℌ and v ∈ Y(S) by

Ψ(τ; v; w) = e(\frac{1}{2}(S[w]ξ + iP_{v}[w]η)) = exp(πi(S[w]ξ + iP_{v}[w]η)).

We see by computation using (2.8)

(2.9)  (∂_{v}^{k}Ψ)(τ; v; w) = (2πi)^{k}(\overline{τ}S(v, w))^{k}Ψ(τ; v; w).

2.2. SL(2) × SL(2) as an orthogonal group. We realize the product as an orthogonal group of signature (2, 2), and hence this group gives a special case of the orthogonal groups treated in the previous subsection.

Let V = M_{2}(ℝ), and consider the symmetric bilinear form S : V × V → ℝ given by S(x, y) = Tr(xy^{ι}), where yy^{ι} = y^{ι}y = det(y) for 2 × 2 matrices y. We let (a, b) ∈ GL_{2}(ℝ) × GL_{2}(ℝ) act on V by x ↦ axb^{ι}. Then

S(axb^{ι}, ayb^{ι}) = Tr(axb^{ι}by^{ι}a^{ι}) = det(b)Tr(axy^{ι}a^{ι})

= det(b)Tr(xy^{ι}a^{ι}a) = det(a) det(b)S(x, y).

Thus we have an isomorphism

(GL_{2}(ℝ) × GL_{2}(ℝ)) /{±(1, 1)} ↪ G(ℝ)

with ν(a, b) = det(a) det(b). Since the symmetric space of G(ℝ) has dimension 2 over ℂ, the above isomorphism has to be onto on the identity connected component. Since G(ℝ) has four connected components (because Y(S) has two), the above morphism has to be a surjective isomorphism because GL_{2}(ℝ)× GL_{2}(ℝ) has four connected components:

(2.10)  (GL_{2}(ℝ) × GL_{2}(ℝ)) /{±(1, 1)} ≅ G(ℝ).

Since the symmetric domain of GL_{2}(ℝ) × GL_{2}(ℝ) is isomorphic to ℌ × ℌ for the upper half complex plane ℌ = {z ∈ ℂ| Im(z) > 0}, we find that ℨ ≅ ℌ × ℌ.

We are going to make this isomorphism: ℨ ≅ ℌ × ℌ more explicit. We study Y = Y(S) more closely. Since V_{ℂ} = M_{2}(ℂ), writing v = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ M_{2}(ℂ), we have from the definition:

Y = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ M_{2}(ℂ)|ad = bc, a\overline{d} − b\overline{c} + d\overline{a} − c\overline{b} < 0\} .

Pick v = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Y, and suppose that c = 0. Then by the defining equation of Y, ad = 0 ⇒ 0 = a\overline{d} + d\overline{a} < 0, which is a contradiction. Thus c ≠ 0; so, we define for v as above, z = \frac{a}{c} and w = −\frac{d}{c}. Then −zw = \frac{b}{c}, and hence (see [Sh2] II (4.6))

(2.11)  v = cp(z, w) with p(z, w) = ( \begin{matrix}z & −wz \\ 1 & −w \end{matrix} ) = −^{t}(z, 1)(w, 1)ε,

where ε = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ). Again by the equation defining Y,

(2.12) S(p(z, w), \overline{p(z, w)}) = (w − \overline{w})(z − \overline{z}) = −z\overline{w} + zw − \overline{z}w + \overline{z}\overline{w} < 0.

From this, it is clear that Y ≅ ℂ^{×} × (ℌ^{2} ⊔ \overline{ℌ}^{2}). By this isomorphism, for α ∈

G(ℝ), we can define its action α(z, w) ∈ (ℌ^{2} ⊔ \overline{ℌ}^{2}) and a factor μ(α; z, w) ∈ ℂ^{×} of automorphy by

α · p(z, w) = p(α(z, w))μ(α; z, w).

By a direct computation, writing j(v, z) = cz+d for v = ( \begin{matrix}a & b \\ c & d \end{matrix} ) and v(z) = \frac{az+b}{cz+d}, we have, for (α, β) ∈ GL_{2}(ℝ) × GL_{2}(ℝ),

(2.13)  αp(z, w)β^{ι} = p(α(z), β(w))j(α, z)j(β, w).

Thus

(α, β)(z, w) = (α(z), β(w)) and μ((α, β); (z, w)) = j(α, z)j(β, w).

We define a spherical function

(2.14)  v ↦ [v; z, w]^{k} = S(v, p(z, w))^{k}

for a positive integer k > 0. This function is spherical because S[p(z, w)] = 2 det p(z, w) = 0, and we have

(2.15)  [αvβ^{ι}; z, w] = j(α^{ι}, z)j(β^{ι}, w)[v; α^{−1}(z), β^{−1}(w)].

2.3. Growth of theta series. Let F be a totally real field with integer ring O and B be a quaternion algebra over F. The algebra B can be M_{2}(F). Let x ↦ x^{ι} be the main involution of B; so, xx^{ι} = N(x) and x + x^{ι} = Tr(x) for the reduced norm N : B → F and the reduced trace Tr : B → F. We consider the symmetric bilinear form S : B×B → F given by S(x, y) = Tr(xy^{ι}).

Writing I for the set of all archimedean places of F, we split I = I_{B} ⊔ I^{B} so that B ⊗_{F,σ} ℝ ≅ M_{2}(ℝ) ⇔ σ ∈ I_{B}. Thus for σ ∈ I^{B},

B ⊗_{F,σ} ℝ ≅ ℍ = \{( \begin{matrix}a & b \\ −\overline{b} & \overline{a} \end{matrix} ) |a, b ∈ ℂ\} .

We identify B_{σ} = B ⊗_{F,σ} ℝ with M_{2}(ℝ) or ℍ for each σ ∈ I. Thus G(ℚ) = (B^{×} × B^{×})/{±(1, 1)} is the orthogonal group of (B, S). Since S at σ ∈ I^{B} is positive definite, G(ℝ) ≅ (GL_{2}(ℝ)×GL_{2}(ℝ))^{I_{B}} ×(ℍ^{×} ×ℍ^{×})^{I^{B}}/{±(1, 1)}. For each b ∈ B_{∞} = B ⊗_{ℚ} ℝ, writing b = (b_{σ}) for σ–component b_{σ} ∈ B_{σ}, we define

(2.16)  [b; z, w]^{k} = ∏_{σ∈I_{B}}[b_{σ}; z_{σ}, w_{σ}]^{k_{σ}} (k = ∑_{σ∈I_{B}} k_{σ}σ ∈ ℤ[I_{B}]),

where [b_{σ}; z_{σ}, w_{σ}] is as in (2.14) defined for B_{σ} = M_{2}(ℝ). For σ ∈ I^{B}, we pick a homogeneous spherical polynomial φ_{σ} : B_{σ} → ℂ of degree κ_{σ}, and put φ = ∏_{σ∈I^{B}} φ_{σ} and κ = ∑_{σ} κ_{σ} ∈ ℤ[I^{B}]. We define an additive character e_{F} : F_{ℂ} = F ⊗_{ℚ} ℂ → ℂ^{×} by e_{F} (z) = exp(2πi ∑_{σ} z_{σ}) (z = (z_{σ})_{σ∈I}) identifying F_{ℂ} with ℂ^{I} as ℂ–algebras. Writing Tr : F_{ℂ} → ℂ for the trace map, we have e_{F} (z) = e(Tr(z)).

We consider Siegel’s theta series defined for 0 ≤ k ∈ ℤ[I_{B}] and a Schwartz-Bruhat function ϕ : B_{𝔸^{(∞)}} → ℂ:

(2.17) η^{−I}θ_{k}(τ; z, w; v, ϕφ) = ∑_{ℓ∈B}[ℓ; z, w]^{k}(ϕφ)(ℓ)e_{F} (\frac{1}{2}(ξS[ℓ] + iηP_{p(z,w)}[ℓ]))

= ∑_{ℓ∈B}[ℓ; z, w]^{k}(ϕφ)(ℓ)e(\frac{1}{2}Tr(S[ℓ]τ))e (\frac{i}{2} ∑_{σ∈I_{B}} \frac{η_{σ} |[ℓ_{σ}; z_{σ}, w_{σ}]|^{2}}{Im(z_{σ}) Im(w_{σ})} ) ,

where τ = ξ + iη ∈ ℌ^{I}, η^{I}(τ) = ∏_{σ} η_{σ} and the last equality follows from (2.12). Since the majorant P_{p(z,w)} is positive definite, the theta series is rapidly decreasing with respect to τ towards the cusp ∞, as long as φ(0)[0; z, w]^{k} = 0 (in other words, as long as k + κ > 0). Since the infinity type k + κ does not change under the transformation τ ↦ α(τ) for α ∈ SL_{2}(F), the theta series is rapidly decreasing towards any given cusp if k + κ > 0. Otherwise it is slowly increasing (see below Proposition 2.3).

2.4. Partial Fourier transform. We are going to compute in the following subsection the Fourier expansion of the theta series (introduced in the earlier subsections) with respect to (z, w) when B = M_{2}(F). This is non-trivial, because θ is defined by its Fourier expansion with respect to the variable τ. A key idea is to compute the partial Fourier transform of each term of the theta series and to resort to the Poisson summation formula. In this subsection, we describe the computation of the partial Fourier transform.

The Schwartz function on B_{∞} = B ⊗_{ℚ} ℝ = M_{2}(F_{∞}) which gives rise to the theta series θ_{0}(τ; z, w; ϕ) is given by

u ↦ Ψ_{0}(u) = η^{I}e_{F} (det(u)τ + \frac{η}{2yt}|[u; z, w]|^{2})

for τ = ξ + iη, z = x + yi and w = r + ti with ξ, x, r ∈ F_{∞} and η, y, t ∈ F_{∞+}^{×}. Here F_{∞+}^{×} is the identity connected component of F_{∞}^{×}. We define

(2.18) Ψ_{k}(u) = ∏_{σ} Ψ_{k_{σ},σ}(u_{σ}) (0 ≤ k = ∑_{σ} k_{σ}σ ∈ ℤ[I]) and

Ψ_{k_{σ},σ}(u_{σ}) = η_{σ}^{k_{σ}+1}[u_{σ}; z_{σ}, w_{σ}]^{k_{σ}}e(det[u_{σ}]τ_{σ} + i\frac{η_{σ}}{2y_{σ}t_{σ}} |[u_{σ}; z_{σ}, w_{σ}]|^{2}).

We write the variable u = ( \begin{matrix}u_{1} \\ u_{2} \end{matrix} ) for two row vectors u_{j} and write individually u_{1} = (a, b) and u_{2} = (c, d). The partial Fourier transform ϕ^{∗} of ϕ is given by

(2.19)  ϕ^{∗} ( \begin{matrix}a & b \\ c & d \end{matrix} ) = ∫_{F_{∞}^{2}} ϕ ( \begin{matrix}a′ & b′ \\ c & d \end{matrix} ) e_{F} (ab′ − ba′)da′db′,

where ab′ − ba′ = \frac{1}{2}S [( \begin{matrix}a & b \\ a′ & b′ \end{matrix} )] and da′ = ⊗_{σ}da′_{σ} for the Lebesgue measure da′_{σ}

on the σ–component ℝ of F_{∞}. By applying complex conjugation, we have

(2.20)  \overline{ϕ^{∗}} ( \begin{matrix}a & b \\ c & d \end{matrix} ) = (\overline{ϕ})^{∗} ( \begin{matrix}−a & −b \\ c & d \end{matrix} ) .

We compute first the partial Fourier transform to the action of U(F_{∞}) × GL_{2}(F_{∞}), where U(X) is made up of upper unipotent matrices with right shoulder entry in X. We first deal with (1, β) with β ∈ GL_{2}(F_{∞}):

(ϕ ∘ (1, β))^{∗} ( \begin{matrix}a & b \\ c & d \end{matrix} ) = ∫_{F_{∞}^{2}} ϕ (( \begin{matrix}a′ & b′ \\ c & d \end{matrix} ) β^{ι}) e_{F} (−(a′, b′)ε^{t}(a, b))da′db′
=^{(a′,b′)β^{ι}↦(a′,b′)}|N(det(β)|^{−1} ∫_{F_{∞}^{2}} ϕ ( \begin{matrix}a′ & b′ \\ c & d \end{matrix} ) e_{F} (−(a′, b′)β^{−ι}ε^{t}(a, b))da′db′
=|N(det(β)|^{−1} ∫_{F_{∞}^{2}} ϕ ( \begin{matrix}a′ & b′ \\ c & d \end{matrix} ) e_{F} (−(a′, b′)β^{−ι}ε^{t}β^{−1}^{t}(a, b))da′db′
=|N(det(β)|^{−1}ϕ^{∗} ∘ (( \begin{matrix}1 & 0 \\ 0 & det(β) \end{matrix} ) , β^{−1}) ( \begin{matrix}a & b \\ c & d \end{matrix} ) .

We now compute (ϕ ∘ (α, 1))^{∗} for α ∈ U(F_{∞}):

\begin{matrix}(ϕ ∘ (( \begin{matrix}1 & x \\ 0 & 1 \end{matrix} ) , 1))^{∗} ( \begin{matrix}a & b \\ c & d \end{matrix} ) = & ∫_{F_{∞}^{2}} ϕ ( \begin{matrix}a′+xc & b′+xd \\ c & d \end{matrix} ) e_{F} (ab′ − ba′)da′db′ \\ =^{(a′+xc,b′+xd)↦(a′,b′)} & ∫_{F_{∞}^{2}} ϕ ( \begin{matrix}a′ & b′ \\ c & d \end{matrix} ) e_{F} (ab′ − ba′)da′db′e_{F} (−x(ad − bc)) \\ = & e_{F} (−x(ad − bc))ϕ^{∗} ( \begin{matrix}a & b \\ c & d \end{matrix} ) . \end{matrix}

Summarizing the above computation, we get for (( \begin{matrix}1 & x \\ 0 & 1 \end{matrix} ) , 1) ∈ U(F_{∞})×SL_{2}(F_{∞})

(2.21)  (ϕ ∘ (( \begin{matrix}1 & x \\ 0 & 1 \end{matrix} ) , β))^{∗}(u) = e_{F} (−x det(u))ϕ^{∗} ∘ (1, β^{−1})(u).

By (2.15), for (α, β) ∈ SL_{2}(F_{∞}) × SL_{2}(F_{∞}), we have

[αuβ^{ι}; z, w] = S(αuβ^{ι}; p(z, w)) = S(u; α^{−1}p(z, w)β^{−ι})

= [u, α^{−1}(z), β^{−1}(w)]j(α^{−1}, z)j(β^{−1}, z).

To compute the partial Fourier transform of Ψ_{k}, we may therefore assume that r = x = 0. Then the computation for Ψ_{0}^{∗} is reduced to, writing u′ = ( \begin{matrix}a′ & b′ \\ c & d \end{matrix} ) (and omitting the subscript σ),

(2.22) ∫_{F_{σ}^{2}} Ψ_{0,σ}(u′)e(ab′ − ba′)da′db′ =

∫_{ℝ^{2}} ηe (ξ det u′ + \frac{iη}{2} (\frac{ta′^{2}}{y} + \frac{b′^{2}}{yt} + \frac{yd^{2}}{t} + ytc^{2})) e(ab′ − ba′)da′db′.

We then invoke the following formula:

∫_{−∞}^{∞} exp(−πza′^{2})e(a′b)da′ = z^{−1/2} exp(−\frac{πb^{2}}{z}),

where z ∈ ℂ − ℝ_{−} (ℝ_{−}: the negative real line) and z^{−1/2} is the branch of the square root which is positive real if z is positive real. Then (2.22) is equal to

(2.23) y_{σ} exp(−πη^{−1} (\frac{y_{σ}}{t_{σ}} (dξ_{σ} − b)^{2} + y_{σ}t_{σ}(cξ_{σ} − a)^{2}) e (\frac{iη_{σ}}{2} (\frac{y_{σ}}{t_{σ}} d^{2} + y_{σ}t_{σ}c^{2}))

= y_{σ} exp (−π \frac{y_{σ}}{η_{σ}} (\frac{1}{t_{σ}} |dτ_{σ} − b|^{2} + t|cτ_{σ} − a|^{2})) .

By computation, we have

(2.24)  t|τc − a|^{2} + t^{−1}|τd − b|^{2} = t^{−1}|[u; τ, it]|^{2} + 2η det(u).

Thus we get

Φ_{0}(u) = Ψ_{0}^{∗}(u) = ∏_{σ} Ψ_{0,σ}^{∗}(u_{σ}),

Φ_{0,σ}(u) = Ψ_{0,σ}^{∗} ( \begin{matrix}a & b \\ c & d \end{matrix} ) = y_{σ} exp (−π \frac{y_{σ}}{η_{σ}} (\frac{1}{t_{σ}} |dτ_{σ} − b|^{2} + t_{σ}|cτ_{σ} − a|^{2}))
= y_{σ} exp (−2πy_{σ} det(u) − π \frac{y_{σ}}{η_{σ}t_{σ}} |[u; τ_{σ}, it_{σ}]|^{2}) .

(2.25)

In order to compute the partial Fourier transform of Ψ_{k}, we consider the following differential operator

(2.26) ∂_{σ} = S (p(τ_{σ}, w_{σ}),^{t} ( \begin{matrix}\frac{∂}{∂a} & \frac{∂}{∂b} \\ \frac{∂}{∂c} & \frac{∂}{∂d} \end{matrix} )^{ι}) = τ_{σ} \frac{∂}{∂a} − w_{σ}τ_{σ} \frac{∂}{∂b} + \frac{∂}{∂c} − w_{σ} \frac{∂}{∂d}.

Since we have, for u = ( \begin{matrix}a′ & b′ \\ c & d \end{matrix} ),

τ_{σ} \frac{∂}{∂a}e(ab′ − ba′) = 2πiτ_{σ}b′e(ab′ − ba′)
−w_{σ}τ_{σ} \frac{∂}{∂b}e(ab′ − ba′) = 2πiw_{σ}τ_{σ}a′e(ab′ − ba′)
\frac{∂}{∂c}Ψ_{0,σ}(u) = (−2πib′τ_{σ} − π \frac{η_{σ}}{y_{σ}t_{σ}} (w_{σ}z_{σ}\overline{[u; z_{σ}, w_{σ}]} + \overline{w}_{σ}\overline{z}_{σ}[u; z_{σ}, w_{σ}]))Ψ_{0,σ}(u)

−w_{σ} \frac{∂}{∂d}Ψ_{0,σ}(u)

= −(2πia′τ_{σ}w_{σ} − π \frac{η_{σ}}{y_{σ}t_{σ}} (w_{σ}z_{σ}\overline{[u; z_{σ}, w_{σ}]} + w_{σ}\overline{z}_{σ}[u; z_{σ}, w_{σ}]))Ψ_{0,σ}(u).

Taking the fact that \overline{w}_{σ} − w_{σ} = 2it_{σ}, z_{σ} = iy_{σ} and

∂_{σ}([u; z_{σ}, w_{σ}]) = ∂_{σ}(S_{σ}(u, p(z_{σ}, w_{σ})) = S_{σ}(p(τ_{σ}, w_{σ}), p(z_{σ}, w_{σ})) = 0

into account, we have

(2.27)  ∂_{σ}(Ψ_{j,σ}(u)e(ab′ − ba′)) = 2πΨ_{j+1,σ}(u)e(ab′ − ba′)

for all integers j ≥ 0.

To complete the computation, we need to compute ∂_{σ}Φ_{j,σ}(u). We have, noting that we are restricting ourselves to w_{σ} = it_{σ}:

τ_{σ} \frac{∂}{∂a}Φ_{0,σ}(u) = π \frac{y_{σ}t_{σ}}{η_{σ}} (τ_{σ}(c\overline{τ}_{σ} − a) + τ_{σ}(cτ_{σ} − a))Φ_{0,σ}(u)
−it_{σ}τ_{σ} \frac{∂}{∂b}Φ_{0,σ}(u) = −πi\frac{y_{σ}}{η_{σ}} (τ_{σ}(d\overline{τ}_{σ} − b) + τ_{σ}(dτ_{σ} − b))Φ_{0,σ}(u)
\frac{∂}{∂c}Φ_{0,σ}(u) = −π \frac{y_{σ}t_{σ}}{η_{σ}} (τ_{σ}(c\overline{τ}_{σ} − a) + \overline{τ}_{σ}(cτ_{σ} − a))Φ_{0,σ}(u)
−it_{σ} \frac{∂}{∂d}Φ_{0,σ}(u) = πi\frac{y_{σ}}{η_{σ}} (τ_{σ}(d\overline{τ}_{σ} − b) + \overline{τ}_{σ}(dτ_{σ} − b))Φ_{0,σ}(u).

From this we get, taking the fact:

it_{σ}(cτ_{σ} − a) + dτ_{σ} − b = [u; τ, it_{σ}]

into account, we have

∂_{σ}Φ_{0,σ}(u) = 2πy_{σ}[u; τ_{σ}, it_{σ}]Φ_{0,σ}(u).

Since ∂_{σ}([u; τ, w]) = 0, we again obtain, when z = iy and w = it,

(2.28)  ∂_{σ}(Φ_{j,σ})(u) = 2πΦ_{j+1}(u),

where Φ_{j,σ}(u) = y_{σ}^{j+1}[u; τ, w]^{j}Φ_{0,σ}(u). By (2.27) and (2.28) combined, we get, at this moment for z = iy and w = it,

(2.29)  (Ψ_{k})^{∗} (u) = Φ_{k}(u),

where Φ_{k}(u) = ∏_{σ} Φ_{k_{σ},σ}(u_{σ}) and Ψ_{k}(u) = ∏_{σ} Ψ_{k_{σ},σ}(u_{σ}).

We are going to compute the partial Fourier transform for general (z, w) and show that (2.29) is valid in general under a suitable description of Φ for general (z, w): To do this, we write

Ψ_{j,σ}^{z_{σ},w_{σ},τ_{σ}}(u) = η_{σ}^{j+1}[u; z_{σ}, w_{σ}]^{j}e (det(u)τ_{σ} + i\frac{η_{σ}}{2y_{σ}t_{σ}} |[u; z_{σ}, w_{σ}]|^{2}) .

Since [u, α(z_{σ}), β(w_{σ})]j(α, z)j(β, w) = [α^{−1}uβ^{−ι}; z_{σ}, w_{σ}] by (2.13) and (2.14) combined, we have

Ψ_{j,σ}^{z_{σ},w_{σ},τ_{σ}} = Ψ_{j,σ}^{iy_{σ},it_{σ},τ_{σ}} ∘ (( \begin{matrix}1 & −x_{σ} \\ 0 & 1 \end{matrix} ) , ( \begin{matrix}1 & −r_{σ} \\ 0 & 1 \end{matrix} )) .

Then by (2.21),

(ϕ ∘ (( \begin{matrix}1 & −x_{σ} \\ 0 & 1 \end{matrix} ) , ( \begin{matrix}1 & −r_{σ} \\ 0 & 1 \end{matrix} )))^{∗} (u) = e(x_{σ} det(u))ϕ^{∗} ∘ (1, ( \begin{matrix}1 & r_{σ} \\ 0 & 1 \end{matrix} ))

and applying this to Ψ_{j,σ}^{z_{σ},w_{σ},τ_{σ}}, we get from (2.29)

(Ψ_{j,σ}^{z_{σ},w_{σ},τ_{σ}})^{∗} (u) = e(x_{σ} det(u))Φ_{j,σ}^{iy_{σ},it_{σ},τ_{σ}} (u ( \begin{matrix}1 & −r_{σ} \\ 0 & 1 \end{matrix} )) ,

where

Φ_{j,σ}^{iy_{σ},it_{σ},τ_{σ}}(u) = [u_{σ}; τ_{σ}, it_{σ}]^{j}y_{σ}^{j+1} exp (−2πy_{σ} det(u) − π \frac{y_{σ}}{η_{σ}t_{σ}} |[u; τ_{σ}, it_{σ}]|^{2}) .

Define

Φ_{k}(u) =Φ_{k}^{z,w,τ} (u) = ∏_{σ} Φ_{k_{σ},σ}^{z_{σ},w_{σ},τ_{σ}}(u) for

Φ_{j,σ}(u) =Φ_{j,σ}^{z_{σ},w_{σ},τ_{σ}}(u) = y_{σ}^{j+1}[u; τ_{σ}, w_{σ}]^{j}e (det(u)z_{σ} + \frac{iy_{σ}}{2η_{σ}t_{σ}} |[u; τ_{σ}, w_{σ}]|^{2}) .

(2.30)

Using this definition, (2.29) is valid for general (z, w, τ) ∈ ℌ^{I} × ℌ^{I} × ℌ^{I}. In other words, we have the reciprocal formula:

(2.31)  Φ_{k}^{z,w,τ} = Ψ_{k}^{τ,w,z} and (Ψ_{k}^{z,w,τ})^{∗} (u) = Ψ_{k}^{τ,w,z}(u).

By (2.20) (and (2.15)), we also have

(2.32) (\overline{Ψ_{k}^{z,w,τ}} )^{∗} (u)

= ∏_{σ} (y_{σ}^{k_{σ}+1}[u_{σ}; −\overline{τ}_{σ}, \overline{w}_{σ}]^{k_{σ}}e (det(u_{σ})\overline{τ}_{σ} + \frac{y_{σ}}{2η_{σ}t_{σ}} |[u_{σ}; −\overline{τ}_{σ}, \overline{w}_{σ}]|^{2})) .

2.5. Fourier expansion of theta series. Write V = M_{2}(F). We choose on F_{𝔸^{(∞)}} = F ⊗_{ℚ} 𝔸^{(∞)} the standard additive Haar measure da so that

∫Ô da = 1 for Ô = O ⊗_{ℤ} ℤ̂ (ℤ̂ = ∏_{p} ℤ_{p}).

At infinity, we choose the Lebesgue measure ⊗_{σ}da_{σ} on F_{∞} = ∏_{σ∈I} ℝ. Then we take the tensor product measure du = da ⊗ db ⊗ dc ⊗ dd for u = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ V_{𝔸}.

Let ϕ : V_{𝔸} = M_{2}(F_{𝔸}) → ℂ be a Schwartz-Bruhat function, and assume that ϕ = ∏_{v} ϕ_{v} for ϕ_{v} : V ⊗ ℚ_{v} → ℂ. We define the partial Fourier transform of ϕ for ϕ : V_{𝔸} → ℂ by the same formula as in (2.19):

(2.33)  ϕ^{∗} ( \begin{matrix}a & b \\ c & d \end{matrix} ) = ∫_{F_{𝔸}^{2}} ϕ ( \begin{matrix}a′ & b′ \\ c & d \end{matrix} ) e_{𝔸}(ab′ − ba′)da′db′,

where e_{𝔸} : F_{𝔸}/F → ℂ^{×} is the additive character with e_{𝔸}(x_{∞}) = e_{F} (x_{∞}) for x_{∞} ∈ F_{∞}. We further assume that ϕ_{∞} = Ψ_{k}^{z,w,τ} studied in the previous subsection. Then we define

(2.34)  Θ(ϕ) = ∑_{ℓ∈V} ϕ(ℓ).

Writing ϕ^{(∞)} for the finite part of ϕ and regarding it as a function on V ⊂ V_{𝔸^{(∞)}}, we find

Θ(ϕ) = η^{k}θ_{k}(τ; z, w; ϕ^{(∞)}).

Since ∫_{F_{𝔸}/F} da = \sqrt{|D|} for the discriminant D of F, the measure |D|^{−1}da′db′ has volume 1 for the quotient F_{𝔸}^{2}/F^{2}. Thus |D|^{−1}ϕ^{∗} gives the partial Fourier transform with respect to volume 1 measure |D|^{−1}da′db′. The Poisson summation formula (with respect to the discrete subgroup F^{2} ⊂ F_{𝔸}^{2}) is valid for the volume 1 measure (cf. [LFE] Section 8.4), we have the following result:

Proposition 2.2. We have Θ(ϕ) = |D|^{−1}Θ(ϕ^{∗}). In terms of θ_{k}, we have

η^{k}θ_{k}(τ; z, w; ϕ^{(∞)}) = |D|^{−1}y^{k}θ_{k}(z; τ, w; ϕ^{∗}^{(∞)}).

We could say that the right-hand-side of this formula gives the Fourier expansion of the theta series in terms of the variable z.

Proposition 2.3. Let

\begin{matrix}Γ^{τ} (ϕ^{∗}) = \{γ ∈ SL_{2}(F)|ϕ^{∗}^{(∞)}(γu) = χ_{τ} (γ)ϕ^{∗}^{(∞)}(u)\} \\ Γ^{z,w}(ϕ) = \{(γ, δ) ∈ SL_{2}(F)^{2}|ϕ^{(∞)}(γuδ^{−1}) = χ_{z,w}(γ, δ)ϕ^{(∞)}(u)\}. \end{matrix}

for characters χ_{τ} : Γ^{τ} (ϕ^{∗}) → ℂ^{×} and χ_{z,w} : Γ^{z,w}(ϕ) → ℂ^{×} Suppose that ϕ_{∞} = Ψ_{k}^{z,w,τ}. Then for (α, β, γ) ∈ Γ^{τ} (ϕ^{∗}) × Γ^{z,w}(ϕ), we have

Θ(ϕ)(α(τ); β(z), γ(w))

= Θ(ϕ)(τ; z, w)χ_{τ} (α)^{−1}χ_{z,w}(β, γ)^{−1}j(α, τ)^{−k}j(β, z)^{−k}j(γ, w)^{−k}.

More generally, for general α ∈ SL_{2}(F), we have

Θ(ϕ)(α(τ); z, w)j(α, τ)^{k} = |D|^{−1}Θ(ϕ^{∗} ∘ α) = Θ(Φ),

where ϕ^{∗} ∘ α(u) = ϕ^{∗}(αu) and Φ ( \begin{matrix}a & b \\ c & d \end{matrix} ) = (ϕ^{∗} ∘ α)^{∗} ( \begin{matrix}−a & −b \\ c & d \end{matrix} ). Similarly, for (β, γ) ∈ SL_{2}(F), we have

Θ(ϕ)(τ; z, w)j(β, z)^{k}j(γ, w)^{k} = Θ(ϕ ∘ (β, γ)),

where ϕ ∘ (β, γ)(u) = ϕ(βuγ^{−1}).

Proof. Since the argument is similar, we prove the formula in details for the action on τ. Write Γ = Γ^{τ} (ϕ^{∗}). We use the expression Θ(ϕ) = |D|^{−1}Θ(ϕ^{∗}). By (2.15), we have

\frac{|[γ^{−1}ℓ; τ_{σ}, w_{σ}]|^{2}}{η(τ_{s})} = \frac{|[ℓ; γ(τ_{σ}), w_{σ}]|^{2}}{η(γ(τ_{s}))} , [γ^{−1}ℓ; τ, w]^{k} = [ℓ; γ(τ), w]^{k}j(γ, τ)^{k}.

Then, up to y^{k+I}e_{F} (det(ℓ)z) (independent of τ), Θ(ϕ^{∗}) is the sum of the following terms over ℓ ∈ Γ\M_{2}(F) and γ ∈ Γ:

χ_{τ} (γ)ϕ^{∗}(ℓ)Y_{ℓ}(γ(τ))j(γ, τ)^{k},

where Y_{ℓ}(τ) = [ℓ; τ, w]^{k} exp(−π ∑_{σ} \frac{y_{σ}}{t_{σ}} \frac{|[ℓ;τ_{σ},w_{σ}]|^{2}}{η_{σ}(τ_{σ})} ). Thus we need to prove the automorphic property with respect to τ for

f(τ) = ∑_{γ∈Γ/Γ_{ℓ}} χ_{τ} (γ)Y_{ℓ}(γ(τ))j(γ, τ)^{k},

where Γ_{ℓ} ⊂ Γ is the stabilizer of ℓ. We see

f(α(τ)) = ∑_{γ∈Γ/Γ_{ℓ}} χ_{τ} (γ)Y_{ℓ}(γα(τ))j(γ, α(τ))^{k}

= ∑_{γ∈Γ/Γ_{ℓ}} χ_{τ} (γ)Y_{ℓ}(γα(τ))j(γα, τ)^{k}j(α, τ)^{−k} =^{γα↦γ} χ_{τ} (α)^{−1}f(τ)j(α, τ)^{−k}.

This shows the first assertion for τ. As for the assertion with respect to (z, w), we argue similarly looking into the terms of Θ(ϕ).

For the action of general α, the argument is similar for Θ(ϕ^{∗}). To return to Θ(ϕ), we need to use the Fourier inversion formula (ϕ^{∗})^{∗} ( \begin{matrix}a & b \\ c & d \end{matrix} ) = ϕ ( \begin{matrix}−a & −b \\ c & d \end{matrix} ). We leave the details to the attentive readers. □

3. q–Expansion of Shimura series

The Shimura series for GL(2) × GL(2) is defined for 0 < k ∈ ℤ[I] and 0 ≤ m ∈ ℤ[I] in [Sh2] II (4.11) by

(3.1) H(z, w; s) = H_{k,m}(z, w; s; ϕ^{(∞)}, f)

= [U] ∑_{0≠α∈M_{2}(F)/U} ϕ^{(∞)}(α)a(− det(α), f)| det(α)|^{m}[α; z, w]^{−k} |[α; z, w]|^{−2sI}

for (z, w) ∈ ℌ^{I} ×ℌ^{I}. When m = 0, we simply write H_{k} for H_{k,0}. The positivity of k means that k ≥ 0 and k_{σ} > 0 for at least one σ ∈ I. Here f is a Hilbert modular form given by the Fourier expansion: ∑_{ξ∈F} a(ξ, f)e_{F} (ξτ) for τ ∈ ℌ^{I} of weight ℓ (e_{F} (ξτ) = exp(2πi ∑_{σ} ξ^{σ}τ_{σ})) with a(ξ, f) = 0 if ξ^{σ} < 0 for some σ ∈ I, U is a subgroup of finite index of the group O_{+}^{×} of all totally positive units for which each term of the above sum is invariant, [U] = [O_{+}^{×} : U]^{−1} and ϕ^{(∞)} : M_{2}(F_{𝔸^{(∞)}}) → ℂ is a locally constant compactly supported function (a Schwartz-Bruhat function). To have invariance of the terms under the unit group U, we need to assume

(3.2) k − ℓ − 2m = [k − ℓ − 2m]I (I = ∑_{σ∈I} σ) for an integer [k − ℓ − 2m].

The series (3.1) converges absolutely and locally uniformly with respect to all variables s, z, w if

(3.3)  Re(s) > n + 2 + 2θ(f) − [k − ℓ − 2m]

as was shown in [Sh2] I Proposition 5.1 and Theorem 5.2, where θ(f) = −1 when f is a constant, and otherwise, θ(f) = θ ≥ −\frac{1}{2} with |a(ξ, f)ξ^{−ℓ/2}| = O(|N(ξ)|^{θ}) for the norm map N = N_{F/ℚ}. This series is a generalization of Eisenstein series, because if we take f = 1 (so ℓ = 0 and m = 0), the series gives an Eisenstein series for GL(2) × GL(2) over F.

We are going to compute the Fourier expansion of the Shimura series. We summarize here how we proceed. We have already computed the Fourier expansion of Θ(ϕ)(τ; z, w) with respect to z, and it is equal to |D|^{−1}Θ(ϕ^{∗})(z; τ, w) for the partial Fourier transform ϕ^{∗} of ϕ. By the integral expression of the series given in [Sh2] I Section 7, the series (actually its complex conjugate) is the Rankin-Selberg convolution product of Θ(ϕ) and f with respect to the variable τ. Since integration with respect to τ preserves Fourier expansion of Θ(ϕ) with respect to z, what we need to compute is

∫_{Γ\ℌ^{I}} Θ(ϕ^{∗})(z; τ, w)f(τ)E(τ; 0)dμ(τ)

for the invariant measure dμ(τ) for a suitable holomorphic Eisenstein series E(τ; 0). This has been actually done, though without referring the result as the Fourier expansion of the series H_{k}(z, w; 0), in [Sh2] II Proposition 5.1 (replacing f(w) and variable w there by E(τ; 0)f(τ) and τ). We recall the integral expression in Subsection 3.1 and the computation of Proposition 5.1 in [Sh2] II in Subsection 3.2. We shall do this to formulate our result in a manner optimal for our later use.

3.1. Integral expression. Let Γ be a congruence subgroup of SL_{2}(F) which leaves θ_{k}(τ; z, w; ϕ^{(∞)}) and f fixed; thus, Γ ⊂ Γ^{τ} (ϕ^{∗}). The stabilizer Γ_{∞} of the infinity cusp has the following canonical exact sequence:

(3.4)  \begin{aligned}0 → & 𝔞 & ⟶ & Γ_{∞} & ⟶ & U & → 1 \\ & a & ↦ & ( \begin{matrix}1 & a \\ 0 & 1 \end{matrix} ) \\ & & & ( \begin{matrix}ϵ & a \\ 0 & ϵ^{−1} \end{matrix} ) & ↦ & ϵ \end{aligned}

for a fractional ideal 𝔞 and a subgroup U ⊂ O^{×} of finite index. By shrinking Γ a little, we may assume that U ⊂ O_{+}^{×}. We recall the integral expression of the Shimura series involving Siegel’s theta series given in [Sh2] I (7.2) and II (6.5b):

(3.5) [U]N(𝔞)^{−1}\sqrt{|D|}^{−1} ∫_{F_{∞+}^{×}/U^{2}} (∫_{F_{∞}/𝔞} Θ(ϕ)d^{m}f(τ)dξ) η^{(s−1)I}d^{×}η,

where d^{m} = ∏_{σ} (\frac{1}{2πi} \frac{∂}{∂τ_{σ}} )^{m_{σ}}, ϕ(u) = ϕ^{(∞)}(u^{(∞)})Ψ_{k}^{z,w,τ} (u_{∞}) and d^{×}η is the

multiplicative Haar measure given by ⊗_{σ}(η_{σ}^{−1}dη_{σ}). We first compute the inner integral: if Re(s) ≫ 0,

\begin{matrix}N(𝔞)^{−1}\sqrt{|D|}^{−1} ∫_{F_{∞}/𝔞} Θ(ϕ)d^{m}f(τ)dξ = \\ ∑_{α∈V,β∈F} ϕ^{(∞)}(α)a(β, f)|β|^{m}[α; z, w]^{k} exp(−π(2β + P_{z,w}(α))η)η^{k+I}δ_{det(α),−β}, \end{matrix}

because for C = N(𝔞)^{−1}\sqrt{|D|}^{−1}

C ∫_{F_{∞}/𝔞} e_{F} ((det(α) + β)ξ)dξ = δ_{det(α),−β} = \begin{cases}1 & if det(α) = −β, \\ 0 & otherwise. \end{cases}

To compute the outer integral, when det(α) = −β, we note from (2.2) that P_{z,w}[α] = S[α] + \frac{|[α;z,w]|^{2}}{yt} for S[α] = 2 det(α) and that

exp(−π(2β+P_{z,w}(α))η) = exp(π(2 det(α)−P_{z,w}(α))η) = exp(−π \frac{|[α; z, w]|^{2}}{yt} η).

Here we have integrated term wise (with respect to) the summation of Θ and the Fourier expansion of f, which can be justified by the locally uniform and absolute convergence of the Fourier expansions of Θ and f as long as the resulting series is absolutely convergent (Lebesgue’s term wise integration theorem). The convergence of the series is guaranteed by (3.3) if Re(s) is large. Again spreading the integral ∫_{F_{∞+}^{×}/U^{2}} ∑_{ϵ∈U} Φ(ε^{2}η)d^{×}η to the whole F_{∞+}^{×} for

Φ(η) = exp(−π \frac{|[α;z,w]|^{2}η}{yt}), we see that (as long as the latter integral is absolutely convergent) the integral (3.5) is equal to

[U] ∑_{α∈V/U} ϕ^{(∞)}(α)a(− det(α), f)| det(α)|^{m}[α; z, w]^{k}

× ∫_{F_{∞+}^{×}} exp(−πTr(\frac{|[α; z, w]|^{2}}{yt} η))η^{k+sI}d^{×}η.

(3.6)

We know

[α; z, w]^{k} ∫_{F_{∞+}^{×}} exp(−π \frac{|[α; z, w]|^{2}}{yt} η)η^{k+sI}d^{×}η

= 2^{1−[F:ℚ]}π^{−k−sI}Γ_{F} (k + sI)y^{k+sI}t^{k+sI}[α; z, w]^{k}|[α; z, w]|^{−2s−2k}

= 2^{1−[F:ℚ]}π^{−k−sI}Γ_{F} (k + sI)y^{k+sI}t^{k+sI}[α; \overline{z}, \overline{w}]^{−k}|[α; z, w]|^{−2s},

where Γ_{F} (k) = ∏_{σ} Γ(k_{σ}), and as for the factor 2^{1−[F:ℚ]}, see [LFE] page 271. Thus we conclude

2^{1−[F:ℚ]}π^{−k−sI}Γ_{F} (k + sI)y^{k+sI}t^{k+sI} \overline{H_{k,m}(z, w; \overline{s}; \overline{ϕ}^{(∞)}, f_{c})}

= [U]N(𝔞)^{−1}\sqrt{|D|}^{−1} ∫_{F_{∞+}^{×}/U^{2}} ∫_{F_{∞}/𝔞} Θ(ϕ)d^{m}f(τ)η^{(s−1)I}dξd^{×}η,

where f_{c}(z) = \overline{f(−\overline{z})}. In other words, by taking complex conjugation, we have, for ϕ̃ = \overline{ϕ}^{(∞)}ϕ_{∞},

(3.7) 2^{1−[F:ℚ]}π^{−k−sI}Γ_{F} (k + sI)y^{k+sI}t^{k+sI}H_{k,m}(z, w; s; ϕ^{(∞)}, f)

= [U]N(𝔞)^{−1}\sqrt{|D|}^{−1} ∫_{F_{∞+}^{×}/U^{2}} ∫_{F_{∞}/𝔞} \overline{Θ(ϕ̃)}\overline{d^{m}f_{c}(τ)}η^{(s−1)I}dξd^{×}η.

The above formula (3.7) is only valid for s satisfying (3.3). However, by Rankin-Selberg convolution, we can analytically continue the function H to a meromorphic function on the whole s–plane (see [Sh2] I Section 7). We recall the process. We first assume that m = 0. Since Γ_{∞}\ℌ^{I} ≅ (F_{∞+}^{×}/U^{2}) × (F_{∞}/𝔞), we can rewrite the above integral as

2^{1−[F:ℚ]}π^{−k−sI}Γ_{F} (k + sI)y^{k+sI}t^{k+sI}H_{k}(z, w; s; ϕ^{(∞)}, f)

= [U]N(𝔞)^{−1}\sqrt{|D|}^{−1} ∫_{Γ_{∞}\ℌ^{I}} \overline{Θ(ϕ̃)}\overline{f_{c}(τ)}η^{sI}dμ(τ)

= N(𝔞)^{−1}\sqrt{|D|}^{−1} ∫_{Γ\ℌ^{I}} ∑_{γ∈Γ_{∞}\Γ} (\overline{Θ(ϕ̃)}\overline{f_{c}(τ)}η^{sI}) ∘ γdμ(τ)

= N(𝔞)^{−1}\sqrt{|D|}^{−1} ∫_{Γ\ℌ^{I}} \overline{Θ(ϕ̃)}\overline{f_{c}(τ)}\overline{E(τ; \overline{s})}dμ(τ), (3.8)

where dμ(τ) is the invariant measure η^{−2I}dξdη on ℌ^{I}, and by Proposition 2.3,

(3.9)  E(τ; s) = E_{k−ℓ}(τ; s) = η^{sI} ∑_{γ∈Γ_{∞}\Γ} j(γ, τ)^{ℓ−k}|j(γ, τ)|^{−2sI}.

In general, if m ≠ 0, we use the formula (see [Sh2] I (1.16a)):

d^{m} = ∑_{0≤j≤m} \binom{m}{j}\frac{Γ_{F} (ℓ + m)}{Γ_{F} (ℓ + j)}(4πη)^{j−m}δ_{τ}^{j} (ℓ)

for δ_{τ}^{σ}(j) = \frac{1}{2πi} (\frac{j}{τ_{σ}−\overline{τ}_{σ}} + \frac{∂}{∂τ_{σ}} ) and

δ_{τ}^{j} (ℓ) = ∏_{σ} (δ_{τ}^{σ}(ℓ_{σ} + 2j_{σ} − 2) ⋯ δ_{τ}^{σ}(ℓ_{σ} + 2)δ_{τ}^{σ}(ℓ_{σ})) .

The binomial coefficients \binom{m}{j} is the product of individual ones \binom{m_{σ}}{j_{σ}} over σ ∈ I. Since δ_{τ}^{j} (ℓ) preserves automorphy (but not holomorphy), we can write d^{m}f_{c} as a linear combination of δ_{ℓ}^{u}f_{c}, which is an automorphic form of weight ℓ+2u on the same Γ, and therefore the above computation still works.

The integral (3.8) (in general for m ≥ 0) is convergent for all s ∈ ℂ except for s giving rise to a singularity of the Eisenstein series, because Θ(ϕ) for k > 0 does not have constant term at any cusp; so, it is rapidly decreasing. Thus the integral of (3.8) converges absolutely for any slowly increasing automorphic form f(τ) as long as E(τ; s) is finite. This is the proof of the analytic continuation given in [Sh1] Section 13. This proof works well even when k = 0 for cusp forms f.

3.2. Computation of q–expansion. We assume that m = 0. We are going to compute the Fourier expansion of ∫_{Γ\ℌ^{I}} Θ(ϕ)g(τ)dμ(τ) for an eigenform g(τ) of Laplacian Δ_{σ}: Δ_{σ}g = (s_{σ}^{2} − \frac{1}{4})g (s_{σ} ∈ ℂ) for all σ ∈ I, where Δ_{σ} =

η_{σ}^{2} (\frac{∂^{2}}{∂ξ_{σ}^{2}} + \frac{∂^{2}}{∂η_{σ}^{2}} ). We assume that (Θ(ϕ)g)(γ(τ)) = (Θ(ϕ)g)(τ) for all γ ∈ Γ.

By (2.2), Θ(ϕ) = |D|^{−1}Θ((ϕ)^{∗}) is the sum of the following terms:

∏_{σ} y_{σ}^{k_{σ}+1}[α^{σ}; τ_{σ}, w_{σ}]^{k_{σ}}e (det(α^{σ})z_{σ} + \frac{iy_{σ}}{2η_{σ}t_{σ}} |[α^{σ}; τ_{σ}, w_{σ}]|^{2}) .

By (2.15), we have, for γ ∈ Γ,

log(Y_{σ}(τ_{σ})) = \frac{−πy_{σ}}{η_{σ}t_{σ}} |[(γα)^{σ}; τ_{σ}, w_{σ}]|^{2}

= \frac{−πy_{σ}}{η_{σ}t_{σ}} |[α^{σ}; γ^{−1}(τ_{σ}), w_{σ}]|^{2}|j(γ^{−1}, z)|^{2}

= \frac{−πy_{σ}}{Im(γ^{−1}(τ_{σ}))t_{σ}} |[α^{σ}; γ^{−1}(τ_{σ}), w_{σ}]|^{2}.

This shows

y^{−(k+I)} ∫_{Γ\ℌ^{I}} Θ(ϕ)g(τ)dμ(τ)

= ∑_{α∈Γ\M_{2}(F)} e_{F} (det(α)z)ϕ^{(∞)}(α) ∫_{Γ\ℌ^{I}} ∑_{γ∈Γ_{α}\Γ}[α; γ(τ), w]^{k}g(γ(τ))Y (γ(τ))dμ

= ∑_{α∈Γ\M_{2}(F)} e_{F} (det(α)z)ϕ^{(∞)}(α) ∫_{Γ_{α}\ℌ^{I}} [α; τ, w]^{k}Y (τ)g(τ)dμ,

(3.10)

where Γ_{α} = {γ ∈ Γ|γα = α} and Y (z) = ∏_{σ} e(Y (z_{σ})). If det(α) ≠ 0, then Γ_{α} = {1}.

We first compute the general term: ∫_{ℌ^{I}} [α; τ, w]^{k}Y (τ)g(τ)dμ(τ). For that, we recall [Sh2] Lemma 5.2 and the discussion after the lemma:

Lemma 3.1. Let α ∈ GL_{2}(F). Let P (τ, w) = exp(− ∑_{σ} \frac{u_{σ}}{η_{σ}t_{σ}} |[α, \overline{τ}_{σ}, w_{σ}]|^{2}) for τ, w ∈ ℌ^{I} with 0 < u_{σ} ∈ ℝ. Assume that the integral ∫_{ℌ^{I}} P (τ, w)g(τ)dμ(τ) is convergent. If Δ_{σ}g = (s_{σ}^{2} − \frac{1}{4})g and det(α) is totally positive, we have

(3.11) ∫_{ℌ^{I}} P (τ, w)g(τ)dμ(τ)

= π^{[F:ℚ]/2}(det(α)u^{−1})^{I/2} exp(−2 ∑_{σ} det(α^{σ})u_{σ})K(det(α)u, s)g|_{k}α(w)

for the modified Bessel function:

K(u, s) = ∏_{σ} ∫_{0}^{∞} exp(−u_{σ}(x_{σ} + x_{σ}^{−1}))x_{σ}^{s_{σ}−1}dx_{σ},

where g|_{k}α(w) = det(α)^{k−I}g(α(w))j(α, z)^{−k}. If det(α) is not totally positive and g is holomorphic, the integral (3.11) vanishes, as long as it converges.

By the above lemma, taking g = f (so, g = f is holomorphic), only non-trivial case is when det(α) is totally negative, and noting the fact that K(u, \frac{1}{2}) = π^{[F:ℚ]/2}u^{−I/2} exp(−2 ∑_{σ} u_{σ}), we have

∫_{ℌ^{I}} exp(−π ∑_{σ} \frac{y_{σ}}{η_{σ}(τ_{σ})t_{σ}} |[α^{σ}; τ_{σ}, w_{σ}]|^{2})[α; τ, w]^{k}f(τ)dμ(τ)

=^{τ↦α(\overline{τ})} ∫_{ℌ^{I}} exp(−π ∑_{σ} \frac{y_{σ}}{|η_{σ}(α(τ_{σ}))|t_{σ}} |[α^{σ}; α(\overline{τ}_{σ}), w_{σ}]|^{2})[α; α(\overline{τ}), w]^{k}f(τ)dμ(τ)

=^{(2.15)} ∫_{ℌ^{I}} exp(−π ∑_{σ} \frac{| det(α^{σ})|y_{σ}}{η_{σ}(τ_{σ})t_{σ}} |[1; \overline{τ}_{σ}, w_{σ}]|^{2})

× [1; \overline{τ}, w]^{k} det(α)^{k}j(α, \overline{τ})^{−k}f(α(\overline{τ}))dμ(τ)

=^{s_{σ}=1/2} (−1)^{[F:ℚ]}(−2i)^{k}t^{k}y^{−I} exp(−4π ∑_{σ} | det(α^{σ})|y_{σ})f|_{k}α(\overline{w}).

If α ≠ 0 and det(α) = 0, then Γ_{α} is equal to Γ ∩ βU(F)β^{−1} for β ∈ GL_{2}(F). By a variable change, we may assume that α = ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ). Then Γ_{α} = Γ ∩ U(F), and we have an isomorphism: 𝔞 ≅ Γ_{α} by 𝔞 ∋ a ↦ ( \begin{matrix}1 & a \\ 0 & 1 \end{matrix} ), where 𝔞 is a fractional ideal of F. In this case, [α; τ, w] = −w. We then have

(3.12) ∫_{Γ_{α}\ℌ^{I}} [α; τ, w]^{k}Y (τ)f(τ)dμ(τ)

= ∫_{F_{∞+}^{×}}[α; τ, w]^{k}Y (τ) ∫_{F_{∞}/𝔞} f(ξ + iη)dξη^{−2I}dη

= N(𝔞)\sqrt{|D|}a(0, f) ∫_{F_{∞+}^{×}}(−w)^{k} exp(−π ∑_{σ} \frac{y_{σ}}{η_{σ}t_{σ}} |w|^{2})η^{−2I}dη

=^{η↦η^{−1}} N(𝔞)\sqrt{|D|}a(0, f)(−w)^{k} ∫_{F_{∞+}^{×}} exp(−π ∑_{σ} \frac{η_{σ}y_{σ}}{t_{σ}} |w|^{2})dη

= π^{−1}N(𝔞)\sqrt{|D|}a(0, f)(−w)^{k} \frac{t_{σ}}{y_{σ}} |w|^{−2I},

where f(τ) = ∑_{δ∈F} a(δ, f)e_{F} (δτ).

Thus we obtain the following version of [Sh2] II Proposition 5.1 for B = M_{2}(F):

Theorem 3.2. Suppose that f is a holomorphic cusp form of weight k > 0. Let Γ be a congruence subgroup of SL_{2}(F) fixing f(τ)Θ(ϕ)(τ). Then we have

(−1)^{[F:ℚ]}|D| ∫_{Γ\ℌ^{I}} Θ(ϕ)(τ; z, w)f(τ)dμ(τ)

= (−2i)^{k}t^{k}y^{k} ∑_{α∈Γ\M_{2}(F);det(α)≪0} ϕ^{∗}^{(∞)}(α)e_{F} (det(α)\overline{z})f|_{k}α(\overline{w}),

where f|α(\overline{w}) = det(α)^{k−I}f(α(\overline{w}))j(α, \overline{w})^{−k} for α ∈ M_{2}(F) with totally negative determinant.

Taking complex conjugate of the above expansion and replacing the pair (ϕ, f) in the above theorem by (ϕ̃ = \overline{ϕ}^{(∞)}ϕ_{∞}, f_{c}E(w; 0)), we get Corollary 3.3. We have, if f is a holomorphic cusp form of weight ℓ with k − ℓ = [k − ℓ]I for an integer [k − ℓ] > 0,

H_{k}(z, w; 0; ϕ^{(∞)}, f) = 2^{[F:ℚ]−1}[U]|D|^{−3/2}N(𝔞)^{−1} \frac{(2πi)^{k}}{Γ_{F} (k)}

× ∑_{α∈Γ\M_{2}(F);det(α)≫0} ϕ^{∗}^{(∞)}(ϵα)e_{F} (det(α)z)(fE_{k−ℓ}(w; 0))|_{k}α(w),

where ϵ = ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ).

We can apply the above theorem to the following integral:

∫_{Γ\ℌ^{I}} Θ(ϕ̃)(s − 1)E_{k−ℓ}(τ; s)\overline{f}_{c}(τ)dμ(τ)

at s = 1 when k = ℓ, because E(τ, s) has a simple pole at s = 1 whose residue is a constant c_{Γ} ≠ 0 (independent of τ). We then have

Corollary 3.4. We have, if f is a holomorphic cusp form of weight k,

Res_{s=1}H_{k}(z, w; s; ϕ^{(∞)}, f) = c_{Γ}(−i)^{[F:ℚ]}2^{−1}[U]|D|^{−3/2}N(𝔞)^{−1} \frac{(2πi)^{k+I}}{Γ_{F} (k + I)}

× y^{−I}t^{−I} ∑_{α∈Γ\M_{2}(F);det(α)≫0} ϕ^{∗}^{(∞)}(ϵα)e_{F} (det(α)z)f|_{k}α(w),

where c_{Γ} = Res_{s=1}E(w; s).

For the exact value of the constant c_{Γ} ≠ 0, see [H99] (RES3) page 173.

4. Evaluation at CM points

We follow [Sh2] I Sections 5 and 8 to write down the evaluation of the Shimura series at some special CM points in terms of Rankin-Selberg L–functions.

4.1. CM points. We fix the “identity” embedding (σ_{0} : F → \overline{ℚ}) ∈ I. Let (z_{0}, w_{0}) be a point in ℌ^{I} such that M = F [z_{0,σ_{0}}] and L = F [w_{0,σ_{0}}] are totally imaginary quadratic extensions of F (so, CM fields). Let Y = M ⊗_{F} L, and we embed Y into M_{2}(F) ⊗_{F} M_{2}(F) = M_{4}(F) by (a, b) ↦ ρ_{M}(a) ⊗ ρ_{L}(b) with

( \begin{matrix}z_{0}a \\ a \end{matrix}) = ρ_{M}(a) ( \begin{matrix}z_{0} \\ 1 \end{matrix} ) and ( \begin{matrix}w_{0}b \\ b \end{matrix} ) = ρ_{L}(b) ( \begin{matrix}w_{0} \\ 1 \end{matrix} ) .

We see easily that ρ_{M}(a)^{ι} = ρ_{M}(a^{c}) and ρ_{L}(b)^{ι} = ρ_{L}(b^{c}) for complex conjugation c. We regard V = M_{2}(F) as a Y –module for the multiplicative semi-group Y via ρ_{M} ⊗ ρ_{L}; in other words, (a, b)v = ρ_{M}(a)vρ_{L}^{ι}(b).

We have four distinct Y –eigenvectors p(z_{0,σ}, w_{0,σ}), p(z_{0,σ}, \overline{w}_{0,σ}), p(\overline{z}_{0,σ}, w_{0,σ}) and p(\overline{z}_{0,σ}, \overline{w}_{0,σ}) in M_{2}(ℂ) = V ⊗_{F,σ} ℂ, whose eigenvalues of (a, b) are (aσ̃b^{σ̃c}), (aσ̃bσ̃), (a^{σ̃c}b^{σ̃c}) and (a^{σ̃c}bσ̃), respectively, for an extension σ̃ of σ to the composite LM. Since V ⊗_{F,σ} ℂ is free of rank 1 over Y_{σ} = Y ⊗_{F,σ} ℂ, V = M_{2}(F) is free of rank 1 over Y (because ℂ is faithfully flat over F). Thus we find v ∈ V such that V = Y v. Then S_{Y} : (y, y′) ↦ S(yv, y′v) gives a non-degenerate symmetric F –bilinear form on Y with S_{Y} (yy′, y′′) = S_{Y} (y′, y^{c}y′′), and we can write S_{Y} (x, y) = Tr_{Y/F} (δxy^{c}) for δ ∈ Y^{×} with δ^{c} = δ.

Suppose now that L = M. Then Y ≅ M ⊕Y_{0} with Y_{0} ≅ M, the first projection to M is given by a ⊗ b ↦ ab^{c} and the second to Y_{0} is given by a ⊗ b ↦ ab. Since c is an automorphism of M, p(z, w) and p(\overline{z}_{0,σ}, \overline{w}_{0,σ}) belongs to Mv ⊗_{F,σ} ℂ ⊂ Y_{σ}v = V_{σ}. The vectors p(z_{0,σ}, w_{0,σ}) and p(\overline{z}_{0,σ}, \overline{w}_{0,σ}) are orthogonal to Y_{0} ⊗_{F,σ} ℂ. In other words,

Y_{0} = {y ∈ Y |σ̃(y) = cσ̃(y) = 0 for all σ ∈ I} .

Thus Σ = ∑_{σ∈I} σ̃ gives rise to a CM type of M (with Σ⊔Σc giving all complex embeddings of M). This shows: writing V ∋ α = av ⊕ bv with a ∈ M and b ∈ Y_{0}

\begin{matrix}[α^{σ}; z_{0,σ}, w_{0,σ}] = S(α^{σ}, p(z_{0,σ}, w_{0,w_{0}})) = S(av_{σ}, p(z_{0,σ}, w_{0,w_{0}})) \\ = S(v_{σ}, a^{c}p(z_{0,σ}, w_{0,w_{0}})) = a^{σ̃c}[v_{σ}; z_{0,σ}, w_{0,σ}], \\ [α; z_{0}, w_{0}]^{−k} |[α; z_{0}, w_{0}]|^{−2sI} = C^{−kΣ}|C^{Σ}|^{−2s}a^{−ckΣ}N(a)^{−s}, \end{matrix} ; (4.1)

where C = [v_{σ}; z_{0,σ}, w_{0,σ}] and N(a) is the absolute norm of a ∈ M. Here we have written kΣ = ∑_{σ∈I} k_{σ}σ̃ and ckΣ = ∑_{σ∈I} k_{σ}σ̃c.

Since p(z_{0,σ}, w_{0,σ}) and p(\overline{z}_{0,σ}, \overline{w}_{0,σ}) span (by the definition of Y(S) in Subsection 2.1) a scalar extension to ℂ of a subspace on which S_{σ} is negative definite, S is totally positive definite on W = Y_{0}v, because every vector in W is orthogonal to p(z_{0,σ}, w_{0,σ}) and p(\overline{z}_{0,σ}, \overline{w}_{0,σ}). We write S_{W} for the restriction of S to W. By this fact, writing δ = −δ_{M} ⊕ δ_{0} for δ_{M} ∈ M and δ_{0} ∈ Y_{0}, then δ_{M} is a totally positive element of F; so, we may assume that δ_{M} = \frac{1}{2} by changing v if necessary. Similarly, we may choose δ_{0} = \frac{1}{2}.

4.2. Special values of Shimura series. As we have explained already, we choose v as in previous subsection so that

(4.2)  S_{Y} ((a, b), (a′, b′)) = \frac{1}{2}Tr_{M/F} (−aa′^{c} + bb′^{c}).

We see, supposing

(4.3)  ϕ^{(∞)}(u) = ϕ_{M} ⊗ ϕ_{0}

for functions ϕ_{M} : Mv → ℂ and ϕ_{0} : Y_{0}v → ℂ,

\begin{aligned}C^{kΣ}|C^{Σ}|^{2s} & H_{k}(z_{0}, w_{0}; s; ϕ^{(∞)}, f) \\ & = [U] ∑_{α∈M/U} ϕ_{M}(αv) ∑_{β∈Y_{0}/U} ϕ_{0}(βv)a(αα^{c} − ββ^{c}, f)α^{−ckΣ}N(α)^{−s}, \end{aligned}

where C is as in (4.1). We now define θ(ϕ_{0}) = ∑_{β∈Y_{0}} ϕ_{0}(βv)e_{F} (ββ^{c}z). Then for f′(z) = θ(ϕ_{0})f(z) = ∑_{ξ∈F} a(ξ, f′)e_{F} (ξz), we have

a(ξ, f′) = ∑_{β∈Y_{0}/U} a(ξ − ββ^{c}, f)ϕ_{0}(βv),

which is a finite sum because {x ∈ Y_{0} ⊗_{F} ℝ|x^{σ}x^{σc} < ξ^{σ} ∀σ} is a compact set. Thus we have, under (4.2) and (4.3)

\begin{aligned}C^{kΣ}|C^{Σ}|^{2s} & H_{k}(z_{0}, w_{0}; s; ϕ^{(∞)}, f) \\ & = [U] ∑_{α∈M/U} ϕ_{M}(αv)a(αα^{c}, f′)α^{−ckΣ}N(α)^{−s}. \end{aligned} ; (4.4)

In general, ϕ^{(∞)}| det^{m} | is a constant linear combination of the functions satisfying (4.3); so, H(z_{0}, w_{0}; s) is a linear combination of the series of the above type. The series (4.4) is the Rankin convolution of f′ and the theta series θ(ϕ_{k,M}) of the norm form of M for ϕ_{k,M}(α) = α^{kΣ}ϕ_{M}(αv) (see (4.9)).

4.3. An explicit formula of Petersson inner product. For a given theta series θ_{M}(ϕ) of weight k + I of a CM field M/F, we are going to write down the inner product ⟨θ_{M}(ϕ), f′_{c}⟩ for a special value of a modular form on GL(2) × GL(2), taking f′ = fθ_{M}(ϕ′) for another theta series θ_{M}(ϕ′) of weight I of M. Here f′_{c}(z) = \overline{f′(−\overline{z})}; so, f′_{c} is a holomorphic modular form whose Fourier coefficients (at the infinity) are the complex conjugate of those of f′. The modular form is given by, up to an explicit constant,

Res_{s=1}H_{k}(z, w; s; ϕ′ ⊗ ϕ^{(∞)}, f).

We will later in Section 7 deduce from this the integrality of \frac{π^{2k+2I}⟨g,θ(ϕ)⟩}{Ω^{2(k+I)}} for the period Ω of the Néron differential of the abelian variety of CM-type sitting at the evaluation point (z_{0}, w_{0}).

Let f and g be Hilbert modular forms on Γ ⊂ SL_{2}(F) with Fourier expansion f = ∑_{ξ∈F} a(ξ, f)e_{F} (ξτ) and g = ∑_{ξ∈F} a(ξ, g)e_{F} (ξτ) for z ∈ ℌ^{I}. We take the ideal 𝔞 ⊂ F and the unit group U ⊂ O_{+}^{×} as in (3.4). Let ℓ and κ be the weights of f and g respectively. We suppose that one of f and g is a cusp form so that \overline{f}g is rapidly decreasing.

We let ϵ ∈ U act on ℌ^{I} by τ ↦ ϵ^{2}τ. Then f(ϵ^{2}τ) = ϵ^{−ℓ}f(τ) and g(ϵ^{2}τ) = ϵ^{−κ}g(τ). Then the function \overline{f}g(τ)η^{(ℓ+κ)/2} is U–invariant. We then consider (4.5)

D(s; f, g) = [U^{2}]N(𝔞)^{−1}\sqrt{|D|}^{−1} ∫_{F_{∞}/𝔞} ∫_{F_{∞+}^{×}/U^{2}} \overline{f}(τ)g(τ)η^{sI+(ℓ+κ)/2}dξd^{×}η.

We now assume that

(4.6)  ℓ ≡ κ mod 2ℤ[I] + ℤI.

Thus we find m ∈ ℤ[I] such that ℓ − κ − 2m ∈ ℤI. Replacing Γ by

\{γ ∈ Γ|(\overline{f}gη^{(ℓ+κ)/2} ∘ γ)(τ) = (\overline{f}gη^{(ℓ+κ)/2})(τ)j(γ, \overline{τ})^{ℓ}j(γ, τ)^{κ}|j(γ, τ)^{−ℓ−κ}|\} if necessary, we have

(4.7) [U^{2}]^{−1}N(𝔞)\sqrt{|D|}D(s; f, g)

= ∫_{Γ\ℌ^{I}} \overline{f}(τ)g(τ)η^{ℓ−m}E_{[ℓ−κ−2m]I,m}(τ; s + 1 − \frac{[ℓ − κ − 2m]}{2})dμ(τ),

where

E_{nI,m}(τ; s) = η^{sI} ∑_{γ∈Γ_{∞}\Γ} (\frac{j(γ, \overline{τ})}{j(γ, τ)})^{m} j(γ, τ)^{−nI}|j(γ, τ)|^{−2sI}.

When m = 0, we write simply E_{nI} for E_{nI,0} (n ∈ ℤ). Since E_{nI,m}(τ; s) has meromorphic continuation on the whole s–plane as a slowly increasing function (outside its singularity), the above integral gives an analytic continuation of D(s; f, g) to the whole complex s–plane. In particular if ℓ = κ, the L–function D(s; f, g) can have a pole at s = 0 because in that case, we can choose m = 0 and E_{0}(τ, s) has a simple pole at s = 1 with a constant residue.

By the same calculation as in Subsection 3.1, we have

\begin{matrix}2^{[F:ℚ]−1}(4π)^{sI+(ℓ+κ)/2}D(s; f, g) \\ = [U^{2}]Γ_{F} (sI + (ℓ + κ)/2) ∑_{0≪ξ∈F^{×}/U^{2}} \overline{a(ξ, f)ξ^{−ℓ/2}}a(ξ, g)ξ^{−κ/2}N(ξ)^{−s} \end{matrix} ; (4.8)

if Re(s) > θ(f) + θ(g) + 1 for θ(f) as in Section 3.

Let us recall the theta series defined below (4.4):

θ(ϕ_{k,M}) = ∑_{a∈M} ϕ_{k,M}(a)e_{F} (aa^{c}τ)

for ϕ_{k,M}(a) = a^{kΣ}ϕ_{M}(a) as in (4.4). We compute D(s; f′_{c}, θ(ϕ_{k,M})) for a cusp form f′ of weight ℓ + I:

2^{[F:ℚ]−1} \frac{(4π)^{sI+(ℓ+k+2I)/2}}{Γ_{F} (sI + (ℓ + k + 2I)/2)}D(s; f′_{c}, θ(ϕ_{k,M}))

= [U^{2}] ∑_{0≪ξ∈F^{×}/U^{2}} a(ξ, f′)ξ^{−(ℓ+I)/2}a(ξ, θ(ϕ_{k,M}))ξ^{−(k+I)/2}N(ξ)^{−s}

= [U^{2}] ∑_{α∈M^{×}/U} ϕ_{M}(α)a(αα^{c}, f′)α^{kΣ}(αα^{c})^{−(k+ℓ+2I)/2}N(α)^{−s}

= [U^{2}] ∑_{α∈M^{×}/U} ϕ_{M}(α)a(αα^{c}, f′)α^{kΣ}(α^{−kΣ}α^{−kcΣ})N(α)^{−s−1+(k−ℓ)/2}

= [U^{2}] ∑_{α∈M^{×}/U} ϕ_{M}(α)a(αα^{c}, f′)α^{−kcΣ}N(α)^{−s−1+(k−ℓ)/2}

Documenta Mathematica · Extra Volume Coates (2006) 465–532

From this, we get under the notation and the assumption of (4.4)

\begin{aligned}2^{1−[F:ℚ]}C^{kΣ}|C^{Σ}|^{2s}(4π)^{−sI−k} & Γ_{F} (sI + k)H_{k}(z_{0}, w_{0}; s; ϕ^{(∞)}, f) \\ & = [U : U^{2}]D(s − 1 + \frac{[k − ℓ]}{2}; f′_{c}, θ(ϕ_{k,M})), \end{aligned} ; (4.9)

where ℓ is the weight of f (so, weight of f′_{c} is ℓ + I). Note here that [U : U^{2}] = 2^{[F:ℚ]−1}.

Since E_{0}(τ; s) has a simple pole at s = 1 with constant residue c_{Γ} ≠ 0, if k = ℓ and ϕ^{(∞)} = ϕ_{M} ⊗ ϕ_{0}, we have from (4.7)

4^{1−[F:ℚ]}C^{kΣ}|C^{Σ}|^{2}(4π)^{−k−I}Γ_{F} (k + I)Res_{s=1}H_{k}(z_{0}, w_{0}; s; ϕ^{(∞)}, f)
= Res_{s=1}D(s − 1; f′_{c}, θ(ϕ_{k,M}))
= [U^{2}]N(𝔞)^{−1}\sqrt{|D|}^{−1}c_{Γ}⟨θ(ϕ_{k,M}), f′_{c}⟩_{Γ},

(4.10)

where

⟨g, f⟩_{Γ} = ∫_{Γ\ℌ^{I}} g(τ)\overline{f(τ)}η^{k}dμ(τ).

Let Ψ_{f}(z, w) be the modular from on GL(2) × GL(2) given by the Fourier expansion:

Ψ_{f}(z, w) = ∑_{α∈Γ\M_{2}(F),det(α)≫0} ϕ^{∗}^{(∞)}(ϵα)e_{F} (det(α)z)f|_{k}α(w)

as in Corollary 3.4. Then taking Γ sufficiently small and combining Corollary 3.4 and (4.10), we get the following explicit formula:

Theorem 4.1. Let f be a Hilbert modular cusp form of weight k. Then we have

⟨θ(ϕ_{k,M}), f′_{c}⟩_{Γ} = 2^{−k−2I}|D|^{−1}C^{kΣ}|C^{Σ}|^{2}i^{k} Im(z_{0})^{−I} Im(w_{0})^{−I}Ψ_{f}(z_{0}, w_{0})

under the notation of (4.4).

This type of results enabled Shimura to get a rationality result of the Petersson inner product of quaternionic cusp forms of CM type with respect to CM periods (for example, see [Sh2] II Section 3).

5. Jacquet-Langlands-Shimizu correspondence

It is a well known result of Jacquet-Langlands and Shimizu that if we choose level appropriately, the space of quaternionic automorphic forms can be embedded into the space of Hilbert modular forms keeping the Hecke operator action. We are going to recall the result, scrutinizing integrality of the correspondence.

5.1. Hilbert modular forms and Hecke algebras. Let us recall the definition of the adelic Hilbert modular forms and their Hecke ring of level 𝔑 for an integral ideal 𝔑 of F (cf. [H96] Sections 2.2-4).

We first recall formal Hecke rings of double cosets. We consider the following open compact subgroup of GL_{2}(F_{𝔸^{(∞)}}):

(5.1)  U_{0}(𝔑) = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ GL_{2}(Ô)|c ≡ 0 mod 𝔑Ô\} ,

where Ô = O⊗_{ℤ} ℤ̂ and ℤ̂ = ∏_{ℓ} ℤ_{ℓ}. Then we introduce the following semi-group

(5.2) Δ_{0}(𝔑) = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ GL_{2}(F_{𝔸^{(∞)}}) ∩ M_{2}(Ô)|c ≡ 0 mod 𝔑Ô, d_{𝔑} ∈ O_{𝔑}^{×}\} ,

where d_{𝔑} is the projection of d ∈ Ô to ∏_{𝔩|𝔑} O_{𝔩} for prime ideals 𝔩. Writing T_{0} for the maximal diagonal torus of GL(2)_{/O} and putting

(5.3)  D_{0} = \{( \begin{matrix}a & 0 \\ 0 & d \end{matrix}) ∈ T_{0}(F_{𝔸^{(∞)}}) ∩ M_{2}(Ô)|d_{𝔑} = 1\} ,

we have (e.g. [MFG] 3.1.6)

(5.4)  Δ_{0}(𝔑) = U_{0}(𝔑)D_{0}U_{0}(𝔑).

Formal finite linear combinations ∑_{δ} c_{δ}U_{0}(𝔑)δU_{0}(𝔑) of double cosets of U_{0}(𝔑) in Δ_{0}(𝔑) form a ring R(U_{0}(𝔑), Δ_{0}(𝔑)) under convolution product (see [IAT] Chapter 3 or [MFG] 3.1.6). The algebra is commutative and is isomorphic to the polynomial ring with variables {T (𝔩), T (𝔩, 𝔩)}_{𝔩}, T (𝔩) for primes 𝔩 corresponding to the double coset U_{0}(𝔑) ( \begin{matrix}ϖ_{𝔩} & 0 \\ 0 & 1 \end{matrix} ) U_{0}(𝔑) and T (𝔩, 𝔩) for primes 𝔩 ∤ 𝔑 corresponding to U_{0}(𝔑)ϖ_{𝔩}U_{0}(𝔑), where ϖ_{𝔩} is a prime element of O_{𝔩}.

The double coset ring R(U_{0}(𝔑), Δ_{0}(𝔑)) naturally acts on the space of adelic modular forms whose definition we now recall. Since T_{0}(O/𝔑′) is canonically a quotient of U_{0}(𝔑′), a character ε : T_{0}(O/𝔑′) → ℂ^{×} can be considered as a character of U_{0}(𝔑′). Writing ε ( \begin{matrix}a & 0 \\ 0 & d \end{matrix}) = ε_{1}(a)ε_{2}(d), if ε̃ = ε_{1}ε_{2}^{−1} factors through O/𝔑 for 𝔑|𝔑′, then we can extend the character ε of U_{0}(𝔑′) to U_{0}(𝔑) by putting ε(u) = ε_{2}(det(u))ε̃(a) for u = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ U_{0}(𝔑). Writing ε^{−} = ε^{−1}, ε(u) has another expression ε(u) = ε_{1}(det(u))ε^{−}(d), because they induce the same character on U_{0}(𝔑′) and on U_{0}(𝔑) ∩ SL_{2}(Ô). Hereafter we use the expression ε(u) = ε_{1}(det(u))ε^{−}(d) (although ε(u) = ε_{2}(det(u))ε̃(a) is used in [Fu] and [HMI]; we note that (κ_{1}, κ_{2}) in this paper corresponds to (κ_{2}, κ_{1}) in [HMI] and [PAF]). We fix an arithmetic character ε_{+} : F_{𝔸}^{×}/F^{×} → ℂ^{×} with ε_{+}|Ô^{×} = ε_{1}ε_{2} and ε_{∞}(x) = x^{−(κ_{1}+κ_{2}−I)}. We use the symbol ε for the triple (ε_{1}, ε_{2}, ε_{+}); thus, we may regard ε as a character of U_{0}(𝔑)F_{𝔸}^{×} by ε(uz) = ε(u)ε_{+}(z) for z ∈ F_{𝔸}^{×} and u ∈ U_{0}(𝔑). If we replace ε_{+} by its p–adic avatar ε̂_{+}, we get a p–adic character ε̂ of U_{0}(𝔑)F_{𝔸}^{×}.

We identify the group of algebraic characters X^{∗}(T_{0}) of T_{0} with ℤ[I]^{2} so that

κ = (κ_{1}, κ_{2}) ∈ ℤ[I]^{2} sends ( \begin{matrix}x & 0 \\ 0 & y \end{matrix} ) to x^{−κ_{1}}y^{−κ_{2}} = ∏_{σ∈I}(σ(x)^{−κ_{1,σ}}σ(y)^{−κ_{2,σ}}). To

each κ ∈ X^{∗}(T_{0}), we associate a factor of automorphy:

(5.5) J_{κ}(g, τ) = det(g)^{κ_{2}−I}j(g, τ)^{κ_{1}−κ_{2}+I} for g ∈ GL_{2}(F_{∞}) and τ ∈ ℌ^{I}.

Then we define S_{κ}(𝔑, ε; ℂ) to be the space of functions f : GL_{2}(F_{𝔸}) → ℂ satisfying the following conditions (e.g. [H96] Section 2.2):

(S1) We have f(αxuz) = ε_{+}(z)ε(u)f(x)J_{κ}(u_{∞}, i)^{−1} for all α ∈ GL_{2}(F), z ∈ F_{𝔸}^{×} and u ∈ U_{0}(𝔑)C_{i} for the stabilizer C_{i} in GL_{2}^{+}(F_{∞}) of i = (\sqrt{−1}, …, \sqrt{−1}) ∈ ℨ = ℌ^{I}, where GL_{2}^{+}(F_{∞}) is the identity connected component of GL_{2}(F_{∞});

(S2) Choosing u ∈ GL_{2}(F_{∞}) with u(i) = τ for each τ ∈ ℌ^{I}, define f_{x}(τ) = f(xu_{∞})J_{κ}(u_{∞}, i) for each x ∈ GL_{2}(F_{𝔸^{(∞)}}). Then f_{x} is a holomorphic function on ℨ for all x;

(S3) f_{x}(τ) is rapidly decreasing towards the cusp ∞. If we replace the word: “rapidly decreasing” in (S3) by “slowly increasing”, we get the definition of the space of modular forms M_{κ}(𝔑, ε; ℂ). It is easy to check (e.g. [MFG] 3.1.5 and [HMI] 2.3.5) that the function f_{x} in (S2) satisfies the classical automorphy condition:

(5.6)  f(γ(τ)) = ε(x^{−1}γx)^{−1}f(τ)J_{κ}(γ, τ) for all γ ∈ Γ_{0,x}(𝔑),

where Γ_{0,x}(𝔑) = xU_{0}(𝔑)x^{−1}GL_{2}^{+}(F_{∞}) ∩ GL_{2}(F), and GL_{2}^{+}(F_{∞}) is the subgroup of GL_{2}(F_{∞}) made up of matrices with totally positive determinant. Also by (S3), f_{x} is rapidly decreasing towards all cusps of Γ_{x} (e.g. [MFG] (3.22)). It is well known that M_{κ} = 0 unless κ_{1} + κ_{2} = [κ_{1} + κ_{2}]I for [κ_{1} + κ_{2}] ∈ ℤ. We write simply [κ] for [κ_{1} + κ_{2}] ∈ ℤ if M_{κ} ≠ 0. In [H88a] Section 2, the space S_{κ} is written as S_{k,ŵ}^{∗} for k = κ_{1} − κ_{2} + I and ŵ = I − κ_{2}, and the action of Hecke operators is the same as specified in [H88a] (2.9e), which we recall now.

In order to define the Hecke operator action on the space of automorphic forms, we fix a prime element ϖ_{𝔩} of the 𝔩–adic completion O_{𝔩} of O for each prime ideal 𝔩 of F. We extend ε^{−} : Ô^{×} → ℂ^{×} to F_{𝔸^{(∞)}}^{×} → ℂ^{×} just by putting ε^{−}(ϖ_{𝔩}^{m}) = 1 for m ∈ ℤ. This is possible because F_{𝔩}^{×} = O_{𝔩}^{×} × ϖ_{𝔩}^{ℤ} for ϖ_{𝔩}^{ℤ} = {ϖ_{𝔩}^{m}|m ∈ ℤ}. Similarly, we extend ε_{2} to F_{𝔸^{(∞)}}^{×}. Then we define ε(u) = ε_{1}(det(u))ε^{−}(d) for u = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Δ_{0}(𝔑). Let U be the unipotent algebraic subgroup of GL(2)_{/F} defined by

U(A) = \{( \begin{matrix}1 & a \\ 0 & 1 \end{matrix} ) |a ∈ A\} .

For each U_{0}(𝔑)yU_{0}(𝔑) ∈ R(U_{0}(𝔑), Δ_{0}(𝔑)), we can decompose

U_{0}(𝔑)yU_{0}(𝔑) = ⨆_{t∈T_{0}(F_{𝔸}^{(∞)}),u∈U(Ô)} utU_{0}(𝔑)

for finitely many u and t (see [IAT] Chapter 3 or [MFG] 3.1.6). We define

(5.7)  f|[U_{0}(𝔑)yU_{0}(𝔑)](x) = ∑_{t,u} ε(t)^{−1}f(xut).

It is easy to check that this operator preserves the space M_{κ}(𝔑, ε; ℂ) and S_{κ}(𝔑, ε; ℂ) by verifying (S1-3) for f|[U_{0}(𝔑)yU_{0}(𝔑)]. This action for y with y_{𝔑} = 1 is independent of the choice of the extension of ε to T_{0}(F_{𝔸}). When y_{𝔑} ≠ 1, we may assume that y_{𝔑} ∈ D_{0} ⊂ T_{0}(F_{𝔸}), and in this case, t can be chosen so that t_{𝔑} = y_{𝔑} (so t_{𝔑} is independent of single right cosets in the double coset). If we extend ε to T (F_{𝔸}^{(∞)}) by choosing another prime element ϖ′_{𝔩} and write the extension as ε′, then we have

ε(t_{𝔑})[U_{0}(𝔑)yU_{0}(𝔑)] = ε′(t_{𝔑})[U_{0}(𝔑)yU_{0}(𝔑)]′,

where the operator on the right-hand-side is defined with respect to ε′. Thus the sole difference is the root of unity ε(t_{𝔑})/ε′(t_{𝔑}) ∈ Im(ε|_{T_{0}(O/𝔑)}). Since it depends on the choice of ϖ_{𝔩}, we make the choice once and for all, and write T (𝔩) for [U_{0}(𝔑) ( \begin{matrix}ϖ_{𝔩} & 0 \\ 0 & 1 \end{matrix} ) U_{0}(𝔑)] (if 𝔩|𝔑). By linearity, these action of double cosets extends to the ring action of the double coset ring R(U_{0}(𝔑), Δ_{0}(𝔑)).

To introduce rationality structure on the space of modular forms, we recall Fourier expansion and q–expansion of modular forms (cf. [H96] Sections 2.3–4 and [HMI] Proposition 2.26, where the order of κ_{j} (j = 1, 2) is reversed; so, (κ_{1}, κ_{2}) here corresponds to (κ_{2}, κ_{1}) in [HMI]). We fix an embedding i_{∞} : \overline{ℚ} ↪ ℂ once and for all and identify \overline{ℚ} with the subfield of all algebraic numbers in ℂ. We also choose a differental idele d ∈ F_{𝔸}^{×} with trivial prime-to–𝔡 part: d^{(𝔡)} = 1. Thus dÔ = 𝔡Ô for the absolute different 𝔡 of F. Each member f of M_{κ}(𝔑, ε; ℂ) has Fourier expansion of the following form: (5.8)

f ( \begin{matrix}y & x \\ 0 & 1 \end{matrix} ) = |y|_{𝔸} \{a_{0}(yd, f)|u|_{𝔸}^{[κ_{2}]} + ∑_{0≪ξ∈F} a(ξyd, f)(ξy_{∞})^{−κ_{2}}e_{F} (iξy_{∞})e_{𝔸}(ξx)\} .

Here y ↦ a(y, f) and a_{0}(y, f) are functions defined on y ∈ F_{𝔸}^{×} only depending on its finite part y^{(∞)}. The function a(y, f) is supported by the set (Ô × F_{∞}) ∩ F_{𝔸}^{×}. When f ∈ S_{κ}(𝔑, ε; ℂ), a_{0}(y, f) = 0; so, we just ignore the constant term a_{0}(y, f). When κ_{2} is not in ℤI, we have S_{κ} = M_{κ}; so, we ignore the constant term if [κ_{2}] ∈ ℤ is not well defined. Let F [κ] be the field fixed by \{σ ∈ Gal(\overline{ℚ}/F)|κσ = κ\}, which is the field of rationality of the character κ ∈ X^{∗}(T_{0}). Write O[κ] for the integer ring of F [κ]. We also define O[κ, ε] for the integer ring of the field F [κ, ε] generated by the values of ε (on finite ideles) over F [κ]. We call an idele y ∈ F_{𝔸}^{×} integral if y^{(∞)} ∈ Ô. Then for any F [κ, ε]–algebra A inside ℂ, we define

M_{κ}(𝔑, ε; A) = {f ∈ M_{κ}(𝔑, ε; ℂ)|a_{0}(y, f), a(y, f) ∈ A as long as y is integral}
S_{κ}(𝔑, ε; A) = M_{κ}(𝔑, ε; A) ∩ S_{κ}(𝔑, ε; ℂ).

(5.9)

Using rationality of (the canonical models of) the Hilbert modular variety (studied by Shimura and others), we can interpret S_{κ}(𝔑, ε; A) (resp. M_{κ}(𝔑, ε; A)) as the space of A–rational global sections of a line bundle of the variety defined over A; so, we have, by the flat base-change theorem (e.g.

[GME] Lemma 1.10.2),

(5.10) M_{κ}(𝔑, ε; A) ⊗_{A} ℂ = M_{κ}(𝔑, ε; ℂ) and S_{κ}(𝔑, ε; A) ⊗_{A} ℂ = S_{κ}(𝔑, ε; ℂ)

Since Hecke operators are induced by algebraic correspondences on the product of two Hilbert modular varieties defined over A (e.g. [GME] 4.2.1 and [PAF] 4.2.5), the action of the Hecke operators T (𝔩) and T (𝔩, 𝔩) preserves the A–rational space of modular forms (see below (5.15) for a more concrete argument showing the Hecke operator stability). We define the Hecke algebra h_{κ}(𝔑, ε; A) ⊂ End_{A}(S_{κ}(𝔑, ε; A)) by the A–subalgebra generated by the Hecke operators T (𝔩) and T (𝔩, 𝔩) for all prime ideals 𝔩 (here we agree to put T (𝔩, 𝔩) = 0 if 𝔩|𝔑). In the same manner, we define H_{κ}(𝔑, ε; A) ⊂ End_{A}(M_{κ}(𝔑, ε; A)).

5.2. q–Expansion of p–integral modular forms. We recall the rational prime p and the embedding i_{p} : \overline{ℚ} ↪ \overline{ℚ}_{p}. Then for any \overline{ℚ}_{p}–algebras A, we define (5.11)

M_{κ}(𝔑, ε; A) = M_{κ}(𝔑, ε; \overline{ℚ}) ⊗_{\overline{ℚ},i_{p}} A and S_{κ}(𝔑, ε; A) = S_{κ}(𝔑, ε; \overline{ℚ}) ⊗_{\overline{ℚ},i_{p}} A.

By linearity, y ↦ a(y, f) and a_{0}(y, f) extend to functions on F_{𝔸}^{×} × M_{κ}(𝔑, ε; A) with values in A. Let N : F_{𝔸}^{×}/F^{×} → \overline{ℚ}_{p}^{×} be the p–cyclotomic character defined by N (y) = y_{p}^{−I}|y^{(∞)}|_{𝔸}^{−1}. Then we define the q–expansion coefficients (at p) of f ∈ M_{κ}(𝔑, ε; A) by

(5.12)  a_{p}(y, f) = y_{p}^{−κ_{2}}a(y, f) and a_{0,p}(y, f) = N (yd^{−1})^{[κ_{2}]}a_{0}(y, f).

Here we note that a_{0}(y, f) = 0 unless [κ_{2}] ∈ ℤ is well defined. We now define for any p–adically complete O[κ, ε]–algebra A in ℚ̂_{p} (the p–adic completion of \overline{ℚ}_{p})

M_{κ}(𝔑, ε; A) = {f ∈ M_{κ}(𝔑, ε; ℚ̂_{p})|a_{0,p}(y, f), a_{p}(y, f) ∈ A for integral y}
S_{κ}(𝔑, ε; A) = M_{κ}(𝔑, ε; A) ∩ S_{κ}(𝔑, ε; ℚ̂_{p}).

(5.13)

These spaces have geometric meaning as the space of A–integral global sections of a line bundle of the Hilbert modular variety of level 𝔑 (e.g. [HT1] 1.3 and [HMI] 4.3.7).

The formal q–expansion of f has values in the space of functions on F_{𝔸^{(∞)}}^{×} with values in the formal monoid algebra A[[q^{ξ}]]_{ξ∈F_{+}} of the multiplicative semi-group F_{+} made up of totally positive elements, which is given by

(5.14)  f(y) = N (y)^{−1} {a_{0,p}(yd, f) + ∑_{ξ≫0} a_{p}(ξyd, f)q^{ξ}} .

We choose a complete representative set {a_{i}}_{i=1,...,h} in finite ideles for the strict

idele class group F^{×}\F_{𝔸}^{×}/Ô^{×}F_{∞+}^{×}. Let 𝔞_{i} = a_{i}O. Write t_{i} = ( \begin{matrix}a_{i}d^{−1} & 0 \\ 0 & 1 \end{matrix} ) and

consider f_{i} = f_{t_{i}} as defined in (S2). The collection (f_{i})_{i=1,...,h} determines f, because

GL_{2}(F_{𝔸}) = ⨆_{i=1}^{h} GL_{2}(F)t_{i}U_{0}(𝔑)GL_{2}^{+}(F_{∞})

by the approximation theorem. Then as observed in [H88a] Section 4 (and [PAF] 4.2.10), f(a_{i}d^{−1}) gives the q–expansion over A of f_{i} at the Tate abelian variety with a_{i}O–polarization Tate_{𝔞_{i}^{−1},O}(q) as in [HT1] 1.7. Thus by the q– expansion principle ([HT1] 1.7 and [HMI] 4.2.8), the q–expansion: y ↦ f(y) determines f uniquely (for any algebra A for which the space of A–integral modular forms is well defined).

We write T (y) for the Hecke operator acting on M_{κ}(𝔑, ε; A) corresponding to the double coset decomposition of

T (y) = {x ∈ Δ_{0}(𝔑)| det(x)Ô = yÔ}

for y ∈ Ô ∩ F_{𝔸^{(∞)}}^{×}. We renormalize T (y) to create a new operator 𝕋(y) by 𝕋(y) = y_{p}^{−κ_{2}}T (y). Since this only affects T (y) with y_{p} ≠ 1, 𝕋(𝔩) = T (ϖ_{𝔩}) = T (𝔩) if 𝔩 ∤ p. However 𝕋(𝔭) ≠ T (𝔭) for primes 𝔭|p. This renormalization is optimal to have the stability of the A–integral spaces under Hecke operators. We define ⟨𝔩⟩ = N(𝔩)T (𝔩, 𝔩) for 𝔩 ∤ 𝔑. This new action also preserves the integrality as long as [κ] ≥ 0 (cf. [H96] Section 2.2 and [HMI] Theorem 2.28). We have the following formula of the action of T (𝔩) and T (𝔩, 𝔩) (e.g. [H96] Section 2.4):

(5.15)  a_{p}(y, f|𝕋(𝔩)) = \begin{cases}a_{p}(yϖ_{𝔩}, f) + a_{p}(yϖ_{𝔩}^{−1}, f|⟨𝔩⟩) & if 𝔩 ∤ 𝔑 \\ a_{p}(yϖ_{𝔩}, f) & if 𝔩|𝔑. \end{cases}

From this, it is plain that T (𝔩) preserves the space S_{κ}(𝔑, ε; A) if either p|𝔑 or [κ] ≥ 0, because a_{p}(yϖ_{𝔩}^{−1}, f|⟨𝔩⟩) = ϖ_{𝔩,p}^{−2κ_{2}}N(𝔩)ε_{+}(𝔩)a_{p}(y, f). We hereafter assume

(5.16)  Either p|𝔑 or [κ] ≥ 0 and κ_{1} − κ_{2} ≥ I.

We define h_{κ}(𝔑, ε; A) again by the A–subalgebra of End_{A}(S_{κ}(𝔑, ε; A)) generated by 𝕋(𝔩) and ⟨𝔩⟩ over A for all primes 𝔩 (for a p–adically complete O[κ, ε]–algebra A).

We can think of the subgroup U(𝔑) of U_{0}(𝔑) made of matrices u ∈ U_{0}(𝔑) whose reduction modulo 𝔑 are upper unipotent. Then for any subgroup U with U(𝔑) ⊂ U ⊂ U_{0}(𝔑), we can think of the space of cusp forms S_{κ}(U, ε; ℂ) made up of cusp forms satisfying (S1-3) for U in place of U_{0}(𝔑). We have Hecke operators 𝕋(y) corresponding to (U · D_{0}U) ∩ T (y) acting on S_{κ}(U, ε; A). Then in the same manner of S_{κ}(𝔑, ε; A), we define S_{κ}(U, ε; A) and the Hecke algebra h_{κ}(U, ε; A) as the A–subalgebra of End_{A}(S_{κ}(U, ε; A)) generated by 𝕋(y) and ⟨𝔩⟩.

Proposition 5.1. Let A be an O[κ, ε]–algebra for which the space of cusp forms S_{κ}(𝔑, ε; A) is well defined (by (5.9) or (5.11) or (5.13)). Write H = h_{κ}(𝔑, ε; A) and S = S_{κ}(𝔑, ε; A). Let V be an H–module and V′ be an A– module of finite type with an A–bilinear product ⟨ , ⟩ : V × V′ → A. Then we have:

(1) The formal q–expansion for v ∈ V and w ∈ V′:

f(v ⊗ w)(y) = N (y)^{−1} {∑_{ξ≫0}⟨v|𝕋(ξyd), w⟩q^{ξ}}

gives a unique element of S.

(2) The map v ⊗ w ↦ f(v ⊗ w) gives an A–linear map of V ⊗_{A} V′ into S with f((v|𝕋(y)) ⊗ w) = f(v ⊗ w)|𝕋(y). If further V′ is an H–module and ⟨v|h, w⟩ = ⟨v, w|h⟩ for all v ∈ V, w ∈ V′ and h ∈ H, then the map f induces an H–linear map: V ⊗_{H} V′ → S_{κ}(𝔑, ε; A).

(3) Suppose that R is an A–algebra direct summand of H, and put V (R) = RV and S(R) = RS. If V (R) is R–free of finite rank and Hom_{A}(V (R), A) is embedded into V′ by the pairing ⟨ , ⟩, then the map f : V (R) ⊗_{A} V′ → S(R) is surjective.

The formulation of this proposition is suggested by the expression of the theta correspondence given in [Sh2] II, Theorem 3.1.

Proof. We have an isomorphism ι : Hom_{A}(H, A) ≅ S given by a_{p}(y, ι(ϕ)) = ϕ(𝕋(y)) (see [H88a] Theorem 5.11, [H91] Theorem 3.1 and [H96] Section 2.6), which is an H–linear map (that is, ι(ϕ ∘ h) = ι(ϕ)|h). Since V is an H– module, h ↦ ⟨v|h, w⟩ gives an element of Hom_{A}(H, A) and hence an element in S. The element has the expression as in (1) by the above explicit form of ι. The assertion (2) is then clear from (1). As for (3), by the isomorphism Hom_{A}(V (R), A) ↪ V′, each element of Hom(R, A) ≅ S(R) is a finite A–linear combination of h ↦ ⟨v|h, w⟩ for v ∈ V (R) and w ∈ V′; so, the surjectivity follows. □

5.3. Integral correspondence. In order to create a proto-typical example of the module V in Proposition 5.1, we study here cohomology groups on quaternionic Shimura varieties. See [H94] and [H88a] for more details of such cohomology groups.

Let B be a quaternion algebra over F. We write G for the algebraic group defined over ℚ such that G(A) = (B ⊗_{ℚ} A)^{×} for each ℚ–algebra A. Let d(B)^{2} be the discriminant of B. We assume that p ∤ d(B) and that

(5.17)  B ⊗_{F,σ} ℝ ≅ \begin{cases}M_{2}(ℝ) & if σ ∈ I_{B} \\ ℍ & if σ ∈ I − I_{B} = I^{B}, \end{cases}

where ℍ is the Hamilton quaternion algebra over ℝ.

We fix once and for all an extension of σ : F ↪ \overline{ℚ} to σ : \overline{F} ≅ \overline{ℚ} for an algebraic closure \overline{F} /F. We take a quadratic extension K/F inside \overline{F} so that K ⊗_{F,σ} ℝ ≅ ℝ × ℝ as F –algebras for σ ∈ I_{B}, K ⊗_{F} F_{𝔭} ≅ F_{𝔭} × F_{𝔭} for primes 𝔭|p and B ⊗_{F} K ≅ M_{2}(K). We can always choose such a quadratic extension K as long as p ∤ d(B). These condition automatically implies K⊗_{F} ℝ ≅ ℂ for σ ∈ I^{B}.

We identify B ⊗_{F} K with M_{2}(K) by the above isomorphism. We fix maximal orders O_{B} and O_{K} of B and K, respectively, and we suppose that

(5.18)  O_{B} ⊗_{O} O_{K} ⊂ M_{2}(O_{K}).

We fix an isomorphism O_{B,𝔩} ≅ M_{2}(O_{𝔩}) so that for the p–adic place 𝔭|p induced

by i_{p} ∘ σ, this isomorphism coincides with the one: O_{B} ↪ M_{2}(O_{K}) ⟶^{i_{p}∘σ} M_{2}(O_{𝔭}). For an integral ideal 𝔑_{0} of F prime to d(B), putting 𝔑 = 𝔑_{0}d(B), we define

(5.19)  U_{0}^{B}(𝔑) = \{x ∈ G(𝔸)|x_{𝔑_{0}} = ( \begin{matrix}a & b \\ c & d \end{matrix} ) with c ∈ 𝔑_{0}O_{𝔑_{0}}\} ,

where O_{𝔑_{0}} = ∏_{𝔩|𝔑_{0}} O_{𝔩}. Similarly we define Δ_{0}^{B}(𝔑) ⊂ B ⊗_{ℚ} 𝔸^{(∞)} so that it is the product of local components Δ_{𝔩} which coincide with the local components of Δ_{0}(𝔑) as long as 𝔩 ∤ d(B) and Δ_{𝔩} = O_{B,𝔩} if 𝔩|d(B). Again we can think of the double coset ring R(U_{0}^{B}(𝔑), Δ_{0}^{B}(𝔑)). We have T (𝔩) and T (𝔩, 𝔩) in R(U_{0}^{B}(𝔑), Δ_{0}^{B}(𝔑)) for 𝔩 ∤ d(B), because the local component at 𝔩 of Δ_{0}^{B}(𝔑) is identical to that of Δ_{0}(𝔑). For 𝔩|d(B), we take α_{𝔩} ∈ O_{B,𝔩} so that its reduced norm generates 𝔩O_{𝔩}. Then we define T (𝔩) = −U_{0}^{B}(𝔑)α_{𝔩}U_{0}^{B}(𝔑) for 𝔩|d(B), and we have

(5.20)  R(U_{0}(𝔑), Δ_{0}(𝔑)) ≅ R(U_{0}^{B}(𝔑), Δ_{0}^{B}(𝔑)).

The above isomorphism brings T (𝔩) and T (𝔩, 𝔩) to the corresponding elements in the right-hand-side.

For a given ring A, we consider the following module L(κ; A) over the multiplicative semi-group M_{2}(A): Let n = κ_{1}−κ_{2}−I ∈ ℤ[I]. We suppose that n ≥ 0 (implying n_{σ} ≥ 0 for all σ ∈ I), and we consider polynomials with coefficients in A of (X_{σ}, Y_{σ})_{σ∈I} homogeneous of degree n_{σ} for each pairs (X_{σ}, Y_{σ}). The collection of all such polynomials forms an A–free module L(κ; A) of rank ∏_{σ}(n_{σ} + 1).

Suppose that A is a closed O_{K}[κ, ε]-algebra (via i_{p}) of ℚ̂_{p}. Then i_{p}(σ(δ_{p})) (which we write simply σ(δ_{p})) for δ ∈ G(𝔸) can be regarded as an element in M_{2}(A). We let Δ_{0}^{B}(𝔑) act on L(κ; A) as follows:

(5.21)  δΦ (( \begin{matrix}X_{σ} \\ Y_{σ} \end{matrix} )) = ε(δ)N(δ)^{κ_{2}}Φ ((σ(δ^{ι})( \begin{matrix}X_{σ} \\ Y_{σ} \end{matrix} )) .

Here N(δ) is the reduced norm of B. We also let z ∈ F_{𝔸}^{×} act on L(κ; A) through scalar multiplication by ε̂_{+}(z) = ε_{+}(z)z_{p}^{−κ_{1}−κ_{2}+I} (the p–adic avatar of ε_{+}). We write L(κε; A) for the module L(κ; A) with this Δ_{0}^{B}(𝔑)F_{𝔸}^{×}–action. By the condition: κ_{1} + κ_{2} ∈ ℤI, if U ⊂ U_{0}^{B}(𝔑) is sufficiently small open compact subgroup, central elements in Γ_{x} = xUx^{−1} ∩ G(ℚ) acts trivially on L(κε; A).

We let g ∈ G(ℝ) with N(g) ≫ 0 act on ℌ^{I_{B}} (by the linear fractional transformation) component-wise via g_{σ} = σ(g) ∈ GL_{2}(K ⊗_{K,σ} ℝ) = GL_{2}(ℝ). We put C_{σ+} for the stabilizer of \sqrt{−1} in the identity connected component of (B ⊗_{F,σ} ℝ)^{×} and define

C_{∞+} = ∏_{σ∈I_{B}} C_{σ+} × ∏_{σ∈I^{B}}(B ⊗_{F,σ} ℝ)^{×}.

Thus we have ℌ^{I_{B}} ≅ G(ℝ)^{+}/C_{∞+} by g(i) ↔ g (i = (\sqrt{−1}, …, \sqrt{−1}) ∈ ℌ^{I_{B}}) for the identity connected component G(ℝ)^{+} of G(ℝ). For any open compact subgroup U ⊂ U_{0}^{B}(𝔑), we think of the complex manifold associated to the Shimura variety:

Y (U) = G(ℚ)\G(𝔸)/F_{𝔸}^{×}U · C_{∞+}.

We write simply Y_{0}^{B}(𝔑) for Y (U_{0}^{B}(𝔑)).

If U is sufficiently small so that the image \overline{Γ}_{U,x} of Γ_{U,x} = xUx^{−1}G^{+}(ℝ) ∩ G(ℚ) in G(ℝ)/F_{∞}^{×} acts freely on ℌ^{I_{B}} for all x ∈ G(𝔸^{(∞)}), and the action of Γ_{U,x} on L(κε; A) factors through \overline{Γ}_{U,x}. Then we can define an étale space over Y (U):

L(κε; A) = G(ℚ)\ (G(𝔸) × L(κε; A)) /F_{𝔸}^{×}U · C_{∞+},

where γ(x, Φ)uz = (γxuz, u^{ι}ε̂_{+}(z)Φ) for u ∈ U · C_{∞+}, z ∈ F_{𝔸}^{×} and γ ∈ G(ℚ). This étale space gives rise to the sheaf L(κε; A)_{/Y (U)} of locally constant sections. We consider the sheaf cohomology group H^{q}(Y (U), L(κε; A)).

Since Y (U) ≅ ⊔_{x}\overline{Γ}_{x}\ℌ^{I_{B}} for finitely many x with x_{p} = 1, we have a canonical isomorphism (cf. [H94] page 470):

(5.22)  H^{q}(Y (U), L(κε; A)) ≅ ⨁_{x} H^{q}(\overline{Γ}_{U,x}, L(κε; A)),

where the right-hand-side is the direct sum of the group cohomology of the \overline{Γ}_{x}–module L(κε; A). The kernel E = Ker(Γ_{U,x} → \overline{Γ}_{U,x}) is a subgroup of units O^{×}. Since κ_{1}+κ_{2} ∈ ℤI, the action of ϵ ∈ E on L(κε; A) is the multiplication by ε̂_{+}(ϵ)N(ϵ)^{[κ]+1} = 1. Even if \overline{Γ}_{U,x} does not act freely on the module L(κε; A), we still have Y (U) ≅ ⨆_{x} \overline{Γ}_{x}\ℌ^{I_{B}} for finitely many x with x_{p} = 1, we can define the left-hand-side of (5.22) by the right hand side of (5.22).

We choose U sufficiently small as above so that [U_{0}^{B}(𝔑) : U] is prime to p (this is a condition on p). Then we have the trace map Tr (that is, the transfer map in group cohomology) and the restriction map Res:

\begin{aligned}Tr : H^{q}(Y^{B}(U), L(κε; A)) & → H^{q}(Y_{0}^{B}(𝔑), L(κε; A)) \\ Res : H^{q}(Y_{0}^{B}(𝔑), L(κε; A)) & → H^{q}(Y^{B}(U), L(κε; A)). \end{aligned}

Since Tr ∘ Res is the multiplication by [U_{0}(𝔑) : U], we have

(5.23) H^{q}(Y_{0}^{B}(𝔑), L(κε; A)) = H^{q}(Y^{B}(U), L(κε; A))/ Ker(Tr) = Im(Res).

We can always choose a multiple 𝔑′ = 𝔑𝔮 (by a prime 𝔮) of 𝔑 so that \overline{Γ}_{0,x}(𝔑′) acts freely on ℌ^{I_{B}}.

As defined in [H88a] Section 7 and [H94] Section 4, where L(κε; A) is written as L(n, v, ε; A) for v = κ_{2} and n = κ_{1} − κ_{2} − I, we have a natural action of the ring R(U_{0}^{B}(𝔑), Δ_{0}^{B}(𝔑)) on the cohomology group H^{q}(Y_{0}^{B}(𝔑), L(κε; A)). For our later use, we recall the definition of the action when q = 0: In this case, we may regard each cohomology class as a global section f : B_{𝔸}^{×} → L(κε; A) with f(αxu) = u^{ι}f(x) for α ∈ B^{×} and u ∈ U_{0}^{B}(𝔑)B_{∞}^{×}. Decomposing

U_{0}^{B}(𝔑) ( \begin{matrix}y & 0 \\ 0 & 1 \end{matrix} ) U_{0}^{B}(𝔑) = ⨆_{ϖ} ϖU_{0}^{B}(𝔑), we have

(5.24)  f|𝕋(y) = y_{p}^{−κ_{2}} ∑_{ϖ} ϖf(xϖ^{−ι}).

Let W be a valuation ring as in the introduction. We assume that h_{κ}(𝔑, ε; W) is well defined and O_{K}[κ, ε] is embedded into W via i_{p}. Let V be the image of H^{q}(Y_{0}^{B}(𝔑), L(κε; W)) in H^{q}(Y_{0}^{B}(𝔑), L(κε; W ⊗_{ℤ} ℚ)). By the Eichler-Shimura isomorphism (between the space of cusp forms on G(𝔸) and the cohomology group; e.g. [H94] Proposition 3.1 and (10.4)) combined with the Jacquet-Langlands-Shimizu correspondence (e.g. [H88a] Theorem 2.1, Proposition 2.3 and [H81] 2.12), the above cohomology group and its compactly supported version (denoted by H_{c}^{q}(Y_{0}^{B}(𝔑), ·)) are the module over the Hecke algebra H_{κ}(𝔑, ε; W ⊗_{ℤ} ℚ). Since

H_{κ}(𝔑, ε; W ⊗_{ℤ} ℚ) = h_{κ}(𝔑, ε; W ⊗_{ℤ} ℚ) ⊕ E

as an algebra direct sum for the Eisenstein part E, for the idempotent 1_{h} of the cuspidal part h_{κ}(𝔑, ε; W ⊗_{ℤ} ℚ), we can define the cuspidal cohomology groups by

H_{cusp}^{q}(Y_{0}^{B}(𝔑), L(κε; W ⊗_{ℤ} ℚ)) = 1_{h}H^{q}(Y_{0}^{B}(𝔑), L(κε; W ⊗_{ℤ} ℚ)).

The natural map from compactly supported cohomology group into the cohomology group without support condition actually induces an isomorphism

1_{h}H_{c}^{q}(Y_{0}^{B}(𝔑), L(κε; W ⊗_{ℤ} ℚ)) ≅ H_{cusp}^{q}(Y_{0}^{B}(𝔑), L(κε; W ⊗_{ℤ} ℚ)).

We then put

(5.25) H_{cusp}^{q}(Y_{0}^{B}(𝔑), L(κε; W)) = H_{cusp}^{q}(Y_{0}^{B}(𝔑), L(κε; W ⊗_{ℤ} ℚ)) ∩ Im(i)

for the natural morphism

i : H^{q}(Y_{0}^{B}(𝔑), L(κε; W)) → H^{q}(Y_{0}^{B}(𝔑), L(κε; W ⊗_{ℤ} ℚ)).

We consider the duality pairing [ , ] on L(κε; A) (for ℚ–algebra A) introduced in [H94] Section 5:

(5.26) [ ∑_{0≤j≤n} b_{j}X^{n−j}Y^{j}, ∑_{0≤j≤n} a_{j}X^{n−j}Y^{j}] = ∑_{j}(−1)^{j}\binom{n}{j}^{−1}b_{n−j}a_{j} ∈ A,

where n = κ_{1} − κ_{2} − I, \binom{n}{j} = ∏_{σ∈I} \binom{n_{σ}}{j_{σ}} and for example X^{j} = ∏_{σ∈I} X_{σ}^{j_{σ}}. As U_{0}(𝔑)F_{𝔸}^{×}-modules, this pairing satisfies:

(5.27)  [uzΦ, uzΦ′] = ε^{2}(u)ε̂_{+}(z)^{2}N_{B/F} (u_{p})^{κ_{1}+κ_{2}−I}[Φ, Φ′],

where N_{B/F} : B → F is the reduced norm map.

Define κ^{∗} = (−κ_{2}, 1 − κ_{1}) and ε^{∗} = ε^{−1}. Thus [κ^{∗}] ≤ 1 ⇔ [κ] ≥ 0. Then the pairing [ , ] induces U_{0}(𝔑)F_{𝔸}^{×}–equivariant pairing

[ , ] : L(κε; A) × L(κ^{∗}ε^{∗}; A) → A.

We now choose q = |I_{B}| = dim_{ℂ} ℌ^{I_{B}}. Then the cup product pairing induces ([H94] (5.3)) a non-degenerate pairing:

( , ) : H_{cusp}^{q}(Y_{0}^{B}(𝔑), L(κε; W)) × H_{cusp}^{q}(Y_{0}^{B}(𝔑), L(κ^{∗}ε^{∗}; W)) → W ⊗_{ℤ} ℚ.

Thus we obtain from Proposition 5.1 the following result:

Proposition 5.2. Let V = H_{cusp}^{q}(Y_{0}^{B}(𝔑), L(κε; W)). Let V^{∗} be the dual W – lattice of V in H_{cusp}^{q}(Y_{0}^{B}(𝔑), L(κ^{∗}ε^{∗}; W ⊗_{ℤ} ℚ)) under the Poincaré duality:

( , ) : H_{cusp}^{q}(Y_{0}^{B}(𝔑), L(κε; W)) × H_{cusp}^{q}(Y_{0}^{B}(𝔑), L^{∗}(κ^{∗}ε^{∗}; W)) → W ⊗_{ℤ} ℚ.

Then we have a h_{κ}(𝔑, ε; W)–linear map

f : V ⊗_{W} V^{∗} → S_{κ}(𝔑, ε; W)

defined by the q–expansion:

f(v ⊗ w) = N (y)^{−1} ∑_{0≪ξ}(v|𝕋(ξyd), w)q^{ξ},

where we regard V ⊗_{W} V^{∗} as an h_{κ}(𝔑, ε; W)–module through the left factor V.

A similar fact for the matrix coefficients of T (y) in place of (v|T (y), w) has been proven in [Sh2] Theorem 3.1 by analytic means without using the Jacquet-Langlands-Shimizu correspondence.

We have H_{cusp}^{q}(Y_{0}^{B}(𝔑), L(κε; W)) = H^{q}(Y_{0}^{B}(𝔑), L(κε; W)) under the following two conditions:

(V1) The character κε : T_{0}(Ô) → W^{×} does not factor through the reduced norm map N : T_{0}(Ô^{(d(B)})) ↪ G(Ô^{(d(B)}) → (Ô^{(d(B))})^{×}. In particular, if κ_{1} ≠ κ_{2}, this condition is satisfied.

(V2) The quaternion algebra B is a division algebra. In particular, this condition is satisfied if |I_{B}| < [F : ℚ].

6. Ordinary cohomology groups

We are going to prove that the morphism f : V (R) ⊗_{W} V (R) → S(R) in Proposition 5.1 for V in Proposition 5.2 is surjective for the nearly ordinary local ring R (associated to a mod p irreducible Galois representation), when B is unramified at every finite place and q = |I_{B}| ≤ 1. A key to the proof is the R–freeness of V (R) proven by Fujiwara [Fu] (see [HMI] Corollary 3.42). Another important ingredient of the proof is the self duality of V (R) over W.

6.1. Freeness as Hecke modules. We recall here a special case of Fujiwara’s result in [Fu] “Freeness Theorem” of the component V (R) for a local ring R of the Hecke algebra h_{κ}(𝔑, ε; W) (see also [HMI] Corollary 3.42). To state the result, we need to have a good description of the modular nearly ordinary Galois representation; so, we recall the description. We call a local ring R of h_{κ}(𝔑, ε; W) nearly ordinary if the projection of 𝕋(p) to R is a unit. We hereafter always assume (ord) R is nearly ordinary with κ_{1} − κ_{2} ≥ I, that is, κ_{1,σ} − κ_{2,σ} ≥ 1 for all

σ. (unr) F/ℚ is unramified at ℚ. We write 𝔑′ for the product of primes 𝔩 ∤ p for which one of ε_{1} and ε_{2} ramifies; so, 𝔑′ ⊂ 𝔑^{(p)}. For a W –algebra homomorphism λ : h_{κ}(𝔑, ε; W) → W factoring through R (such a λ is called nearly ordinary), we have a Galois representation ρ = ρ_{λ} : Gal(\overline{F} /F) → GL_{2}(W) (e.g. [H96] 2.8 and [MFG] 5.6.1) such that

(G1) ρ is continuous and is absolutely irreducible over W ⊗_{ℤ} ℚ;

(G2) ρ is unramified outside 𝔑′p;

(G3) For primes 𝔩 outside 𝔑′p, we have

det(1_{2} − ρ(F rob_{𝔩})X) = 1 − λ(T (𝔩))X + λ(⟨𝔩⟩)X^{2};

(G4) For the decomposition group D_{𝔭} ⊂ Gal(\overline{F} /F) at each prime 𝔭|p, we have an exact sequence of D_{𝔭}–modules: 0 → ϵ_{𝔭} → ρ|_{D_{𝔭}} → δ_{𝔭} → 0 with one dimensional character δ_{𝔭} satisfying δ_{𝔭}([y; F_{𝔭}]) = λ(𝕋(y)) for the local Artin symbol [y; F_{𝔭}] of y ∈ F_{𝔭}^{×}.

Writing 𝔽 for the residue field of W, the semi-simplification \overline{ρ} = \overline{ρ}_{R} of the reduction of ρ modulo the maximal ideal 𝔪_{W} of W is independent of λ by (G2-3) (cf. [MFG] Corollary 2.8 combined with the Chebotarev density). In particular, if \overline{ρ} is irreducible, the isomorphism class of ρ mod 𝔪_{W} for the maximal ideal 𝔪_{W} is unique, and always we have (ρ mod 𝔪_{W}) ≅ \overline{ρ}.

We shall recall some terminology from (formal) deformation theory of Galois representations. See [MFG] Section 2.3 for basics of formal deformation theory of representations. Let H be a subgroup of Gal(\overline{F} /F). We call a representation ρ : H → GL_{2}(A) for a local proartinian W –algebra A with residue field 𝔽 a deformation over H of \overline{ρ} if ρ ≡ \overline{ρ}|_{H} mod 𝔪_{A}. Let χ = det(ρ_{λ})/N^{[κ]} for the p–adic cyclotomic character N. Then χ is of finite order. For any character φ : D_{𝔩} → A^{×}, let C(φ) denote the conductor of φ; thus, C(φ) = 1 if φ is unramified, and C(φ) = 𝔩^{m} if y ↦ φ([y, F_{𝔩}]) factors through F_{𝔩}^{×}/(1 + 𝔩^{m}O_{𝔩}) but not F_{𝔩}^{×}/(1 + 𝔩^{m−1}O_{𝔩}) for m > 0. We assume the following four conditions on ρ_{λ}:

(H1) χ is of order prime to p.

(H2) For primes 𝔩|𝔑p, write D_{𝔩} for the decomposition group at 𝔩. Then we have ρ|_{D_{𝔭}} ≅ ( \begin{matrix}ϵ_{𝔩} & ∗ \\ 0 & δ_{𝔩} \end{matrix} ) with δ([y, F_{𝔩}]) = λ(𝕋(y)). This condition actually follows for 𝔩|p from near ordinarity of λ as already remarked in (G4).

(H3) If a prime 𝔩|𝔑 but 𝔩 ∤ p, then the restriction of δ_{𝔩} and ϵ_{𝔩} to the inertia subgroup I_{𝔩} of D_{𝔩} is of order prime to p.

(H4) If ϵ_{𝔭} ≡ δ_{𝔭}N mod 𝔪_{W} on I_{𝔭} for a prime 𝔭|p, the following five conditions have to be met: (i) the character ϵ_{𝔭} is of order prime to p, (ii) κ = (I, 0), (iii) ρ_{λ}|_{I_{𝔭}} is associated to a p–divisible group over an unramified extension of O_{𝔭}, (iv) 𝔭 ∤ 𝔑, and (v) ϵ_{𝔭}δ_{𝔭}^{−1}(y)y^{−I} = 1 for all y ∈ O_{𝔭}^{×}.

We write \overline{δ}_{𝔩} = (δ_{𝔩} mod 𝔪_{W}) and \overline{ϵ}_{𝔩} = (ϵ_{𝔩} mod 𝔪_{W}). We assume the following two local conditions on \overline{ρ}.

(H5) For all 𝔭|p, \overline{δ}_{𝔭} ≠ \overline{ϵ}_{𝔭}.

(H6) For 𝔩|𝔑 and 𝔩 ∤ p, the 𝔩–primary part of 𝔑 coincides with C(\overline{ϵ}_{𝔩}\overline{δ}_{𝔩}^{−1}). Thus \overline{ρ} could ramify at a prime 𝔩 ∤ 𝔑, and by (H3), 𝔑′ gives the product of primes (outside p) at which \overline{ρ} ramifies. We assume the following global condition on \overline{ρ}:

(H7) \overline{ρ} is absolutely irreducible over Gal(\overline{F} /F [\sqrt{p^{∗}}]) for p^{∗} = (−1)^{(p−1)/2}p. We choose a quaternion algebra B_{/F} so that d(B) = 1 and ramified at most infinite places (that is I^{B} is as large as possible). This implies:

(6.1)  I_{B} = {σ_{1}} if [F : ℚ] is odd, and I_{B} = ∅ if [F : ℚ] is even.

We now quote the following special case of “Freeness Theorem” in Section 0 in [Fu] (see [HMI] Corollary 3.42 for a proof of this Fujiwara’s result):

Theorem 6.1. Suppose the conditions (6.1), (ord), (unr), (H1-7) and p > 3. Then V (R) for V = H^{q}(Y_{0}^{B}(𝔑), L(κε; W)) (q = |I_{B}|) is free of rank 2^{q} over the local ring R. Even if we ease the condition (H4) to allow the case where the 𝔭–primary part of 𝔑 is equal to 𝔭 for primes 𝔭|p, the same assertion holds as long as [F : ℚ] is even.

This is a special case of Fujiwara’s result. In particular, we do not need to assume unramifiedness of p in F, but we use the assumption (unr) anyway in our later application; so, we have imposed it.

Proof. Here is a brief account of how to deduce the above theorem either from [HMI] Corollary 3.42 or from [Fu], because the set of the assumptions imposed in these works appears different. In [HMI] Corollary 3.42, the theorem is proven under the assumptions:

(A) [F : ℚ] is even;

(B) κ = (I, 0); (C) the assumptions (H1–3) and (H5–7); (D) the milder condition than (H4) as stated in the theorem.

As can be easily seen, the conditions (A–D) implies the assumptions actually stated in Corollary 3.42 of [HMI]: the absolute irreducibility of \overline{ρ} over F [μ_{p}] (written as (ai_{F} [μ_{p}]) in [HMI]) which follows from (H7), the conditions (h1–4) in [HMI] 3.2.1, (ds_{Q}) which is (H5) and (H6), and the conditions (Q1–6) (for Q = ∅) in [HMI] Section 3.2.1. These conditions exhaust all the assumptions of Corollary 3.42 of [HMI] except for the condition (sm1). The condition: p > 3 and the unramifiedness of p in F/ℚ implies [F [μ_{p}] : F] > 2, which is the last assumption (sm1) in Corollary 3.42 of [HMI]. We only use this theorem under the four conditions (A–D); so, logically, for the proof of the main theorem of this paper, it is sufficient to quote [HMI] Corollary 3.42.

For the sake of completeness, we now reduce the theorem in the case not covered under (A–D) to [Fu] (the version of 1999). Recall that 𝔑′ is the product of all primes (outside p) at which \overline{ρ} ramifies. We consider an open compact subgroup U(\overline{ρ}) = ∏_{𝔩} U_{𝔩}(\overline{ρ}) ⊂ U_{0}(𝔑) and a character ν_{𝔩} of U_{𝔩}(\overline{ρ}) with values in W^{×} defined as follows:

(1) U_{𝔩}(\overline{ρ}) = GL_{2}(O_{𝔩}) in B_{𝔩}^{×} if 𝔩 ∤ 𝔑p, and ν_{𝔩} is the trivial character;

(2) Suppose that 𝔩|𝔑′. If \overline{ϵ}_{𝔩} ≠ \overline{δ}_{𝔩} on I_{𝔩}, then 𝔩|𝔑,

U_{𝔩}(\overline{ρ}) = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ U_{0}(𝔑)_{𝔩}|a ≡ 1 mod 𝔩, (d mod 𝔩) is of p–power order\}

and ν_{ℓ}(u) = δ_{𝔩}([det(u), F_{𝔩}]). If \overline{ϵ}_{𝔩} = \overline{δ}_{𝔩} on I_{𝔩}, U_{𝔩}(\overline{ρ}) = GL_{2}(O_{𝔩}) (so 𝔩 ∤ 𝔑) and ν_{𝔩}(u) = δ_{𝔩}([det(u), F_{𝔩}]).

(3) For 𝔭|p, define ν_{𝔭}(u) = ϵ_{𝔭}([det(u), F_{𝔭}])(det(u))^{−κ_{2}} for u ∈ GL_{2}(O_{𝔭}), which is a finite order character and can be regarded as a character with values in W^{×}. If \overline{ϵ}_{𝔭} ≠ \overline{δ}_{𝔭}\overline{ω} on I_{𝔭} for \overline{ω} = (N mod 𝔪_{W}), then 𝔭|𝔑 and

U_{𝔭}(\overline{ρ}) = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ U_{0}(p)_{𝔭}|a ≡ 1 mod 𝔭, (d mod 𝔭) is of p–power order\} .

If \overline{ϵ}_{𝔭} = \overline{δ}_{𝔭}\overline{ω}_{𝔭} on I_{𝔭}, then U_{𝔭} = GL_{2}(O_{𝔭}) (and 𝔭 ∤ 𝔑).

Let U = Ker(∏_{𝔩} ν_{𝔩} : U(\overline{ρ}) → W^{×}). Since the restriction of operators of h_{κ}(U, ε; W) to S_{κ}(𝔑, ε; W) induces a surjective algebra homomorphism π : h_{κ}(U, ε; W) ↠ h_{κ}(𝔑, ε; W), we have a unique local ring R_{U} of h_{κ}(U, ε; W) through which λ ∘ π factors. Then R_{U} is nearly ordinary. For a smaller open compact subgroup U′ with U(𝔑𝔮) ⊂ U′ ⊂ U for a suitable prime 𝔮 outside 𝔑′p, it is proven in [Fu] that

(1) For all x ∈ G(𝔸^{(∞)}), \overline{Γ}_{U′,x} is torsion-free and acts on ℌ^{I_{B}} freely;

(2) The action of Γ_{U′,x} on L(κε; A) factors through \overline{Γ}_{U′,x} for all x ∈ G(𝔸^{(p∞)});

(3) The Hecke algebra h_{κ}(U′, ε; W) has a local ring R′ with R′ ≅ R as W –algebras;

(4) Let V_{U} = H^{q}(Y^{B}(U), L) and V′ = H^{q}(Y^{B}(U′), L) for L = L(κε; W)) and q = |I_{B}|. Then the restriction map Res : H^{q}(Y_{0}^{B}(𝔑), L) → H^{q}(Y^{B}(U), L) composed with the multiplication by the idempotent of R′ induces a W –linear map: V_{U}(R_{U}) ≅ V′(R′) which is 𝕋(y)– equivariant as long as y_{𝔮} = 1;

(5) R_{U} is generated by 𝕋(y) with y_{𝔮} = 1;

(6) V′(R′) ≅ R′^{r} for some r.

In [Fu], U′ and U are written as K_{D,y} and K_{D}, respectively. This is enough to conclude that V (R_{U}) is R_{U}–free. On the other hand, for the Sylow p–subgroup S of U_{0}(𝔑)/U, R_{U} is W [S]–free of finite rank. Then R ≅ R_{U} ⊗_{W [S],ε} W, where ε is the algebra homomorphism W [S] → W induced by the character ε of S.

This fact follows from the freeness of the Hecke algebra over the group algebra (under (unr) and p > 2), for example, [H02] Corollary 4.3, [H05a] Corollary 9.3 or [PAF] 4.2.11–12. In the above papers, the symbol N is used for the primeto-p–part of the present level 𝔑. Similarly, V_{U} is W [S]–free of finite rank by [H89] Theorem 3.8. Thus we have

V (R) = V_{U}(R) ⊗_{W [S],ε} W ≅ (R_{U} ⊗_{W [S],ε} W)^{r} ≅ R^{r}

for a suitable integer r. Actually r = 2^{q} ≤ 2, because V ⊗ ℚ is of rank 2^{q} over the (rational) Hecke algebra.

As for the easing of the condition (H4) on 𝔑, it follows from the same argument, replacing 𝔑 by 𝔑 ∩ ∏_{𝔭∈P} 𝔭, because this is the case where the deformation is unrestricted at 𝔭 ∈ P, which has been dealt with in [Fu] assuming that, for example, [F : ℚ] is even (see [HMI] Section 3.2). □

By the theory of p–adic analytic families of nearly ordinary cusp forms (see [H89], [H96] Section 2.7 and [HMI] 3.2.8, 3.3.4 and 4.3.9), we can ease slightly the conditions necessary to have freeness of V (R) over R. We shall describe this generalization for our later use. Let G = G(𝔑′) = Cl_{F}^{+}(𝔑′p^{∞}) × (O_{p} × O/𝔑′^{(p)})^{×}, where Cl_{F}^{+}(𝔑′p^{n}) is the strict ray class group modulo 𝔑′p^{n} of F, and

Cl_{F}^{+}(𝔑′p^{∞}) = \underleftarrow{lim}_{n} Cl_{F}^{+}(𝔑′p^{n}) = F_{𝔸}^{×}/F^{×}U_{F} (𝔑′)^{(p)}F_{∞+}^{×}

with U_{F} (𝔑′) = Ô^{×}∩(1+𝔑′Ô). We have a natural homomorphism ι : T_{0}(O_{p}) → G sending (a, b) to (a^{−1}, a^{−1}b). Each element (z, y) ∈ G acts on f ∈ S_{κ}(U, ε; A) by f|(z, y)(x) = f|𝕋(y)(xz) (for U ⊂ U_{0}(𝔑′)). Let Γ_{0} be the maximal torsionfree quotient of G (which is independent of 𝔑′ up to isomorphisms), and fix a splitting G = Γ_{0} × G_{tor}. We consider the Iwasawa algebra W [[Γ_{0}]]. For an integral domain 𝕀 finite flat over W [[Γ_{0}]], we define

A(𝕀) = \{P ∈ Hom_{W} (𝕀, \overline{ℚ}_{p})|P ∘ ι ∼ κ with κ_{1} − κ_{2} ≥ I and [κ] ≥ 0\} ,

where φ ∼ ψ if φ = ψ locally on T_{0}(O_{p}) (in other words, φψ^{−1} is of finite order). For each P ∈ A(𝕀), we write κ(P) and ε_{P} for the corresponding algebraic character of T_{0} and the character of

g = ( \begin{matrix}a & 0 \\ 0 & b \end{matrix}) z ∈ T_{0}(O_{p} × (O/𝔑′^{(p)})F_{𝔸}^{×} ⊂ GL_{2}(F_{𝔸})

given by G ∋ g ↦ P (T (ab^{−1}))P (⟨bz⟩)ε_{tor}(g), where ε_{tor} is the restriction of ε to the torsion part 𝔾_{tor} (regarded as a character of G). Thus we can form a triple (ε_{P,1}, ε_{P,2}, ε_{P +}) out of ε_{P} so that ε_{P} (g) = ε_{P,1}(a)ε_{P,2}(b)ε_{P +}(z). For a given nearly ordinary Hecke eigenform f ∈ S_{κ}(𝔑, ε; W) with κ_{1} − κ_{2} ≥ I, decomposing G into a product Γ_{0} ×Δ for a finite subgroup Δ, we write ε(P) = ε_{P} ε|_{Δ}. Thus for a suitable P ∈ Spec(𝕀)(\overline{ℚ}_{p}) whose weight is κ, we find ε(P) = ε. Then there exist 𝕀_{/W [[Γ_{0}]]} as above and a unique family of Hecke eigenforms {f_{P} }_{P ∈A(𝕀)} containing f and satisfying the following two conditions:

(1) f_{P} ∈ S_{κ(P)}(𝔑_{P} , ε(P); W [ε_{P}]) for the conductor 𝔑_{P} of the character ε(P)^{−}, where W [ε_{P}] is a subring of \overline{ℚ}_{p} generated over W by the values

of ε_{P};

(2) There exists a function a : F_{𝔸}^{×} → 𝕀 such that a_{p}(y, f_{P}) = a(y)(P) for

all y ∈ F_{𝔸}^{×} and all P ∈ A(𝕀).

Corollary 6.2. Let {f_{P} }_{P ∈A(𝕀)} be the family of nearly p–ordinary Hecke eigenforms as above. Write R_{P} be the local ring of h_{κ(P)}(𝔑_{P} , ε(P); W [ε_{P}]) through which the algebra homomorphism λ_{P} of the Hecke algebra given by f_{P} |𝕋(y) = λ_{P} (𝕋(y))f_{P} factors. If one member f ∈ S_{κ}(𝔑, ε; W) satisfies the assumptions (H1-7), V (R_{P}) is R_{P} –free of rank 2^{q}, where V = H^{q}(Y_{0}^{B}(𝔑_{P}), L(κ(P)ε(P); W [ε_{P}])) and q = 0, 1 by (6.1).

Proof. We choose U′ as in the proof of Theorem 6.1 and write U′_{0}(𝔑′) = U′ ∩ U_{0}(𝔑′). We consider the limit V = \underrightarrow{lim}_{n} H_{n.ord}^{q}(Y^{B}(U′ ∩ U(p^{n})), L(κε; W) ⊗_{ℤ_{p}} ℚ_{p}/ℤ_{p}), where H_{n.ord}^{q} = eH^{q} for the idempotent e = lim_{n→∞} 𝕋(p)^{n!}. The module V is naturally a module over G(𝔑′) and hence over W [[Γ_{0}]]. Then in the same manner in [H89] Corollary 3.5 and Theorem 3.8, we can prove that for the Pontryagin dual V′ of V,

V′/P V′ ≅ H_{n.ord}^{q}(Y^{B}(U′_{0}(𝔑_{P})), L(κ(P)ε(P); W [ε_{P}]))

as Hecke modules and that V′ is W [[Γ_{0}]]–free module of finite rank. We write V′_{P} for the Hecke module of the right-hand-side of the above formula. Then we define h′ ⊂ End_{W [[Γ_{0}]]}(V′) by the W [[Γ_{0}]]–subalgebra generated by 𝕋(y) for all integral ideles y. As proved under (unr) and p > 3 in [PAF] Corollaries 4.31–32 or [H02] Corollary 4.3 (where the assumption is p > 2 and N denotes the prime-to–p part of the present 𝔑′), h′ is W [[Γ_{0}]]–algebra free of finite rank, whose rank is equal to rank_{W [ε_{P}]} h′_{P} for h′_{P} = h_{κ(P)}^{n.ord}(U′_{0}(𝔑𝔭^{e(P)}), ε(P); W [ε_{P}]). Since they have the same generators 𝕋(y)’s, h′/P h′ surjects down to h′_{P}. By comparing their rank over W [ε_{P}], we find h′/P h′ ≅ h′_{P} canonically sending 𝕋(y) to 𝕋(y). Since R′ is the direct summand of h′_{P_{0}} ⊂ h_{κ}(𝔑, ε; W), by Hensel’s lemma (cf. [BCM] III.4.6), h′ has a unique local ring R′ ⊂ h′ with R′/P_{0}R′ ≅ R′. We put V′(R′) = R′V′, which is W [[Γ_{0}]]–free module of finite rank. Since V′(R′)/P_{0}V′(R′) ≅ V′(R′), which is a free of finite rank over R′ = R′/P_{0}R′, we choose a lift {v_{1}, …, v_{r}} in V′ of a base of V′(R′) over R′. Then the R′–linear map π : R′^{r} → V′(R′) given by (h_{1}, …, h_{r}) ↦ ∑_{j} h_{j}v_{j} is surjective by Nakayama’s lemma applied to R′ and ideal P_{0}. By comparing the rank over W [[Γ_{0}]], we find that π is an isomorphism. Thus V′(R′) is free of rank r = 2^{q} over R′.

We define R′_{P} by R′_{P} = R/P R ⊂ h′_{P}. Then R′_{P} ≅ R_{P} canonically, and V′_{P} (R′_{P}) ≅ V_{P} (R_{P}) for V_{P} = H_{n.ord}^{q}(U′_{0}(𝔑_{P}), L(κ(P)ε(P); W [ε_{P}]) in the same manner as in the proof of Theorem 6.1. This finishes the proof. □

6.2. Induced representations. We are going to verify the assumption of the freeness theorem: Theorem 6.1 for induced representations from CM fields.

We first recall a cusp form f on GL_{2}(F_{𝔸}) with complex multiplication by a CM field M. Let M/F be a CM field with integer ring R and choose a CM type Σ:

I_{M} = Hom_{field}(M, \overline{ℚ}) = Σ ⊔ Σc

for complex conjugation c. To assure the assumption (ord), we need to assume that the CM type Σ is p–ordinary, that is, the set Σ_{p} of p–adic places induced by i_{p} ∘ σ for σ ∈ Σ is disjoint from Σ_{p}c (its conjugate by the generator c of Gal(M/F)). The existence of such an ordinary CM type implies that all prime factors of p in F split in M/F. For each k ∈ ℤ[I], we write kΣ = ∑_{σ∈Σ} k_{σ|F} σ.

We choose κ_{1} > κ_{2} with κ_{1} + κ_{2} = [κ]I for an integer [κ]. We then choose a Hecke character λ of conductor ℭ𝔓^{e} (ℭ prime to p) such that

λ((α)) = α^{κ_{1}Σ+cκ_{2}Σ} for α ∈ M^{×} with α ≡ 1 mod ℭ𝔓^{e},

where 𝔓^{e} = ∏_{𝔓∈Σ_{p}} (𝔓^{e(𝔓)}𝔓^{ce(𝔓^{c})}) for e = ∑_{𝔓∈Σ_{p}⊔Σ_{p}c} e(𝔓)𝔓. We also decompose ℭ = ∏_{𝔏} 𝔏^{e(𝔏)} for prime ideals 𝔏 of M. We extend λ to a p–adic idele character λ̂ : M_{𝔸}^{×}/M^{×}M_{∞}^{×} → \overline{ℚ}_{p}^{×} so that λ̂(a) = λ(aO)a_{p}^{−κ_{1}Σ−cκ_{2}Σ}. By class field theory, we may regard λ̂ as a character of Gal(\overline{F} /M). Any character φ of Gal(\overline{F} /M) of the form λ̂ as above is called “of weight κ”. For a prime ideal 𝔏 of M outside p, we write λ_{𝔏} for the restriction of λ̂ to M_{𝔏}^{×}. For 𝔓 ∈ Σ_{p}, we define λ_{𝔓}(x) = λ̂(x)x^{κ_{1}Σ} for x ∈ M_{𝔓}^{×} and λ_{𝔓^{c}}(x) = λ̂(x)x^{cκ_{2}Σ} for x ∈ M_{𝔓^{c}}^{×}. Then λ_{𝔏} for all prime ideals 𝔏 is a continuous character of M_{𝔏}^{×} with values in \overline{ℚ} whose restriction to R_{𝔏}^{×} is of finite order. By the condition κ_{1} > κ_{2}, λ̂ cannot be of the form λ̂ = ϕ ∘ N_{M/F} for an idele character ϕ : F_{𝔸}^{×}/F^{×}F_{∞+}^{×} → \overline{ℚ}_{p}^{×}.

We define a function F_{𝔸}^{×} ∋ y ↦ a_{p}(y, θ(λ)) supported by integral ideles by

(6.2)  a_{p}(y, θ(λ)) = ∑_{x∈M_{𝔸}^{×},xx^{c}=y,x_{Σp} =1} λ̂(x) if y is integral.

where x runs over elements in M_{𝔸^{(∞)}}^{×}/(R̂^{(pℭℭ^{c})})^{×} satisfying the following three conditions: (i) xR is an integral ideal of M, (ii) N_{M/F} (x) = y and (iii) x_{𝔔} = 1 for primes 𝔔 in Σ_{p} and 𝔔|ℭ. The q–expansion determined by the coefficients a_{p}(y, θ(λ)) gives a unique element θ(λ) ∈ S_{κ}(𝔑′, ε′_{λ}; \overline{ℚ}) ([HT1] Theorem 6.1), where 𝔑′ = N_{M/F} (ℭ𝔓^{e})d(M/F) for the discriminant d(M/F) of M/F and ε′_{λ} is a suitable “Neben” character.

We decompose ℭ = 𝔉𝔉^{c}ℑ so that 𝔉𝔉_{c} is a product of split primes and ℑ for the product of inert or ramified primes, 𝔉 + 𝔉_{c} = R and 𝔉 ⊂ 𝔉_{c}^{c}. We put 𝔣 = 𝔉 ∩ F and 𝔦 = ℑ ∩ F. Assuming that λ^{−} has split conductor, we describe the Neben character ε_{λ} of the minimal form f(λ) in the automorphic representation π(λ) generated by θ(λ). The character ε_{λ} is possibly different from ε′_{λ} and is given as follows:

(1) For 𝔩|𝔣, we identify T_{0}(O_{𝔩}) = O_{𝔩}^{×}×O_{𝔩}^{×} with R_{𝔏^{c}}^{×} ×R_{𝔏}^{×} with this order for the prime ideal 𝔏|(𝔩R ∩𝔉). We define ε_{λ,𝔩} by the restriction of λ_{𝔏^{c}} ×λ_{𝔏} to T_{0}(O_{𝔩}).

(2) For 𝔭|p, identify T_{0}(O_{𝔭}) with R_{𝔓^{c}}^{×} × R_{𝔓}^{×} for 𝔓|𝔭 in Σ_{p}, we define ε_{λ,𝔭} by the restriction of λ_{𝔓^{c}} × λ_{𝔓} to T_{0}(O_{𝔭}).

(3) For 𝔩|𝔦d(M/F), we choose a character ϕ_{𝔩} : F_{𝔩}^{×} → ℂ^{×} such that λ_{𝔏} = ϕ_{𝔩} ∘ N_{M_{𝔏}/F_{𝔩}} (this is possible because λ^{−} has split conductor). Then

we define ε_{λ,1,𝔩}(a) = ϕ_{𝔩} and ε_{λ,2,𝔩}(d) = (\frac{M_{𝔏}/F_{𝔩}}{d} ) λ_{𝔏}(d), where 𝔏 is the

prime factor of 𝔩 in M and (\frac{M_{𝔏}/F_{𝔩}}{d} ) is the quadratic residue symbol for M_{𝔏}/F_{𝔩}.

(4) The central character ε_{λ+} is given by the product of the restriction of λ to F_{𝔸}^{×} and the quadratic character ^{(\underline{M/F} )} of the CM field M/F.

We now give an explicit description of f(λ) without assuming that λ^{−} has split conductor. Let Ξ_{pr} be the set of prime factors 𝔩 of 𝔑′ = d(M/F)N_{M/F} (ℭ𝔓^{e}) where π_{𝔩} is principal. If λ^{−} has split conductor, Ξ_{pr} is the full set of prime factors of 𝔑′. Otherwise, 𝔩 ∈ Ξ_{pr} if and only if either 𝔩|𝔣 or 𝔩|𝔦 and

(6.3)  λ_{𝔏}(x) = ϕ_{𝔩}(xx^{c}) for a character ϕ_{𝔩} : F_{𝔩}^{×} → ℂ^{×}.

For 𝔩 ∈ Ξ_{pr}, taking a prime 𝔏|𝔩 in M, we have

(6.4)  π_{𝔩}(λ) ≅ \begin{cases}π(λ_{𝔏^{c}}, λ_{𝔏}) & if 𝔩|𝔣 and 𝔏|𝔉, \\ π(ϕ_{𝔩},^{(\underline{M_{𝔏}/F_{𝔩}} )} ϕ_{𝔩}) & if 𝔩|𝔦. \end{cases}

We split 𝔑′ into a product 𝔑_{1}𝔑_{2} of co-prime ideals so that 𝔑_{1} is made up of primes in Ξ_{pr}. Writing π_{𝔩}(λ) = π(η_{𝔩}, η′_{𝔩}) for characters η_{𝔩}, η′_{𝔩} : F_{𝔩}^{×} → ℂ^{×}, we write C_{𝔩} for the conductor of η_{𝔩}^{−1}η′_{𝔩}. Define the minimal level of π(λ) by

𝔑(λ) = 𝔑_{2} ∏_{𝔩∈Ξ_{pr}} C_{𝔩}.

We write Ξ = {𝔏|𝔏 ⊃ 𝔉𝔓^{Σ}, 𝔏 ⊃ 𝔑(λ)} for primes 𝔏 of M and define

(6.5)  a_{p}(y, f(λ)) = \begin{cases}∑_{xx^{c}=y,x_{Ξ}=1} λ̂(x)x_{p}^{(κ_{1}−κ_{2})Σ} & if y is integral, \\ 0 & otherwise, \end{cases}

where x runs over (R̂ ∩ M_{𝔸^{(∞)}}^{×}/(R^{(Ξ)})^{×} with x_{𝔏} = 1 for 𝔏 ∈ Ξ. The value λ̂(x) is well defined modulo (R^{(Ξ)})^{×} as long as x_{Ξ} = 1 for the following reason: For primes 𝔩|𝔑(λ) non-split in M/F, by the condition xx^{c} = y, x is determined up to a unit u with uu^{c} = 1. Since λ_{𝔏}(u) = ϕ_{𝔩}(uu^{c}) = 1, the value λ_{𝔏}(x_{𝔏}) is well defined. For 𝔏 ∈ Ξ, by imposing x_{𝔏} = 1, the condition xx^{c} = y implies x_{𝔏^{c}} = y_{𝔩}; so, the value λ_{𝔏}(x_{𝔩}) is again well defined. As for a split prime 𝔩 ∤ 𝔑(λ) but 𝔩|N_{M/F} (ℭ), we have λ_{𝔏}|_{O_{𝔩}^{×}} = λ_{𝔏^{c}}|_{O_{𝔩}^{×}}, so λ_{𝔏}(u_{𝔏})λ_{𝔏^{c}}(u_{𝔏^{c}}) = 1 because uu^{c} = 1 implies u_{𝔏} = u_{𝔏^{c}}^{−1} identifying R_{𝔏} and R_{𝔏^{c}} with O_{𝔩}. As for 𝔭|p with 𝔭 ∤ 𝔑(λ), if (uu^{c}) = 1, we have

λ̂(u)u^{(κ_{1}−κ_{2})Σ} = u^{−κ_{1}Σ−cκ_{2}Σ+(κ_{1}−κ_{2})Σ} = (uu^{c})^{−κ_{2}} = 1.

So again, λ̂(x)x_{p}^{(κ_{1}−κ_{2})Σ} is well-defined modulo such local units.

For a principal series representation π(η′, η) of GL_{2}(F_{𝔩}), if η|_{O_{𝔩}^{×}} = η′|_{O_{𝔩}^{×}}, we have π(η′, η) ≅ η ⊗ π(η^{−1}η′, 1) and π(η^{−1}η′, 1) is spherical; thus we have a unique spherical vector v ≠ 0 in π(η^{−1}η′, 1) with v|T (𝔩) = (1+η^{−1}η′(ϖ_{𝔩}))v. The corresponding vector v′ = v ⊗ η in π(η′, η) has minimal level fixed by SL_{2}(O_{𝔩}) with v′|T (y) = (η(y)+η′(y))v′. If the conductor C_{𝔩} of η^{−1}η′ is non-trivial, again by the same argument, we find v′ ≠ 0 in π_{𝔩}(λ) such that v′|T (y) = η(y)v′ and

v′|u = ε(u)v′ (u ∈ U_{0}(C_{𝔩})_{𝔩}), where ε(u) = η(det(u))(η^{−1}η′(a)) for u = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈

U_{0}(C_{𝔩})_{𝔩}. This shows that f(λ) is a classical modular form in M_{κ}(𝔑(λ), ε_{λ}; \overline{ℚ}) if λ^{−} has split conductor. The form f(λ) is a common eigenform of Hecke operators 𝕋(y). The p–adic Galois representation ρ_{λ} associated to f(λ) is the induced representation Ind_{M}^{F} λ̂, regarding λ̂ as a character of Gal(\overline{F} /M) by class field theory. By regularity: κ_{1} > κ_{2}, λ̂(cσc^{−1}) ≠ λ̂(σ) for σ ∈ Gal(\overline{F} /M), ρ_{λ} is absolutely irreducible by Mackey’s theorem, andf(λ) is a cusp form.

We take the coefficient ring W to be free of finite rank over ℤ_{p}. Assuming that λ^{−} has split conductor (⇔ π(λ) is principal at every finite place), we shall study when f(λ) satisfies the conditions (H1-7) of Theorem 6.1. We take a character φ of Gal(\overline{F} /M) of order prime to p such that λ̂φ^{−1} ≡ 1 mod 𝔪_{W} and define ψ = φ^{−}. Suppose that λ and φ coincides on R_{𝔏}^{×} if 𝔏 ∤ p. Then the conditions (2) and (3) on ψ in the introduction are an interpretation of principality of π(λ) at every finite place. To interpret the four conditions (1-4) on ψ in the introduction in terms of φ, let G(ℭ) = M_{𝔸}^{×}/\overline{M^{×}U_{M}(ℭ)^{(p)}M_{∞}^{×}}, where

U_{M}(ℭ)^{(p)} = {x ∈ R̂^{×}|x_{p} = 1, x ≡ 1 mod ℭR̂} .

The first conditions (1) on ψ can be stated in terms of φ as follows:

(h1) φ has order prime to p with exact conductor ℭ𝔓^{e} for ℭ prime to p. Thus φ factors through the maximal prime-to-p quotient of G(ℭ) which can be regarded canonically as a subgroup of G(ℭ), because G(ℭ) is almost p–profinite. The conditions (2-4) in the introduction imply the following three assertions:

(h2) For all prime factors 𝔏|ℑ, φ_{𝔏} = ϕ_{𝔩} ∘ N_{M/F} for a character ϕ_{𝔩} : F_{𝔩}^{×} →

W^{×}. (h3) φ_{𝔓} ≠ φ_{𝔓^{c}} for all 𝔓 ∈ Σ_{p}. (h4) Over Gal(\overline{F} /M[\sqrt{p^{∗}}]), we have φ_{c} ≠ φ, where φ_{c}(σ) = φ(cσc^{−1}).

We write G_{tor}(ℭ) for the maximal torsion subgroup of G(ℭ).

Theorem 6.3. Assume (6.1) and the four conditions (h1-4). Let λ_{k} : G(ℭ) → \overline{ℚ}_{p}^{×} be an arithmetic Galoischaracter of weight k_{1}Σ + ck_{2}Σ (k_{j} ∈ ℤ[I]) such that k_{1} > k_{2} and λ_{k}|_{G_{tor}(ℭ)} = φ. Then for the local ring R of h_{k}(𝔑(λ_{k}), ε_{λ_{k}}; W [λ_{k}]) corresponding to f(λ_{k}), the R–component V (R) of V = H^{q}(Y_{0}^{B}(𝔑(λ_{k})), L(kε_{λ_{k}}; W [λ_{k}])) is R–free of rank 2^{q}, where W [λ_{k}] is the complete discrete valuation ring inside \overline{ℚ}_{p} generated by the values of λ_{k} over W and q = |I_{B}| ≤ 1.

Proof. We take a sufficiently large κ with κ_{1} > κ_{2} and κ_{1} + κ_{2} = [κ]I for 0 ≤ [κ] ∈ ℤ so that ζ^{κ_{1}Σ+cκ_{2}Σ} = 1 for all ζ ∈ R^{×} and κ ≡ 0 mod (Q − 1)ℤ[I] for Q = |𝔽|. Then we can find a Hecke character λ with the following properties:

(1) We have λ((α)) = α^{κ_{1}Σ+cκ_{2}Σ}φ((α)) for all α ∈ M^{×} prime to ℭ𝔓^{e};

(2) λ ≡ φ mod 𝔪_{W}.

We are going to show for f(λ) the assumptions (H1-7) except for (H4) of Theorem 6.1. Thus if (H4) is not applicable to Ind_{M}^{F} λ̂ , we get the result for f(λ_{k}) by Corollary 6.2, because f(λ) is a member of the p–adic family of modular forms determined by f(λ_{k}). Otherwise, we modify the choice of λ.

We verify condition (H1-3) and (H5-7) one by one. We always have a character λ_{1} of conductor 1 with λ_{1}((α)) = α^{κ_{1}Σ+cκ_{2}Σ} for all α ∈ M^{×} and λ_{1} ≡ 1 mod 𝔪_{W} by our choice of κ; so, λ/λ_{1} ≡ φ mod 𝔪_{W}. We may assume that λ/λ_{1} = φ.

• By the above choice of λ_{1}, we have det ρ_{λ_{1}} = N^{[κ]} ^{(\underline{M/F} )} and det ρ_{λ} =

N^{[κ]}φ̃^{(\underline{M/F} )}, where φ̃ is the Galois character corresponding to the pull

back of φ as a Hecke character of M_{𝔸}^{×} to F_{𝔸}^{×}. Then χ in (H1) is given by φ̃^{(\underline{M/F} )}, which has order prime to p because p > 2. This shows (H1).

• By (h2), we have for 𝔩|𝔑(λ)p,

ρ_{λ}|_{D_{𝔩}} ≅ \begin{cases}( \begin{matrix}λ̃ & 0 \\ 0 & λ̃_{c} \end{matrix} ) & if 𝔩 = 𝔏\overline{𝔏} (𝔏 ≠ \overline{𝔏}) in M, \\ ( 0 λ̃_{𝔩}(^{\underline{M/F}})^{λ̃_{𝔩} 0} ) & if 𝔩 is inert or ramified in M/F. \end{cases}

We can choose λ̂_{c} to corresponds to λ̂_{𝔓^{c}} for 𝔓 ∈ Σ_{p} with 𝔓|𝔩 if 𝔩|p. Then by construction (or the definition of κ_{2}), we have δ_{𝔩} = λ̂_{𝔓^{c}}. This shows (H2).

• Since λ_{1} is of conductor 1, we find that λ̂|_{I_{𝔩}} = φ|_{I_{𝔩}}, which is of order prime to p. This shows (H3).

• Since λ̂ ≡ φ mod 𝔪_{W}, (h3) implies that \overline{δ}_{𝔭} ≠ \overline{ε}_{𝔭}; so, (H5) follows from (h3).

• The condition (H6) follows from the definition of 𝔑(λ) and (h1), because C(\overline{ε}_{𝔩}\overline{δ}_{𝔩}^{−1}) is equal to C(ε_{𝔩}δ_{𝔩}^{−1}) by (H3) already verified. By our definition of 𝔑(λ), its 𝔩 part coincides with C(ε_{𝔩}δ_{𝔩}^{−1}).

• The condition (H7) follows from (h4) by Mackey’s theorem.

Thus as long as λ̂ ≢ λ̂_{c}N mod 𝔪_{W} on I_{𝔭} for every 𝔭|p, we have verified the theorem.

Now assume that

P = {𝔭|p|λ̂ ≡ λ̂_{c}N mod 𝔪_{W} on I_{𝔭}}

is non-empty. Let \overline{R^{×}} (resp. \overline{O^{×}}) be the p–adic closure of R^{×} (resp. O^{×}) in R_{p}^{×} for R_{p} = R ⊗_{ℤ} ℤ_{p}. Since M cannot have p–th root of unity (by ordinarity of Σ and unramifiedness of p in F/ℚ), [R^{×} : O^{×}] is prime to p; so, \overline{R^{×}}/\overline{O^{×}} has order prime to p. We consider the character x ↦ x^{Σ} of R_{p}^{×} → W^{×}, which has values in a complete valuation subring A of W unramified and finite over ℤ_{p}. Let A_{p}^{×} be the maximal p–profinite subgroup of A^{×}, which is canonically a direct factor of A^{×}, because A is unramified over ℤ_{p}. Let x ↦ ⟨x⟩ be the projection of x ∈ A^{×} to A_{p}^{×}. Thus ⟨x⟩ ≡ 1 mod 𝔪_{W} for all x ∈ A^{×} and ⟨ζ⟩ = 1 for all roots of unity ζ in A. Thus x ↦ ⟨x^{Σ}⟩ is a character of R_{p}^{×}/\overline{R^{×}}, which is a subgroup of finite index of G(1). We can extend this character to a character χ̂ of G(1) so that χ̂ ≡ 1 mod 𝔪_{W} on G(1). This is possible for the following reason: We first extend the character to a character χ′ : G(1) → W^{×}, which is always possible, replacing W by its finite extension if necessary. Then we take a Teichmüller lift ε of the reduction (χ′ mod 𝔪_{W}). Then χ̂ = ε^{−1}χ′ gives the desired extension. By our construction, χ̂ is the p–adic avatar of an arithmetic Hecke character χ whose infinity type is Σ.

We now take the Teichmüller lift λ̂_{0} of (λ̂ mod 𝔪_{W}), which is a p–adic avatar of a finite order character λ_{0} : G(ℭ) → W^{×}. Then λ′ = λ_{0}χ is of infinite type Σ and satisfies λ′ ≡ λ ≡ φ mod 𝔪_{W}. For x ∈ R_{p}, we write ω(x) = lim_{n→∞} x^{[R:pR]^{n}} ∈ R_{p} for x ∈ R_{p}. Since p is unramified in M/ℚ, the Teichmüller lift of (x^{k} mod 𝔪_{W}) for k ∈ ℤ[Σ ∪ Σc] is given by ω(x)^{k} (in other words, the operations k and ω commute). Thus, at the place 𝔭 ∈ P, by the above process of construction, λ′^{−}(x_{𝔭}) = N^{−1}(x_{𝔭}) for x_{𝔭} ∈ R_{𝔓} ∩ F_{𝔓}^{×} (𝔓 ∈ Σ_{p} with 𝔓|𝔭), and the level 𝔑(λ′) of f(λ′) is prime to all 𝔭 ∈ P. Thus f(λ′) has weight (I, 0) and its Galois representation satisfies (H4). Then the theorem follows from Corollary 6.2, since f(λ_{k}) comes from the same local ring of the universal nearly ordinary Hecke algebra h as the local ring of the p–adic family of Hecke eigenforms determined by f(λ) or f(λ′). □

For our later use, we shall compute the q–expansion of classical modular forms associated to f(λ). Pick y ∈ F_{𝔸}^{×} with y_{p} = y_{∞} = 1. Then by the definition of λ̂ and (6.5), we get the following formula of the complex Fourier coefficients:

a(ξyd, f(λ)) = ∑_{xx^{c}=ξyd,x_{Ξ}=1} λ(xR),

where xR = F ∩ xR̂ and x runs over (R̂ ∩ M_{𝔸^{(∞)}}^{×})/R^{(Ξ)} for Ξ as in (6.5). This shows that for f_{diag[y,1]} in (S2),

f_{diag[y,1]}(τ) = N(𝔶)^{−1} ∑_{𝔄;𝔄𝔄^{c}∼𝔶𝔡} λ(𝔄)α^{−κ_{2}}θ(λ; 𝔄),

where 𝔄 runs over a complete representative set for ideal classes of M with 𝔄𝔄^{c} = α𝔶𝔡 (𝔶 = yÔ ∩ F) for a totally positive α ∈ F and

(6.6)  θ(λ; 𝔄) = ∑_{ξ∈𝔄^{−1}/μ(M)} λ(ξ^{(∞Ξ)})(ξξ^{c})^{−κ_{2}}q^{αξξ^{c}}.

Here we regard λ as an idele character λ : M_{𝔸}^{×}/M^{×} by putting

λ(x) = λ(xR)x_{∞}^{−κ_{1}Σ−cκ_{2}Σ},

and ξ runs over elements in 𝔄^{−1} such that ξ𝔄 is outside Ξ for Ξ as in (6.5). As a locally constant function on 𝔄̂^{−1}, the p–component of ϕ′_{1} : ξ ↦ λ(ξ^{(Ξ)}) is given by λ_{p}^{−1} restricted to 𝔄_{p}^{−1} by the following reason: ϕ′_{1} is the characteristic function of 𝔄_{𝔩}^{−1} for 𝔩 outside the conductor C(λ), and taking ξ ∈ 𝔄^{−1} with ξ ≡ 1 mod C^{(p)}(λ), we see that ϕ′_{1}(ξ) = λ(ξ^{(Ξ)}) = λ(ξ^{(p)}) = λ(ξ_{p})^{−1}.

The modular form θ(λ; 𝔄) is of weight κε on

Γ_{0}(𝔑(λ); 𝔶) = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(F)|a, d ∈ O, b ∈ 𝔶, c ∈ 𝔑(λ)𝔶^{−1}\} .

6.3. Self-duality. Let L^{∗}(κε; W) be the dual lattice of L(κ; W) under the pairing [ , ] introduced in Subsection 5.3. Then by definition, L^{∗}(κε; W) ⊂ L(κε; W) and the quotient L(κε; W)/L^{∗}(κε; W) is spanned by X^{n−j}Y^{j} for 0 < j < n.

Since

U_{0}(𝔑p)_{p} diag[p, 1]U_{0}(𝔑p)_{p} = ⨆_{u mod pO_{p}} ( \begin{matrix}p & u \\ 0 & 1 \end{matrix} ) U_{0}(𝔑p)_{p},

the action of ( \begin{matrix}p & u \\ 0 & 1 \end{matrix} ) on L(κε; W)/L^{∗}(κε; W) (even after dividing by p^{κ_{2}}) is nilpotent. Thus the projector e = lim_{n→∞} 𝕋(p) kills the cohomology group:

H_{∗}^{r}(Y, L(κε; W)/L^{∗}(κε; W)) (Y = Y_{0}^{B}(𝔑))

for any r ≥ 0, and hence by cohomology sequence, we get a canonical isomorphism for Y = Y_{0}^{B}(𝔑):

(6.7)  H_{∗,n.ord}^{r}(Y, L^{∗}(κε; W)) ≅ H_{∗,n.ord}^{r}(Y, L(κε; W)),

where H_{∗}^{r} is either compactly supported or usual cohomology group. We define the action of Hecke operators 𝕋(y) and ⟨𝔩⟩ on H_{∗}^{r}(Y, L^{∗}(κ^{∗}ε^{∗}; W)) via the adjoint action under [, ] of the semi-group Δ_{0}(𝔑). Then the operator is integral if either p|𝔑 or [κ] ≤ 1 ⇔ [κ] ≥ 0. Thus in the same way, we get

(6.8)  H_{∗,n.ord}^{r}(Y, L(κ^{∗}ε^{∗}; W)) ≅ H_{∗,n.ord}^{r}(Y, L^{∗}(κ^{∗}ε^{∗}; W)).

As we have seen in [H88a] Theorem 10.1, H_{∗}^{r}(Y, L(κε; W) ⊗ (ℚ_{p}/ℤ_{p})) is p– divisible if |I_{B}| ≤ 1. Then by looking into the cohomology sequence attached to the short exact sequence:

0 → L(κε; W) → L(κε; W ⊗ ℚ_{p}) → L(κε; W) ⊗ (ℚ_{p}/ℤ_{p}) → 0, H_{∗}^{r}(Y, L(κε; W)) is free of finite rank over W, and we get a perfect pairing:

( , )_{W} : H_{n.ord}^{q}(Y, L(κε; W)) × H_{c,n.ord}^{q}(Y, L(κ^{∗}ε^{∗}; W)) → W

of W –free modules. For the moment, assume that F ≠ ℚ. Then Y^{B}(U) is compact; so, H_{cusp,n.ord}^{q} = H_{n.ord}^{q} = H_{c,n.ord}^{q}, and we have the perfect duality pairing

(6.9) ( , )_{W} : H_{cusp,n.ord}^{q}(Y, L(κε; W)) × H_{cusp,n.ord}^{q}(Y, L(κ^{∗}ε^{∗}; W)) ⟶ W

As already verified in [H88b] Theorem 3.1 for F = ℚ, the assertion (6.9) holds even for F = ℚ; so, we do not need to assume F ≠ ℚ anymore. We thus have

Corollary 6.4. Under the assumptions and notations of Corollary 6.2, the map (v, w) ↦ f(v ⊗ w) induces a surjective linear map: V (R_{P}) ⊗_{R_{P}} V′(R_{P}) ↠ S(R_{P}) for all P ∈ A(𝕀), where S = S_{κ(P)}(𝔑, ε(P); W [ε_{P}]), V = H^{q}(Y, L(κ(P)ε(P); W [ε_{P}])), V′ = H^{q}(Y, L(κ(P)^{∗}ε(P)^{∗}; W [ε_{P}])). If q = |I_{B}| = 0, f is an isomorphism: V (R_{P}) ⊗_{R_{P}} V′(R_{P}) ≅ S(R_{P}).

7. Proof of the theorem

We shall prove the theorem in the introduction under the assumptions (h1-4) on φ, which are equivalent to the assumptions (1-4) in the introduction once we have chosen φ with ψ = φ^{−}. We first recall integrality results due to Shimura [ACM] Section 32 and Katz [K] II on the values of modular forms and then prepare preliminary results on integral decomposition of quaternionic quadratic spaces. After that, we prove the theorem in the case where the degree [F : ℚ] is even. The odd degree case will be reduced to the even degree case.

7.1. Integrality of values of modular forms. By the approximation theorem,

GL_{2}(F)\GL_{2}(F_{𝔸}^{(∞)})/U_{0}(𝔑) ≅ F^{×}\F_{𝔸^{(∞)}}^{×}/ det(U_{0}(𝔑)) ≅ Cl_{F} via y ↦ det(y)

for the class group Cl_{F} of F. From this, f ∈ S_{κ}(𝔑, ε; W) is determined by the q–expansions {f(y)}_{y}. Writing 𝔶 = yÔ ∩ F for the ideal corresponding to the idele y and setting ỹ = ( \begin{matrix}y & 0 \\ 0 & 1 \end{matrix} ), f(y) is the q–expansion at the Tate AVRM Tate_{𝔶^{∗},O}(q) (in [K] 1.1) of the classical modular form fỹ (of (S2) in Subsection 5.1) of weight k = κ_{1} −κ_{2} +I on the following congruence subgroup:

(7.1)  Γ_{0}(𝔑; 𝔶) = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(F)|a, d ∈ O, b ∈ 𝔶, c ∈ 𝔶^{−1}𝔑\} .

Here 𝔶^{∗} = 𝔶^{−1}𝔡^{−1} for the absolute different 𝔡 of F.

A classical modular form with q–expansion coefficients in W on a slightly smaller Γ_{1}–type congruence subgroup:

(7.2)  Γ(𝔑; 𝔶) = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ_{0}(𝔑; 𝔶)|a ≡ d ≡ 1 mod 𝔑\}

has a moduli theoretic interpretation, which we recall in the following paragraph. We write S_{k}(Γ(𝔑; 𝔶); A) for the space of the classical cusp forms on Γ(𝔑; 𝔶) of weight k with q–expansion coefficients in A.

Let A be a valuation ring with residual characteristic p. An abelian scheme X_{/A′} over an A–algebra A′ is called an abelian variety with real multiplication by O (AVRM) if it has an embedding: O ↪ End(X_{/A′}) of algebras such that H^{0}(X, Ω_{X/O}) = (O ⊗_{ℤ} A′)ω for a nowhere vanishing differential ω. Here we have used the unramifiedness of F at p (otherwise, we need to formulate this condition as H^{0}(X, Ω_{X/O}) = (𝔡^{−1} ⊗_{ℤ} A′)ω). Each Hilbert modular form f ∈ S_{k}(Γ(𝔑; 𝔶); A) can be regarded as a function of quintuples: (X, λ, i, ω, A′) made up of an A–algebra A′, an AVRM X over A′, a polarization λ whose polarization ideal is given by 𝔶^{∗}, an embedding i : μ_{𝔑} ↪ X of group schemes over A′ and a differential ω as above (see, for more details of AVRM’s, [K] 1.0 and [PAF] Section 4.1). Here μ_{𝔑} is the group scheme made up of 𝔑–torsion points of 𝔾_{m} ⊗ 𝔡^{−1}, that is, μ_{𝔑}(A) = {ζ ∈ 𝔾_{m} ⊗ 𝔡^{−1}(A)|𝔑ζ = 0}, regarding 𝔾_{m} ⊗ 𝔡^{−1}(A) as an additive group. Every ingredient of the quintuple has to be defined over A′. As a function of (X, λ, i, ω)_{/A′}, f satisfies the following conditions (see [HMI] 4.2.7):

(M1) f(X′, λ′, i′, ω′) = ρ(f(X, λ, i, ω)) if ρ : A′ → C is an A–algebra homomorphism and (X′, λ′, i′, ω′)_{/C} ≅ (X, λ, i, ω)×_{A′,ρ}C. Here “≅” implies: ϕ : X ×_{A} C ≅ X′_{/C} as AVRM’s, ^{t}ϕ ∘ λ′ ∘ ϕ = λ ×_{A′} C, ϕ ∘ i ≡ i′ and ϕ^{∗}ω′ = ω.

(M2) f vanishes at all cusps, that is, the q–expansion of f at every Tate

quintuple vanishes at q = 0. (M3) f(X, λ, i, αω) = α^{−k}f(X, λ, i, ω) for α ∈ (A′ ⊗_{ℤ} O)^{×}.

The “Neben” character ε : U_{0}(𝔑) → \overline{ℚ}^{×} restricted to U_{0}^{1}(𝔑) = U_{0}(𝔑)∩SL_{2}(Ô) factors through U_{0}^{1}(𝔑)/U^{1}(𝔑) for U^{1}(𝔑) = U(𝔑) ∩ SL_{2}(Ô) (the conductor of ε^{−} is 𝔑), because ε(u) = ε_{1}(det(u))ε^{−}(d) for u = ( \begin{matrix}a & b \\ c & d \end{matrix} ). Thus to evaluate f ∈ S_{κ}(𝔑, ε; A) at an AVRM X of CM type, we only need to specify μ_{𝔑} ↪ X.

Let M/F be the CM quadratic extension in the introduction. Recall the decomposition: ℭ = 𝔉𝔉_{c}ℑ of the conductor of the Hecke character λ such that 𝔉 + 𝔉_{c} = R with 𝔉 ⊂ 𝔉_{c}^{c} and ℑ is made up of primes non-split in M/F. By (h2) (for φ = λ), the prime factors of 𝔑(λ) are either split or ramified over F. If 𝔩|𝔑(λ) and 𝔩 = 𝔏\overline{𝔏} (with 𝔏 ≠ \overline{𝔏}) in M, we may choose 𝔏 so that 𝔏 ⊃ 𝔉. The exponent of 𝔩 in 𝔑(λ) is less than or equal to that of 𝔏 in 𝔉. Thus to evaluate f(λ) at a CM point, we need to specify the level structure for the level d(M/F)𝔣 (𝔣 = 𝔉 ∩ F). Actually we later need the level structure at other primes non-split in M/F; so, we first specify level structure for split primes and then extend the definition to non-split primes. We shall do this first for an abelian variety of CM type Σ with multiplication by R. Hereafter 𝔉 is an integral ideal of R with 𝔉 + 𝔉^{c} = R and prime to p (because we need to be more careful for primes dividing p).

Let \overline{W} be as in the introduction. Define W = i_{p}^{−1}(\overline{W}) ⊂ \overline{ℚ}, which is a valuation ring unramified over ℤ_{(p)} with algebraically closed residue field \overline{𝔽}.

We consider X(R)_{/ℂ} to be the algebraization of the complex torus ℂ^{Σ}/R^{Σ}, where R^{Σ} = {(a^{σ})_{σ∈Σ}|a ∈ R} and ℂ^{Σ} = R^{Σ} ⊗_{ℤ} ℝ. Since X(R) has complex multiplication, it can be defined over \overline{ℚ} and hence over a number field (see [ACM] 12.4). By the main theorem of complex multiplication (see [ACM] 18.6), X(R) and its ℓ–divisible group for any prime ℓ outside p are actually defined over an infinite extension K of ℚ unramified at p. By the criterion of good reduction by unramifiedness of ℓ–power torsion points (see [ST]), we find a model X(R)_{/W} of X(R)_{/K}.

By choosing δ ∈ M with Im(σ(δ)) > 0 for σ ∈ Σ, we have a polarization pairing (x, y) ↦ Tr_{M/ℚ}(δxc(y)). This pairing identifies R ∧_{O} R with 𝔶 for a suitable choice of a fractional ideal 𝔶 ⊂ F (prime to p) and induces a 𝔶^{∗}–polarization λ = λ(R). Thus we have the CM-triple (X(R), λ(R), i(R), ω(R))_{/W}, choosing ω(R) so that H^{0}(X(R), Ω_{X(R)/W}) = (O ⊗_{ℤ} W)ω(R).

Since \overline{W} has algebraically closed residue field, for any integer m prime to p, we have X(R)[m] = \{x ∈ X(R)(\overline{W})|mx = 0\} ≅ (ℤ/mℤ)^{[M:ℚ]} and μ_{m} ≅ ℤ/mℤ as group schemes over \overline{W}. Thus we define the level 𝔣–structure to be

μ_{𝔣} ≅ O/𝔣 ≅ X(R)[𝔉] = \{x ∈ X(R)(\overline{W})|𝔉x = 0\}.

Since the Frobenius map of \overline{𝔽}_{p} acts by multiplication by p (times a unit) on Ω_{X(R)/\overline{W}}, the p–divisible group X(R)[𝔓^{∞}]_{/W} for 𝔓 ∈ Σ_{p} is connected. Since the residue field of W is algebraically closed, we see that X(R)[𝔓^{e}] ≅ μ_{𝔓^{e}} over W (for e = (e(𝔓))_{𝔓∈Σ_{p}}), which gives rise to the level 𝔭^{e}–structure we need.

Since R ∧ R ≅ 𝔶, we can choose a base w_{1} and w_{2} of R so that R = Ow_{1} + 𝔶w_{2}. For any integral ideal 𝔮 prime to p, we choose a generator ϖ_{𝔮} of 𝔮O_{𝔮}. Fixing an isomorphism O/𝔮 ≅ 𝔶/𝔮, we embed O/𝔮 ≅ 𝔶/𝔮/𝔶 ↪ 𝔮^{−1}R_{𝔮}/R_{𝔮} ≅ X(R)[𝔮] by sending x to ϖ_{𝔮}^{−1}xw_{2} ∈ M_{𝔮}/R_{𝔮}, which gives the level 𝔮–structure on X(R). We choose the base w = (w_{1}, w_{2}) so that the level 𝔭^{e}𝔣–structure we have chosen coincides with the one for 𝔮 if 𝔭^{e}𝔣 + 𝔮 is non-trivial. We may always choose w so that w_{0} = w_{1}/w_{2} ∈ ℌ^{I}. Therefore choosing the base (w_{1}, w_{2}) is almost equivalent to the choice of a point w_{0} ∈ ℌ^{I} modulo Γ(𝔑, 𝔶) for 𝔑 = 𝔮 ∩ 𝔣𝔭^{e}. We write the level structure as i(R) : μ_{𝔑} ↪ X(R)[𝔑].

The above definition of the quadruple x(R) = (X(R), Λ(R), i(R), ω(R))_{/W} can be generalized to ideals of an O–order of R. Let 𝔪 be an integral ideal of F prime to p𝔣. Let R′ = O + 𝔪R be the O–order of M of conductor 𝔪. We take a proper fractional ideal 𝔄 of R′ prime to p𝔣𝔮d(M/F). A fractional R′–ideal 𝔄 is called R′–proper if {x ∈ M|x𝔄 ⊂ 𝔄} = R′. The polarization pairing on R (so on M) induces the polarization Λ(𝔄) on 𝔄. We identify 𝔄 ∧ 𝔄 with a fractional ideal 𝔶(𝔄) of F under this pairing. It is easy to verify 𝔶(𝔄) = 𝔶(R)𝔪N_{M/F} (𝔄). Then we can choose a base w of 𝔄 so that 𝔄 = Ow_{1} + 𝔶(𝔄)w_{2} and w_{0}(𝔄) = w_{1}/w_{2} ∈ ℌ^{I}. This choice w gives rise to the level structure i(𝔄) : μ_{𝔑} ↪ X(𝔄)[𝔑]. We can always find an étale constant subgroup C ≅ O/𝔠 (𝔠 prime to 𝔣𝔮d(M/F)p) in X(R) such that the étale quotient X(𝔄) = X(R)/C over W (e.g. [GME] 1.8.3) gives a model over W of ℂ^{Σ}/𝔄^{Σ}. Since 𝔠 is prime to p𝔣, the level structure i(R) and the differential ω(R) induce a unique level structure and a unique differential ω(𝔄) on X(𝔄). We make a choice w so that the two level structures (one coming from i(R) and another from the base w) coincide at primes where the two are well defined. Thus we have a unique point w_{0}(𝔄) ∈ ℌ^{I}/Γ_{1}(𝔑, 𝔶). Having w is equivalent to having the quadruple x(𝔄) = (X(𝔄), Λ(𝔄), i(𝔄), ω(𝔄)) over ℂ.

Supposing that f ∈ S_{k}(Γ(𝔑; 𝔶); W) (and regarding f as a complex modular form), we may interpret the value f(x(𝔄)) in terms of evaluation at a CM point w_{0}(𝔄) ∈ ℌ^{I}. For each z = (z_{1}, z_{2}) with z_{0} := \frac{z_{1}}{z_{2}} ∈ ℌ^{I}, we consider the lattice L_{z} = L_{z}^{𝔶} = 2πi(Oz_{1} + 𝔶z_{2}) ⊂ F_{ℂ} = F ⊗_{ℚ} ℂ. We define a pairing ⟨ , ⟩ : F_{ℂ} ×F_{ℂ} → ℝ by ⟨2πi(az_{1} +bz_{2}), 2πi(cz_{1} +dz_{2})⟩ = ad−bc, which induces a 𝔶^{∗}–polarization λ_{z} = λ_{z}^{𝔶} on the complex torus X_{z} = X_{z}^{𝔶} = F_{ℂ}/L_{z}. Thus we can algebraize X_{z} to an abelian variety X_{z/ℂ}. We have a canonical level 𝔑–structure i_{z} : (𝔡^{−1} ⊗ O/𝔑) ≅ 2πi(𝔶z_{2} ⊗ O/𝔑) ⊂ X_{z}(ℂ) as long as 𝔶 is prime to 𝔑. Then the analytic value of f at z is given by

(7.3)  z_{2}^{−k}f((z_{0}, 1)) = f(z) = f(x_{z}^{𝔶}) for x_{z}^{𝔶} = (X_{z}, λ_{z}, i_{z}, du),

where u is the variable (u_{σ})_{σ∈I} with u_{σ} ∈ ℂ identifying F_{ℂ} with ℂ^{I} as ℂ–algebras.

Defining the canonical period Ω ∈ F_{ℂ}^{×} = (ℂ^{×})^{Σ} by

(7.4)  ω(R) = Ωdu

and choosing 𝔶 so that R = (2πi)^{−1}L_{z_{0}}^{𝔶}, we find x(𝔄) ≅ x_{z}^{𝔶(𝔄)} and

(7.5)  f(x(𝔄)) = \frac{(2πi)^{k}f(z)}{Ω^{k}} ∈ W up to units in W,

because ω(𝔄)/ω(R) ∈ (O ⊗_{ℤ} W)^{×} (see [ACM] Section 32 and [K] II). Here writing Ω = (Ω_{σ}) ∈ ℂ^{Σ}, Ω^{k} = ∏_{σ∈Σ} Ω_{σ}^{k_{σ}}.

Since W–integral modular forms f(z, w) of weight (k, k) for the product of congruence subgroups: Γ(𝔑; 𝔶) × Γ(𝔑′; 𝔶′) classify the pairs of test objects: (x_{z}^{𝔶}, x_{w}^{𝔶′}), the same formula is valid (by the same proof given in [K]): up to units in W,

(7.6)  f(x(𝔄), x(𝔅)) = \frac{(2πi)^{2k}f(z, w)}{Ω^{2k}}.

7.2. Error terms of integral decomposition. Let B be a quaternion algebra over F. Let M/F be a CM field with integer ring R. We are going to compute error terms of O–integral decomposition of an O–lattice of B as an integral quadratic space into a direct sum of two O–lattices of M with its norm form.

We fix a maximal order O_{B} of B. For an embedding i : R ↪ B of O–algebras, since i is an embedding of O–algebras, we have Tr(i(a)) = Tr_{M/F} (a) for the reduced trace Tr of B and i(a)i(a)^{ι} = N_{M/F} (a) = aa^{c}. This shows i(a^{c}) = i(a)^{ι} for the main involution ι of B.

Let L be an O–lattice in B. We consider the two orders:

(7.7)  O_{L}^{l} = {x ∈ B|xL ⊂ L} and O_{L}^{r} = {x ∈ B|Lx ⊂ L}.

We suppose to have two embedding l : R ↪ B and r : R ↪ B. Thus L becomes an R_{l} ⊗_{O} R_{r}–module by (a ⊗ b)ℓ = l(a)ℓr(b), where R_{l} = l^{−1}(l(R) ∩ O_{L}^{l}) and R_{r} = r^{−1}(r(R) ∩ O_{L}^{r}). Since K^{m} ⊗ K^{n} ≅ M_{m×n}(K), we find that M_{m} ⊗_{K} M_{n}(K) ≅ M_{mn}(K) as K–algebras. By extending scalars to M, we find B ⊗_{F} M ≅ M_{2}(M), and the above argument applied to the extended algebra M_{2}(M) shows that the embedding l ⊗ r : R_{l} ⊗_{O} R_{r} ↪ End_{O}(L) is injective. Therefore B is a free M ⊗_{F} M–module of rank 1. When we regard B as an M–vector space, we agree to use right multiplication by α ∈ M given by αb = b · r(α). Therefore M ⊗_{F} M is identified with M ⊕ M by a ⊗ b ↦ (ab, a^{c}b) for the generator c of Gal(M/F). Then we define L^{1} = (1, 0)L and L^{2} = (0, 1)L for the idempotents (1, 0), (0, 1) ∈ M ⊕ M. Since L^{M} = L^{1} ⊕ L^{2} ⊃ L, we can define L_{j} = L^{j} ∩ L. Then L_{M} = L_{1} ⊕ L_{2} ⊂ L. Since (1, 0)B is the eigenspace of M ⊕ M killed by the right factor M, we have

L_{2} = {x ∈ L|S(L_{1}, x) = 0} ,

because multiplication by units in (M ⊗_{F} M)^{×} preserves the inner product S(x, y) = Tr(xy^{ι}) up to scalar similitude. By S, we have the orthogonal projection π_{1} of B to ML_{1} and π_{2} to ML_{2}. Then we may have defined L^{M} = π_{1}(L) ⊕ π_{2}(L). Indeed, π_{1} (resp. π_{2}) is given by the multiplication by (1, 0) (resp. (0, 1) ∈ M ⊗_{F} M). We want to determine primes dividing the index [L^{M} : L_{M}]. Here is the result:

Lemma 7.1. Let d(R_{l}/O) (resp. d(R_{r}/O)) be the relative discriminant of R_{l}/O (resp. of R_{r}/O). Then we have d(R_{l}/O)d(R_{r}/O)L^{M} ⊂ L_{M}.

Proof. The process constructing L^{M} and L_{M} can be done at each localization B_{𝔭} for primes 𝔭 of O. Then L_{i,𝔭} = L_{𝔭} ∩ M_{𝔭}L_{i} and π_{j}(L_{𝔭}) = π_{j}(L)_{𝔭}. If a prime 𝔭 of O is unramified in R_{r} and R_{l}, we have R_{l,𝔭} ⊗_{O_{𝔭}} R_{r,𝔭} ≅ R_{𝔭} ⊕ R_{𝔭}, and hence L_{𝔭}^{M} = L_{M,𝔭} by definition. More generally, by the definition of the discriminant, we have

d(R_{l}/O)d(R_{r}/O)(R ⊕ R) ⊂ R_{l} ⊗ R_{r} ⊂ M ⊗_{F} M.

This shows the desired assertion. □

For a prime 𝔩 outside the discriminant of B/F, identifying B_{𝔩} with M_{2}(F_{𝔩}), we define the Eichler order of level 𝔩^{m} by

Ô_{0}(𝔩^{m})_{𝔩} = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ M_{2}(F_{𝔩})|c ∈ 𝔩^{m}O_{𝔩}\} .

Here Ô_{0}(𝔩^{0})_{𝔩} is the fixed maximal order of B_{𝔩}. We then put for ideals 𝔑 outside the discriminant of B/F, Ô_{0}(𝔑) = ∏_{𝔩} Ô_{0}(𝔩^{e(𝔩)}), where 𝔑 = ∏_{𝔩} 𝔩^{e(𝔩)} is the prime decomposition of 𝔑 (for 𝔩 ∤ 𝔑, we agree to put e(𝔩) = 0).

We identify B_{p} with M_{2}(F_{p}) so that r and l both bring (x, y) ∈ M_{p} = M_{Σ_{p}} × M_{Σ_{p}c} onto ( \begin{matrix}x & 0 \\ 0 & y \end{matrix} ) in M_{2}(F_{p}). For 𝔩|𝔣 = 𝔉 ∩ O, we take the factor 𝔏|𝔩 so that 𝔏|𝔉, and we identify B_{𝔩} with M_{2}(F_{𝔩}) bringing (x, y) ∈ M_{𝔩} = M_{𝔏} × M\overline{𝔏} to ( \begin{matrix}x & 0 \\ 0 & y \end{matrix} ) in M_{2}(F_{𝔩}). For 𝔩|D(M/F), we embed M_{𝔩} by r = l into the O_{𝔩}–order of M_{2}(F_{𝔩}) generated by the scalar in O_{𝔩} and Δ_{0}^{B}(𝔩), that is the Eichler order Ô_{0}(D(M/F))_{𝔩} of level D(M/F)_{𝔩}.

Proposition 7.2. Suppose the following three conditions:

(a) p𝔑 is prime to 𝔇 = d(R_{r}/O)d(R_{l}/O);

(b) L_{𝔣p} = Ô_{0}(𝔣𝔭^{e})_{𝔣p} ⊂ B_{𝔣p} for the conductor 𝔭^{e} = ∏_{𝔭|p} 𝔭^{e(𝔭)} of ε_{2};
(c) ε_{1,p} is trivial on O_{p}^{×} and κ = (I, 0).

Let v ∈ L(κε; \overline{W}) = \overline{W} and w ∈ L(κ^{∗}ε^{∗}; \overline{W}) = \overline{W}. Then ϕ : L → \overline{W} given by ϕ(γ) = [γv, w] is a \overline{W} –integral linear combination of functions of the form ϕ_{1} ⊗ ϕ_{2} for functions ϕ_{j} : L^{j} → \overline{W} such that

(1) ϕ_{1}(x) = ϕ_{1,p}(x_{p})ϕ_{1}^{(p)}(x^{(p)}) (resp. ϕ_{2}(x) = ϕ_{2,p}(x_{p})ϕ_{2}^{(p)}(x^{(p)})), where we embed x ∈ M into M_{p} × M^{(p)} by x ↦ (x_{p}, x^{(p)}) and for a ℤ–module X ⊂ B, X^{(p)} = X ⊗_{ℤ} ℤ̂^{(p)} with ℤ̂^{(p)} = ∏_{ℓ∤p} ℤ_{ℓ};

(2) ϕ_{2,p} ( \begin{matrix}b \\ d \end{matrix} ) = ε_{2}(d) if ( \begin{matrix}b \\ d \end{matrix} ) ∈ O_{p}^{2} and vanishes outside O_{p}×O_{p}^{×} ⊂ O_{p}^{2} = L_{p}^{2};

(3) ϕ_{1,p} is the characteristic function of L_{p}^{1} ≅ O_{p} × 𝔭^{e}O_{p};

(4) ϕ_{j}^{(p)} (j = 1, 2) factors through the finite quotient L^{j}/𝔣𝔇L^{j} of L^{j,(p)};

(5) the function ϕ_{j} is supported on L^{j} and has values in \overline{W}.

Proof. We regard ϕ as a function of B_{𝔸}^{(∞)} = B_{p} × B_{𝔸}^{(p∞)} supported on L̂ so that ϕ(b) = ϕ_{p}(b_{p})ϕ^{(p)}(b^{(p)}) for ϕ_{p} = ϕ|_{B_{p}} and ϕ^{(p)} = ϕ|_{B_{𝔸}^{(p∞)}}. We identify B_{p} with

M_{2}(F_{p}) = M_{p} ⊕ M_{p} = ( \begin{matrix}R_{Σpc} & R_{Σpc} \\ R_{Σp} & R_{Σp} \end{matrix} ) .

Then ϕ_{p} ( \begin{matrix}a & b \\ c & d \end{matrix} ) = ε_{1}(a)ε_{2}(d)[v, w] if ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Ô_{0}(𝔭^{e})_{p}. This shows the desired assertion for ϕ_{p}.

As for the component outside p, we only need to prove that the characteristic function χ_{L^{(p)}} of L^{(p)} is a finite \overline{W} –linear combination of tensor products of \overline{W} –integral locally constant functions. Note that any additive character L^{M}/L_{M} → \overline{W}^{×} is a tensor product of \overline{W} –integral valued additive characters of L^{M}/L_{M} = L^{M,(p)}/L_{M}^{(p)}, because [L^{M} : L] is a product of primes dividing the discriminant 𝔇 by the proposition. We then have χ_{L} = [L^{M} : L]^{−1} ∑_{ψ} ψ, where ψ running through all additive characters of L^{M,(p)}/L^{(p)}. Note that ψ = ψ_{1} ⊗ ψ_{2} with locally constant additive characters of ψ_{j} : L^{j} → \overline{W}^{×}.

Thus we may take ϕ_{1}(x_{p}, x^{(p)}) = ϕ_{1,p}(x_{p})ψ_{1}(x^{(p)})ε_{2}^{(p)}(x^{c}^{(p)}) and ϕ_{2}(y_{p}, y^{(p)}) = ϕ_{2,p}(y_{p})ψ_{1}(y^{(p)})ε_{1}^{(p)}(y^{(p)}) for (x, y) ∈ L^{1} ⊕ L^{2}. Since ψ_{j} (resp. ε_{j}^{(p)}) factors through L^{j}/𝔇L^{j} by Lemma 7.1 (resp. L^{j}/𝔣L^{j} by definition), we conclude that ϕ_{j}^{(p)} factors through L^{j}/𝔣𝔇L^{j}. □

Let B = M_{2}(F). We choose two fractional ideals 𝔅_{1} and 𝔅_{2} of M. Then we decompose 𝔄 = Oz_{1} + 𝔞z_{2} and 𝔅 = Ow_{1} + 𝔟w_{2} with z_{0} = z_{1}/z_{2} ∈ ℌ^{I} and w_{0} = w_{1}/w_{2} ∈ ℌ^{I}. The regular representation l of R on 𝔅_{1} given by l(α) ( \begin{matrix}z_{0} \\ 1 \end{matrix} ) = ( \begin{matrix}z_{0}α \\ α \end{matrix}) gives an embedding of R into

O_{L}^{l} = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) |a, d ∈ O, b ∈ 𝔞, c ∈ 𝔞^{−1}\} .

Similarly we define an embedding r : R ↪ O_{L}^{r} replacing z_{0} by w_{0}, where

O_{L}^{r} = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) |a, d ∈ O, b ∈ 𝔟, c ∈ 𝔟^{−1}\} .

We consider the tensor product: 𝔅_{1} ⊗_{O} 𝔅_{2} and L = l(𝔅_{1})v · r(𝔅_{2}) ⊂ M_{2}(F) for a suitable v ∈ M_{2}(F).

We want to determine the factors of L_{M} and L^{M}. Since L^{1} is the projection of L to the first factor M of M ⊗_{F} M = M ⊕ M, writing the projection to the first factor as (a ⊗ b) ↦ a^{c}b (so the projection to the second factor is given by (a ⊗ b) ↦ a^{c}b^{c}), we have L^{1} ≅ 𝔅_{1}^{c}𝔅_{2} and L^{2} ≅ 𝔅_{1}^{c}𝔅_{2}^{c}.

Since R ⊗_{O} R can be identified with

{(a, b) ∈ R ⊕ R|a ≡ b mod 𝔡(M/F)}

inside R ⊕ R ⊂ M ⊗_{F} M for the relative different 𝔡(M/F) for M/F, we see that L_{1} ≅ 𝔅_{1}^{c}𝔅_{2}𝔡(M/F) and L_{2} ≅ 𝔅_{1}^{c}𝔅_{2}^{c}𝔡(M/F).

Remark 7.1. We analyze the choice of v locally at primes 𝔭|p of F when 𝔅_{j,𝔭} = R_{𝔭} for j = 1, 2. Since the prime ideal 𝔭 is split into 𝔓𝔓^{c} with 𝔓 ∈ Σ_{p} in M, by choosing the base (e_{1}, e_{2}) for e_{1} = (1, 0), e_{2} = (0, 1) of R_{𝔭} = R_{𝔓^{c}} ⊕R_{𝔓} over O_{𝔭}, we may assume that l(α) = r(α) = ( \begin{matrix}α^{c} & 0 \\ 0 & α \end{matrix} ). Then we choose v to be b = ( \begin{matrix}1 & 1 \\ 1 & 1 \end{matrix} ). By computation, we have

l(α)b · r(β^{c}) = ( \begin{matrix}α^{c} & 0 \\ 0 & α \end{matrix} ) b ( \begin{matrix}β & 0 \\ 0 & β^{c} \end{matrix} ) = ( \begin{matrix}α^{c}β & α^{c}β^{c} \\ αβ & αβ^{c} \end{matrix} ) .

This shows that l(R_{𝔭})b · r(R_{𝔭}) = M_{2}(O_{𝔭}), and regarding M_{2}(O_{𝔭}) as an R_{𝔭}– module via αx = l(α)x, we find

M_{2}(O_{𝔭}) = ( \begin{matrix}R_{𝔓^{c}} & R_{𝔓^{c}} \\ R_{𝔓} & R_{𝔓} \end{matrix} ) .

Take O_{𝔭} basis w = (w_{1}, w_{2}) and z = (z_{1}, z_{2}) of R_{𝔭} in M so that w ≡ z ≡ (e_{1}, e_{2}) mod 𝔭^{m} for m ≥ e(𝔭) for e(𝔭) as in Proposition 7.2.

We define p(z, w) = z_{2}w_{2}p(z_{0}, w_{0}) and [u; z, w] = S(u, p(z, w)) (the homogeneous form of [u; z_{0}, w_{0}]). Then we find [b; z, w] = (z_{1} − z_{2})(w_{2} − w_{1}) and that [b; z, w] is a 𝔭–adic unit.

7.3. Proof. We first suppose that [F : ℚ] is even. Then we have a definite quaternion algebra B_{/F} with d(B/F) = 1 and I_{B} = ∅. We write G_{/ℚ} for the algebraic group associated to B^{×}.

We fix a maximal order O_{B} and identify Ô_{B} with M_{2}(Ô) once and for all. Thus Ô_{0}(𝔑) ⊂ Ô_{B} is an open compact subring. We have U_{0}^{B}(𝔑) = Ô_{0}(𝔑)^{×}. We fix complete representative sets {a_{1}, …, a_{h}} for G(ℚ)\G(𝔸)/U_{0}^{B}(𝔑)G(ℝ)F_{𝔸}^{×} with a_{i,𝔑p} = a_{∞} = 1 and Z ⊂ (F_{𝔸}^{×})^{(𝔑p∞)} for Cl_{F} = F_{𝔸}^{×}/F^{×}Ô^{×}F_{∞}^{×}. We consider

(7.8) Δ_{ijz}(𝔑) = a_{i}^{−ι}z · Δ_{0}^{B}(𝔑)a_{j}^{ι} ∩ B, O_{ijz}(𝔑) = a_{i}^{−ι}z · Ô_{0}(𝔑)a_{j}^{ι} ∩ B (z ∈ Z)

and Γ_{0}^{i}(𝔑) = G^{1}(ℚ) ∩ a_{i}U_{0}^{B}(𝔑)a_{i}^{−1}G(ℝ),

where G^{1}(A) = {g ∈ G(A)|gg^{ι} = 1}. Thus Δ_{ijz}(𝔑) ⊂ O_{ijz}(𝔑). Note here that {a_{i}z|z ∈ Z}_{i=1,...,h} gives a complete representative set for G(ℚ)\G(𝔸)/U_{0}^{B}(𝔑)G(ℝ).

Let ϕ ∈ H^{0}(Y_{0}^{B}(𝔑), L(κε; \overline{W})); so, we may regard ϕ : G(𝔸) → L(κε; \overline{W}) with ϕ(γxu) = u^{ι}ϕ(x) for u ∈ U_{0}^{B}(𝔑)F_{𝔸}^{×}G(ℝ)F_{𝔸}^{×} and γ ∈ G(ℚ). Similarly, we choose ϕ^{∗} ∈ H^{0}(Y_{0}^{B}(𝔑), L(κ^{∗}ε^{∗}; \overline{W})). Then

(ϕ, ϕ^{∗})\overline{W} = ∑_{i=1}^{h}[ϕ(a_{i}), ϕ^{∗}(a_{i})].

Pick y ∈ F_{𝔸}^{×} with y_{p} = y_{∞} = 1. Supposing ξyd is integral, we consider 𝕋(ξyd) for 0 ≪ ξ ∈ F. By (unr), we have d_{p} = 1. We choose a decomposition

U_{0}^{B}(𝔑) ( \begin{matrix}ξyd & 0 \\ 0 & 1 \end{matrix} ) U_{0}^{B}(𝔑) = ⨆_{ϖ} ϖU_{0}^{B}(𝔑).

Here we can choose ϖ so that ϖϖ^{ι} = ξyd, because

U\U ( \begin{matrix}y & 0 \\ 0 & 1 \end{matrix} ) U/U = U_{0}^{B}(𝔑)\U_{0}^{B}(𝔑) ( \begin{matrix}y & 0 \\ 0 & 1 \end{matrix} ) U_{0}^{B}(𝔑)/U_{0}^{B}(𝔑)

writing U = {u ∈ U_{0}^{B}(𝔑)|uu^{ι} = 1}. Thus ϖ_{p}ϖ_{p}^{ι} = ξ. Then

ϕ|𝕋(ξyd)(x) = ξ^{−κ_{2}} ∑_{ϖ} ϖ_{p𝔑}ϕ(xϖ^{−ι}).

Since a_{i}ϖ^{−ι} ∈ ⊔_{j,z}G(ℚ)a_{j}z · U_{0}^{B}(𝔑)G(ℝ), we can write a_{i}ϖ^{−ι} = γ_{i}a_{j}u_{i}z for γ_{i}^{−ι} ∈ Δ_{ijz}(𝔑) and u_{i} ∈ U_{0}^{B}(𝔑)G(ℝ). Thus we have, writing 𝔞_{i} = N_{B/F} (a_{i})Ô ∩ F and 𝔶 = yÔ ∩ F, 𝔞_{j}𝔶𝔡𝔷^{2}ξ = N(γ_{i}^{−1})𝔞_{i} 𝔷 = zÔ ∩ F; in other words, 𝔶^{−1}𝔡^{−1}𝔞_{i}𝔞_{j}^{−1}𝔷^{−2} is generated by a totally positive element α_{ijz} ∈ F prime to p𝔑. Thus we have

ξ = α_{ijz}γ_{i}^{−1}γ_{i}^{−ι} up to totally positive units.

Then we see, up to totally positive units,

ϕ|𝕋(ξyd)(a_{i}) = ξ^{−κ_{2}} ∑_{ϖ} ϖ_{p𝔑}ϕ(a_{i}ϖ^{−ι}) = α_{ijz}^{−κ_{2}} ∑_{γ_{i}} N_{B/F} (γ_{i})^{κ_{2}}γ_{i}^{−ι}ϕ(a_{j}).

Here, extending ε : U_{0}(𝔑) → \overline{ℚ}^{×} to ε : U_{0}(𝔑)F_{𝔸}^{×} → \overline{ℚ}_{p}^{×} by the p–adic avatar ε̂_{+} : F_{𝔸}^{×}/F^{×} → \overline{ℚ}_{p}^{×} of the central character ε_{+}, we have

N_{B/F} (γ_{i})^{κ_{2}}γ_{i}^{−ι}ϕ(a_{j}) = ε(γ_{i}^{−ι})ϕ(a_{j}) (σ(γ_{i}^{−1}) ( \begin{matrix}X_{σ} \\ Y_{σ} \end{matrix} )) ,

which is p–integral if ϕ(a_{j}) is in L(κε; \overline{W}).

Since B is totally definite (|I_{B}| ≤ 1 and |I_{B}| ≡ [F : ℚ] mod 2 ⇒ I_{B} = ∅), \overline{Γ}_{0}^{i}(𝔑) = Γ_{0}^{i}(𝔑)/O^{×} is a finite group. We then put e_{i} = |\overline{Γ}_{0}^{i}(𝔑)|. Defining

Θ_{ijz}(v, w) = \frac{1}{e_{i}e_{j}} ∑_{γ∈Δ_{ijz}(𝔑)∩Supp(ε)} N_{B/F} (γ)^{−κ_{2}}[γv, w]q^{α_{ijz}γγ^{ι}}

for v ∈ H^{0}(Γ_{0}^{j}(𝔑), L(κε; \overline{W})) and w ∈ H^{0}(Γ_{0}^{i}(𝔑), L(κ^{∗}ε^{∗}; \overline{W})) (and rewriting γ_{i}^{−ι} as γ), we find for y ∈ F_{𝔸}^{×} with y_{p} = 1

(7.9)  f(ϕ ⊗ ϕ^{∗})(y) = N(y)^{−1} ∑_{i,j,z;𝔞_{i}𝔞_{j}^{−1}𝔷^{−2}∼𝔶𝔡} α_{ijz}^{−κ_{2}}Θ_{ijz}(ϕ(a_{j}), ϕ^{∗}(a_{i})),

where 𝔞 ∼ 𝔟 indicates that the two ideals belong to the same strict class in F. Here Θ_{ijz} is a theta series of the O–lattice Δ_{ijz}(𝔑) and is a Hilbert modular form of weight κε on Γ_{0}(𝔑; 𝔶) for 𝔶 = F ∩ yÔ. Since the pairing: [ , ] is p–integral valued on L(κε; \overline{W}) × L(κ^{∗}ε^{∗}; \overline{W}) and α_{ijz} is prime to p𝔑, the theta series has p–integral Fourier coefficients (except possibly for the constant term). The constant term does not show up if ϕ ∈ H_{n.ord}^{0}(Y_{0}^{B}(𝔑), L(κε; W)) ⊂ H_{cusp}^{0}(Y_{0}^{B}(𝔑), L(κε; W)). Thus restricting ϕ to the ordinary part, f(ϕ⊗ϕ^{∗}) has to be cuspidal (cf. [H88a] Theorem 6.2) and hence, the constant term vanishes. We may forget about the integrality problem stemming from the denominator: e_{i}e_{j}.

We choose an ideal 𝔄 of M with N_{M/F} (𝔄) ∼ 𝔶. We choose α ≫ 0 with 𝔶^{−1}𝔡^{−1}N_{M/F} (𝔄) = (α). Then we consider the theta series defined in (6.6):

θ(λ; 𝔄) = ∑_{γ∈𝔄^{−1}} λ(γ)(γγ^{c})^{−κ_{2}}q^{αγγ^{c}}

for a Hecke character λ of conductor ℭ with λ(α) = α^{κ_{1}Σ+κ_{2}Σc} if α ≡ 1 mod ℭ. Strictly speaking, we need to divide the above series by |μ(M)| (see (6.6)), but |μ(M)| is prime to p by the unramifiedness of p in M/ℚ. So we forget about |μ(M)|. Here we have freedom of choosing 𝔄 in its ideal class (by changing α ≫ 0 suitably).

We define the reversed Petersson inner product (f, g) = ⟨g, f⟩ = \overline{⟨f, g⟩} to make it linear with respect to the right variable g. By the variable change z ↦ −\overline{z}, we have

(7.10)  (f, g) = ⟨f_{c}, g_{c}⟩ for f_{c}(z) = \overline{f(−\overline{z})}.

Unless the following condition is met:

(7.11)  κ = (I, 0) and (λ^{−})^{∗}(𝔓^{c}) ≡ 1 mod 𝔪_{W} for some 𝔓 ∈ Σ_{p},

we have proven in [H05d] Proposition 5.6 the following equality up to units in \overline{W}:

(MT)  L_{p}(λ̂^{−}) = \frac{(2πi)^{2(κ_{1}−κ_{2})}W_{p}(λ^{−})(f(λ), f(λ))_{𝔑}}{Ω^{2(κ_{1}−κ_{2})}} ∈ \overline{W} ,

where W_{p}(λ) = ∏_{𝔓∈Σ_{p}} W (λ_{𝔓}) and

W (λ_{𝔓}) = N(𝔓^{−e(𝔓)})λ(ϖ_{𝔓}^{−e(𝔓)}) ∑_{u∈(R/𝔓^{e(𝔓)})^{×}} λ_{𝔓}(u)e_{M} (\frac{u}{ϖ_{𝔓}^{e(𝔓)}} )

if e(𝔓) > 0 and W (λ_{𝔓}) = 1 otherwise. We would like to show (choosing λ in the p–adic analytic family so that (7.11) does not hold)

(GL)  \frac{(2πi)^{2(κ_{1}−κ_{2})}W_{p}(λ^{−})(θ(λ; 𝔄), Θ_{ijz}(ϕ(a_{j}), ϕ^{∗}(a_{j})))_{Γ}}{Ω^{2(κ_{1}−κ_{2})}} ∈ \overline{W}

for Γ = Γ_{0}(𝔑(λ); 𝔶) and the optimal CM period Ω defined in (7.4), as long as ϕ ∈ V (R) and ϕ^{∗} ∈ V^{∗}(R) for R = R_{P} as in Corollary 6.4 for P associated to λ.

We write O_{i}(𝔑) for O_{iiz}(𝔑) with z = 1. We choose an embedding i_{0} : M ↪ B. We may then realize B as

B = \{( \begin{matrix}a^{c} & b^{c} \\ bη & a \end{matrix} ) |a, b ∈ M\}

with O_{B} containing ( \begin{matrix}a^{c} & b^{c} \\ bη & a \end{matrix} ) if a, b ∈ R. We define i_{1}(a) = ( \begin{matrix}a^{c} & 0 \\ 0 & a \end{matrix} ) ∈ B.

For primes 𝔩 split in M/F, we assume that our identification B_{𝔩} ≅ M_{2}(F_{𝔩}) is induced by completing 𝔏–adically the above expression of B choosing one prime factor 𝔏|𝔩 in M. Taking a_{1} = 1, we find that i_{1}(R) ⊂ O_{1}(𝔑) if 𝔑 is made of primes split in M/F. Suppose now that 𝔑 contains primes non-split in M/F. For a given finite set S of primes, we can conjugate the embedding i_{1} by a norm 1 element u_{𝔩} (𝔩 ∈ S) so that ui_{1}u^{−1}(R_{S}) ⊂ O_{1}(𝔑)_{S} (O_{1}(𝔑)_{S} = O_{1}(𝔑) ⊗_{O} O_{S} for the localization O_{S} = ∏_{𝔩∈S} O_{𝔩}). By the strong approximation theorem, choosing one prime 𝔮 of F, we can write u = γu′ with γ ∈ G(ℚ) and u′ ∈ U_{0}^{B}(𝔑)B_{𝔮}^{×}. Thus changing i_{1} by γi_{1}γ^{−1}, we may assume that for any given 𝔑 that i_{1}(R_{1}) ⊂ O_{1}(𝔑) for an O–order R_{1} ⊂ R of 𝔮–power conductor. We identify M_{𝔸}^{×} with the image in G(𝔸) under i_{1}.

If d(M/F) ≠ 1, we find b_{1}, …, b_{j} in M_{𝔸}^{×} so that N_{M/F} (b_{j}) gives a complete representative set for F^{×}\F_{𝔸}^{×}/Ô^{×}(F_{𝔸}^{×})^{2}. By the reduced norm map: N_{B/F} : G(𝔸) → F_{𝔸+}^{×}, we have a surjection:

G(ℚ)\G(𝔸)/U_{0}^{B}(𝔑)G(ℝ)F_{𝔸}^{×} ↠ F_{+}^{×}\F_{𝔸+}^{×}/Ô^{×}(F_{𝔸}^{×})^{2}.

Thus we can choose {a_{i} = b_{j}s_{k}} = {b_{j}} × {s_{k}} so that N_{B/F} (s_{k}) = 1. Then again by the strong approximation theorem, we can write s_{k} = γ_{k}u_{k} with u_{k} ∈ U_{0}^{B}(𝔑)B_{𝔮}^{×} and γ_{k} ∈ G(ℚ). Since b_{j} commutes with i_{1}(R_{1}), conjugation by b_{j} does not alter i_{1}. Then defining i_{j} : M → B by γ_{k}i_{1}γ_{k}^{−1} and putting R_{j} to be the inverse image under i_{j} of i_{j}(M) ∩ O_{j}(𝔑), we find that R_{j} is an O–order of M of 𝔮–power conductor.

Suppose now that d(M/F) = 1. In this case, the image of M_{𝔸}^{×} in the class group F_{+}^{×}\F_{𝔸+}^{×}/Ô^{×}F_{∞+}^{×} under the norm map is of index two; so, we need to add one more element b′ ∈ G(F_{𝔮}) with N_{M/F} (b) generating 𝔮O_{𝔮}, choosing the prime 𝔮 to be inert in M/F. Then the representatives a_{i} can be chosen as b_{j}s_{k} or b_{j}b′s_{k} for s_{k} ∈ SL_{2}(F_{𝔮}) and b_{j} ∈ M_{𝔸}^{×}. Thus, by the same argument as above, we find again an O–order R_{j} of 𝔮–power conductor and an embedding i_{j} : R_{j} ↪ O_{j}(𝔑). We have now proven:

Lemma 7.3. Let the notation be as above. By choosing a prime ideal 𝔮 of F outside any given finite set of primes, we can embed the order O + 𝔮^{m}R ⊂ M of 𝔮–power conductor into O_{j}(𝔑) for all j = 1, 2, …, h, if the conductor 𝔮^{m} is sufficiently deep.

We write R_{j} for O_{j}(𝔑) ∩ R. By the above lemma, we assume that R_{j} is of conductor 𝔮^{m(j)}. We choose later 𝔮 in a way optimal to our proof. We regard L_{ijz} = O_{ijz}(𝔑) as R_{i} ⊗_{O} R_{j} module by i_{1}: (α, β)b = αbβ. Since M ⊗_{F} M = M ⊕ M, writing 1_{r} (resp. 1_{l}) the idempotent of left and right factors, we split O_{ijz}(𝔑) ⊂ L_{ijz}^{M} = 1_{r}L_{ijz} ⊕ 1_{l}L_{ijz}. The index [L_{ijz}^{M} : L_{ijz}] is a product of a power of 𝔮 and primes ramifying in M/F, which we can choose to be prime to p. Then as studied in Subsection 7.2, we can write Θ_{ijz} of level 𝔑 as a p–integral linear combination of θ(ϕ_{1})θ(ϕ_{2}) of theta series of L_{ijz}^{r} = 1_{r}L_{ijz} and L_{ijz}^{l} = 1_{l}L_{ijz}, respectively. The functions ϕ_{k} (k = 1, 2) can be chosen to be p–integral.

We now bound the level of θ(ϕ_{k}). To make the argument simple, first assume that i_{1}(R) ⊂ O_{1}(𝔑), a_{i} = b_{i′} and a_{j} = b_{j′}, and we choose that b_{i′} so that b_{i′,𝔩} = 1 for all primes 𝔩|𝔑p · d(M/F)𝔮. Note that b_{i′}z · O_{0}(𝔑)b_{j′}^{−1} = 𝔷𝔟_{i′} ⊗ 𝔟_{j′}^{−1} as R ⊗_{O} R–modules for 𝔟_{i′} = (b_{i′} R̂ ∩ M), we find from the discussion at the end of the previous section that L_{ijz}^{1} = 𝔷𝔟_{i′}^{c}𝔟_{j′}^{−1} and L_{ijz}^{2} = 𝔷𝔟_{i′}^{c}𝔟_{j′}^{−c}. Thus we find that 𝔶𝔡 = 𝔞_{i}^{−1}𝔞_{j}𝔷^{2} = N_{M/F} (L_{ijz}^{1}) = N_{M/F} L_{ijz}^{2}.

As explained in the introduction, we take φ with ψ = φ^{−}. We may assume that the weight κ of f(φ) is (I, 0). We than take a weight κ member f(λ) of the p–adic family (associated with φ: λ̂|_{G_{tor}(ℭ)} = φ) with complex multiplication by M. To avoid (7.11) (⇔ (MT)), we choose ε so that it is non-trivial at all 𝔭|p. Replacing φ by φη for a finite order character η : Gal(\overline{ℚ}/F) → \overline{W}^{×} does not alter the anticyclotomic part φ^{−}. By a theorem of Chevalley ([Ch]), we can choose η so that η_{𝔩} = λ_{𝔩}^{−1} on the inertia group at 𝔩 for every prime 𝔩 in any given finite set of prime ideals. Thus we may assume

(7.12)  λ has conductor prime to Σ_{p}c.

Write 𝔑 = 𝔑(λ). Under this assumption, κ = κ^{∗}, ε^{∗} = ε^{−1} and [v, w] = vw by identifying L(κε; \overline{W}) = \overline{W} (on which Δ_{0}(𝔑) acts via multiplication by ε) and L(κ^{∗}ε^{∗}; \overline{W}) = \overline{W}. Then

[γϕ(a_{j}), ϕ^{∗}(a_{i})] = ε(γ_{𝔣p})ϕ(a_{j})ϕ^{∗}(a_{i}).

Regarding the character ε : Δ_{0}^{B}(𝔑)_{𝔣p} → \overline{W}^{×} as a function ε_{ijz} of B ⊗_{ℚ} 𝔸^{(∞)} supported on Δ̂_{ijz}(N) = a_{i}^{ι}zΔ_{0}^{B}(𝔑)a_{j}^{−ι} by ε_{ijz}(x) = ε(x_{𝔣p}) (Δ̂_{ijz}(N)_{𝔣p} = Δ_{0}^{B}(N)_{𝔣p}), the function χ_{ijz} : γ ↦ ε(γ)[γϕ(a_{j}), ϕ^{∗}(a_{i})] is the function ε_{ijz} multiplied by the p–integral constant: ϕ(a_{j})ϕ^{∗}(a_{i}). Write down χ_{ijz} as a sum χ_{ijz} = ∑_{ϕ_{1},ϕ_{2}} ϕ_{1} ⊗ ϕ_{2} for finitely many p–integral locally constant functions ϕ_{1} : L_{ijz}^{1} → \overline{W} and ϕ_{2} : L_{ijz}^{2} → \overline{W}. By Proposition 7.2, ϕ_{2,p}(x_{p}) = λ_{Σ_{p}}(x_{Σ_{p}}) on R_{Σ_{p}c} ×R_{Σ_{p}}^{×} and is supported by (R_{Σ_{p}c} ×R_{Σ_{p}}^{×}) ⊂ L_{ijz,p}^{2} (and ϕ_{1,p} is the characteristic function of L_{ijz,p}^{1} = R_{Σ_{p}c} × 𝔭^{e}R_{Σ_{p}}).

By the proof of Proposition 7.2, we find that ϕ_{k}^{(p)} (k = 1, 2) factors through L_{ijz}^{k}/𝔡(M/F)𝔣L_{ijz}^{k}. Thus θ(ϕ_{k}) is at least automorphic with respect to the congruence subgroup Γ_{0}(𝔑(λ); 𝔶) ∩ Γ(d(M/F)^{2}; 𝔶), where

Γ(𝔑; 𝔶) = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ_{0}(𝔑; 𝔶)|a ≡ d ≡ 1 mod 𝔑\} .

This follows from the fact that ϕ_{k} as above is a linear combination of p–integral functions χ of the lattice (𝔷𝔟_{j′}^{c}𝔟_{i′}^{−1}) modulo (𝔷𝔟_{j′}^{c}𝔟_{i′}^{−1}𝔡(M/F)) for a sufficiently large m and the fact that θ(χ) = ∑_{ξ∈M} χ(ξ)q^{α_{ijz}ξξ^{c}} has the level as described above.

More generally, when a_{i} = b_{i′}s and a_{j} = b_{j′}s′ for s or s′ with norm 1 in B_{𝔮}^{×}, R_{i} and R_{j} could have conductor a power of 𝔮; so, the same argument yields that θ(ϕ_{k}) is on Γ = Γ_{0}(𝔑(λ); 𝔶) ∩ Γ(d(M/F)^{2}𝔮^{m}; 𝔶) for a sufficiently large m.

As seen in (6.6), the 𝔶–component of f(λ) is given by a p–integral finite sum |μ(M)|^{−1} ∑_{𝔄} λ(𝔄)θ(λ; 𝔄) of theta series of the form:

θ(λ; 𝔄) = ∑_{ξ∈𝔄^{−1}} λ(ξ^{(Ξ∞)})q^{αξξ^{c}},

where 𝔄𝔄^{c} = α𝔶𝔡 (with α ≫ 0 in F). Here the sum ∑_{𝔄} λ(𝔄)θ(λ; 𝔄) is over ideal classes of M whose norm isequivalent to 𝔶𝔡. By choosing v ∈ M_{2}(F) and (z_{0}, w_{0}) ∈ ℌ^{I} × ℌ^{I} as in Section 4, we identify M_{2}(F) with M ⊕ M. Then we choose L = 𝔄^{−1} ⊕ L_{ijz}^{2} as an O–lattice of M_{2}(F). Since we have freedom of changing 𝔄 in its ideal class, we may assume that the p–adic completion L_{p} = L ⊗_{ℤ} ℤ_{p} is equal to M_{2}(O_{p}) in M_{2}(F_{p}) = B_{p}, because L_{ijz}^{2} = O_{p} ⊕ O_{p}. Then L^{1} = 𝔄^{−1} and L^{2} = L_{ijz}^{2}. We take ϕ′_{1} : L^{1} → \overline{W} so that θ(ϕ′_{1}) = θ(λ; 𝔄). Then ϕ′_{1}(ξ) = λ(ξ^{(Ξ∞)}) and ϕ′_{1,Σ_{p}} = λ_{Σ_{p}}^{−1}(ξ_{Σ_{p}}), and ϕ′_{1,Σ_{p}c} is the characteristic function of R_{Σ_{p}c}.

We choose two ideals 𝔅_{1} and 𝔅_{2} of M and v ∈ M_{2}(F) very close p–adically to b = ( \begin{matrix}1 & 1 \\ 1 & 1 \end{matrix} ) ∈ M_{2}(F_{p}) as in Remark 7.1 so that L = l(𝔅_{1})v · r(𝔅_{2}) ⊂ L with L/L killed by a power of d(M/F)𝔮. Here we need to introduce another prime 𝔮, because L^{2} is stable only by an O–order in M of 𝔮–power conductor. We choose the base 𝔅_{1} = Oz_{1} + 𝔶z_{2} and 𝔅_{2} = Ow_{1} + 𝔶w_{2} again as in Remark 7.1. Let z = (z_{1}, z_{2}) and w = (w_{1}, w_{2}). Thus z_{0} = z_{1}/z_{2} and w_{0} = w_{1}/w_{2} are both in ℌ^{I}. Thus we have from Theorem 4.1 and (7.10) that:

(θ(ϕ′_{1}), θ(ϕ_{1})θ(ϕ_{2}))_{Γ} = ⟨θ(ϕ′_{1})_{c}, θ(ϕ_{1} ⊗ ϕ_{2})_{c}⟩_{Γ} = CΨ(z, w)

for a constant C ∈ \overline{W}^{×} and a congruence subgroup

Γ = Γ_{0}(𝔑(λ); 𝔶) ∩ Γ(d(M/F)^{2}𝔮^{m}; 𝔶) (m ≫ 0).

Here Ψ(z, w) is the homogeneous version of the modular form:

(7.13)  Ψ(z_{0}, w_{0}) = ∑_{0≪α∈Γ\M_{2}(F)} ϕ^{∗}^{(∞)}(ϵα)e_{F} (det(α)z_{0})θ(ϕ_{1})|_{1}α(w_{0})

for the partial Fourier transform ϕ^{∗} of ϕ = \overline{ϕ}′_{1} ∘ c ⊗ ϕ_{2}, because θ(ϕ)_{c}(z) = \overline{θ(ϕ)(−\overline{z})} = θ(\overline{ϕ ∘ c}). The constant C is prime to p (that is, i_{p}(C) ∈ W^{×}) because of the following reason: Since Ψ is of weight (I, I), the homogeneous form is given by z_{2}^{I}w_{2}^{I}Ψ(z, w) = Ψ(z_{0}, w_{0}). Since v is very close p–adically to b, we may assume that v = b. Then by Theorem 4.1 and [b; z, w] = (z_{1} −z_{2})(w_{2} − w_{1}), we have

C = z_{2}^{I}w_{2}^{I} Im(z_{0})^{−I} Im(w_{0})^{−I}[b; z_{0}, w_{0}]^{I}|[b; z_{0}, w_{0}]|^{2I}

= \frac{(z_{1} − z_{2})(w_{2} − w_{1})|(z_{1} − z_{2})(w_{2} − w_{1})|^{2}}{(z_{1}\overline{z}_{2} − \overline{z}_{1}z_{2})(w_{1}\overline{w}_{2} − \overline{w}_{1}w_{2})} ,

whose image under i_{p} is easily seen to be in W^{×} (by our choice of the base z and w as in Remark 7.1).

The local partial Fourier transform preserves p–integral Schwartz-Bruhat functions on M_{2}(F_{𝔩}) as long as 𝔩 ∤ p. Since M_{p} = M_{Σ_{p}} ⊕ M_{Σ_{p}c}, we find

M_{2}(F_{p}) = M_{p} ⊕ M_{p} = ( \begin{matrix}M_{Σpc} & M_{Σpc} \\ M_{Σp} & M_{Σp} \end{matrix} ) .

The first column is the factor M_{p} carrying \overline{ϕ}′_{1,p} ∘ c. The function \overline{ϕ}′_{1,p} ∘ c is supported on R_{p}. Since complex conjugation interchanges a and c (see Proposition 7.2), we see from (6.6) that (\overline{ϕ}′_{1,p} ∘ c) ( \begin{matrix}a \\ c \end{matrix}) = λ_{Σ_{p}}(a) (because we have assumed that λ has conductor prime to Σ_{p}^{c}: (7.12)). Similarly, ϕ_{2,p} ( \begin{matrix}b \\ d \end{matrix} ) = λ_{Σ_{p}}(d) for b ∈ R_{Σ_{p}c} and d ∈ R_{Σ_{p}}. Thus ϕ_{p}(a, b) equals to χ(a, b)λ_{Σ_{p}}(a) for the characteristic function χ of R_{p} = O_{p}^{×} × O_{p}. The partial Fourier transform is with respect to the variables “(a^{(p)}, b^{(p)})” keeps p–integrality by the Fourier inversion formula. Thus we may concentrate on the p–component. Define for each 𝔓 ∈ Σ, Φ_{λ}(x) to be equal to λ_{𝔓}(x) if x ∈ R_{𝔓}^{×} and 0 outside R_{𝔓}^{×}. Then the

Fourier transform of Φ_{λ} is given by W (λ_{𝔓})λ_{𝔓}(ϖ_{𝔓}^{e(𝔓)})Φ\overline{λ}(ϖ_{𝔓}^{e(𝔓)}x) (see [BNT] Proposition 13 in VII.7). Thus we need to prove W (λ_{𝔓}^{−})W (λ_{𝔓})λ(ϖ_{𝔓}^{e(𝔓)}) ∈ \overline{W}. This can be done as follows: writing e = e(𝔓) and ϖ = ϖ_{𝔓},

(7.14) W (λ_{𝔓}^{−})λ(ϖ^{e})W (λ_{𝔓}) = N(𝔓^{−2e})λ(ϖ^{e})λ(ϖ^{−ce})G(λ_{𝔓}^{−1})G(λ_{𝔓})

= λ_{𝔓}(−1)λ(ϖ^{−ce})λ(ϖ^{e})N(𝔓^{−e}),

where G(χ) = ∑_{u∈R/𝔓^{e}} χ(u)e_{M}(\frac{u}{ϖ^{e}}) for the conductor 𝔓^{e} of χ. Note here that the infinity type of λ is −Σ, and hence λ(ϖ^{e}) is up to unit equal to ϖ^{eΣ} which is equal to N(𝔓^{e}) up to units in \overline{W}. This shows the desired integrality.

Since the partial Fourier transform with respect to the character e_{𝔸}(ab′ − ba′) interchanges (a, b), the support of \overline{ϕ}_{p}^{∗} is contained in

( \begin{matrix}O_{p} & ϖ^{−e}O_{p}^{×} \\ O_{p} & O_{p} \end{matrix} ) = τ^{−1} ( \begin{matrix}O_{p} & O_{p} \\ 𝔭^{e}O_{p} & O_{p}^{×} \end{matrix} ) ⊂ M_{2}(F_{p}),

where τ = ( \begin{matrix}0 & −1 \\ ϖ^{e} & 0 \end{matrix} ).

The function ϕ_{1,p} is the characteristic function of R_{Σ_{p}c} × 𝔭^{e}R_{Σ_{p}}^{×}. Since τ normalizes U_{0}(𝔭^{e})_{p}, we can choose complete representative set R for

U_{0}(𝔭^{e})_{p}\ (( \begin{matrix}O_{p} & O_{p} \\ 𝔭^{e}O_{p} & O_{p}^{×} \end{matrix} ) × GL_{2}(F_{𝔸^{(p∞)}}))

such that α ∈ R can be written as τ^{−1}β with p–component β_{p} is upper triangular (e.g. [MFG] 3.1.6) with p–adic unit at the lower bottom corner. The Hecke operator UxU for x ∈ R preserves the p–integral structure of S_{κ}(Γ; W) (the space of cusp forms on Γ with W –integral Fourier coefficients). This fact follows, for example, [H88a] Theorem 4.11, and actually, if x ∈ GL_{2}(F) has upper triangular p–component with p–adic unit at the lower bottom corner, the action of θ ↦ θ|_{1}x on modular forms preserves p–integrality since it is basically given by θ(z) ↦ θ(az) for totally positive a. Thus the action of β: θ(ϕ_{1})|_{1}τ^{−1} ↦ θ(ϕ_{1})|_{1}τ^{−1}β in (7.13) preserves the p–integrality (see Theorem 4.9 in [H88a]), and θ(ϕ_{1})|_{1}τ^{−1}β has p–integral q–expansion with respect to the variable w if θ(ϕ_{1})|τ^{−1} is p–integral. Thus we need to prove that θ(ϕ)|_{1}τ^{−1} has p–integral q–expansion coefficients, in order to show Ψ(z, w) in (7.13) has p–integral q–expansion. Since θ(ϕ)|τ′ for τ′ = ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) is given by θ(ϕ̂_{1}) for the Fourier transform ϕ̂_{1} of ϕ_{1} regarding it as a function on M_{𝔸}. The p–integrality only depends on the p–part ϕ_{1,p} of ϕ_{1}. By computation, ϕ̂_{1,p} is N(𝔭^{−e}) times the characteristic function of R_{Σ_{p}c} × 𝔭^{−e}R_{Σ_{p}}. Taking ϖ^{−e} in O, we find that θ(ϕ_{1})|τ^{−1} is equal to θ(ϕ̂_{1})|_{1} ( \begin{matrix}1 & 0 \\ 0 & ϖ^{−e} \end{matrix} ) (w) = ϖ^{e}θ(ϕ̂_{1})(ϖ^{e}w) up to a p–adic unit. Since ϖ^{e}N(𝔭^{−e}) is a p–adic unit, we get the desired integrality.

By the q–expansion principle, we conclude from (7.6)

(7.15)  \frac{(2πi)^{2Σ}W_{p}(λ^{−})(θ(λ; 𝔄), θ(ϕ_{1})θ(ϕ_{2}))_{Γ}}{Ω^{2Σ}} ∈ \overline{W} .

Documenta Mathematica · Extra Volume Coates (2006) 465–532

This shows

[\overline{Γ}_{0}(𝔑(λ); 𝔶) : \overline{Γ}]\frac{(2πi)^{2Σ}W_{p}(λ^{−})(θ(λ; 𝔄), Θ_{ijz})_{Γ_{0}(𝔑(λ);𝔶)}}{Ω^{2Σ}} ∈ \overline{W} .

Write (θ(λ; 𝔄), f)_{Γ} for the inner product (θ(λ; 𝔄), f_{𝔶}) for the 𝔶–component f_{𝔶} of f ∈ S_{κ}(𝔑, ε; W). Since any f ∈ S_{κ}(𝔑(λ), ε; \overline{W}) is a \overline{W} –linear combination of Θ_{ijz} by Corollary 6.4, we conclude

[\overline{Γ}_{0}(𝔑(λ); 𝔶) : \overline{Γ}]\frac{(2πi)^{2Σ}W_{p}(λ^{−})(θ(λ; 𝔄), S_{κ}(𝔑(λ), ε; \overline{W}))_{Γ_{0}(𝔑(λ);𝔶)}}{Ω^{2Σ}} ⊂ \overline{W} .

Since [\overline{Γ}_{0}(𝔑(λ); 𝔶) : \overline{Γ}] is a factor of

N_{F/ℚ}(𝔑′)^{2} ∏_{𝔩|𝔑′} (1 − \frac{1}{N_{F/ℚ}(𝔩)^{2}} )

for 𝔑′ = d(M/F)^{2}𝔮^{m}, if p ∤ (N(𝔩) ± 1) for all primes 𝔩|d(M/F)𝔮, we get

(7.16)  \frac{(2πi)^{2Σ}W_{p}(λ^{−})(θ(λ; 𝔄), S_{κ}(𝔑(λ), ε; \overline{W}))_{Γ_{0}(𝔑(λ);𝔶)}}{Ω^{2Σ}} ⊂ \overline{W} .

We can choose 𝔮 (by unramifiedness of p in F/ℚ and p ≥ 5) so that

p ∤ (N_{M/F} (𝔮) ± 1).

Thus if p ∤ (N_{M/F} (𝔩)±1) for all primes 𝔩|d(M/F), we conclude H(φ)|\frac{h(M)}{h(F)}L^{−}(φ) as we explained in the introduction. Here H(φ) is the congruence power series with respect to the nearly ordinary Hecke algebra h(𝔑(φ), ε_{φ}; W) interpolating h_{κ}^{n.ord}(𝔑(λ), ε_{λ}; W) (for all κε ∈ A(𝕀)). Thus H(φ) divides thecongruence power series H in [HT1] but could be smaller if ℭ ∩ ℭ^{c} contains non-trivial prime factor. In [HT1], we had an extra factor Δ(M/F; ℭ) which is equal to the product of the Euler factors of L(s, α)L(s, φ^{−1}φ_{c}) for primes outside p in ℭ ∩ ℭ^{c}. This comes from the formula of the inner product of θ(λ) in [HT1] Theorem 7.1. After doing the same computation for f(λ) of smaller level instead of θ(λ) and writing k = κ_{1} − κ_{2} + I (see [H05d] (5.5)), we get the exact formula, if λ^{−} has split conductor:

(7.17) (f(λ)^{u}, f(λ)^{u})_{𝔑(λ)}

= D · N_{F/ℚ}(𝔑(λ))2^{−2k+1}π^{−(k+I)}Γ_{F} (k + I)L(1, Ad(f(λ)))

under the terminology of [HT1] Section 7 without any error terms. Here D = N(𝔡) is the discriminant of F/ℚ.

Here is how to remove the condition: p ∤ (N_{M/F} (𝔩) ± 1) for primes 𝔩 in the discriminant d(M/F). The idea is to make quadratic base-change (and then descent). As a target of the base-change, we can find a totally real quadratic extension F′/F unramified at p such that d(M′/F′) for the composite M′ = MF′ does not contain prime factors as above. Then for M′/F′, we get the assertion. We later choose F′ more carefully so that we can effectively descend back to F again. Let χ be the character Gal(F′/F) ≅ {±1} restricted to Gal(\overline{ℚ}/M). Suppose that we find a character η of Gal(\overline{ℚ}/M) of conductor ℭ′ such that η^{c−1} = χ.

We can always assume that η is of order prime to p by taking the Teichmüller lift of (η mod 𝔪\overline{W}). Let Γ_{M} be as in the introduction and we consider the universal character φ̃ : G(ℭ) → W [[Γ_{M}]] with branch character φ. Put Ψ : Gal(\overline{F} /F) → GL_{2}(𝕀) be the induced Galois representation Ind_{M}^{F} φ̃. Then we have

Ad(Ψ) ≅ α ⊕ Ind_{M}^{F}(ψ̃) for α = ^{(\underline{M/F} )}.

Thus

Ad(Ψ) ⊗ χ ≅ αχ ⊕ Ind_{M}^{F}(φ̃η^{−1}φ̃η_{c}).

By Fujiwara’s “R = T ” theorem [Fu] (actually its 𝕀–adic version: [HMI] Theorem 3.59), under the assumption (h1-4), the congruence power series H(φ) gives the characteristic power series of the Selmer group

Sel(Ad(Ψ)) = Hom(Cl^{−}, ℚ_{p}/ℤ_{p}) ⊗_{ℤ} 𝕀 ⊕ Sel(ψ),

where Cl^{−} = Cl_{M}/Cl_{F} for the class groups Cl_{M} (resp. Cl_{F}) of M (resp. F).

We need to argue more for the character φη, because φη may not satisfy the condition (h2). We choose F′ so that F′_{𝔩} = M_{𝔩} for all 𝔩|2pd(M/F) and F′/F ramifies outside 2d(M/F) only at primes which split in M/F. This is possible for the following reason: We take an element δ ∈ O so that M = F [\sqrt{δ}]. Then we take a high power 𝔞 = (2pd(M/F))^{m} so that any element u ∈ F with u ≡ 1 mod 𝔞 is a square in F_{𝔩} for all 𝔩|2pd(M/F). Then for the infinite set Ξ = {ε ∈ O|ε ≡ δ mod 𝔞, ε ≫ 0}, we can find an infinite set of primes 𝔮 = (εδ^{−1}) which splits in M/F. Then we define F′ = F [\sqrt{ε}]. By our choice, (ε) = 𝔮(δ), and hence if a prime outside 2d(M/F) ramifies in F′/F, it has to be 𝔮, which splits in M/F.

We shall show that for the above choice of F′, φη satisfies (h2). In fact, suppose that 𝔩 remains prime in M/F. Then if η_{𝔩}^{c−1} = χ_{𝔩} ≠ 1, then χ_{𝔩} has to ramify, and hence F′/F ramifies at 𝔩. By our choice of F′, 𝔩 splits in M/F, a contradiction. If 𝔩 ramifies in M, χ_{𝔩} restricted in Gal(\overline{M}_{𝔩}/M_{𝔩}) is trivial because F′_{𝔩} = M_{𝔩}. This shows that φ_{𝔩}η_{𝔩} is c–invariant, and hence by local class field theory, it is a pull-back of a character of F_{𝔩}^{×} by the norm. Thus φη satisfies (h2), and the congruence power series H(φη) still gives the exact characteristic power series of Sel(Ad(Ψ′)), where Ψ′ = Ind_{M}^{F} φ̃η. This is the beauty of taking level 𝔑(φ) (not the deeper level: N_{M/F} (ℭ)d(M/F) taken in [HT1] and [HT2]). Writing the congruence power series for φ̂ = φ ∘ N_{M′/M} as H(φ̂), by the base change (cf. [H00] Proposition 2.4), we have (by p > 2),

Sel(Ad(Ind_{M′}^{F′} φ̃)) = Sel(Ad(Ψ)) ⊕ Sel(Ad(Ψ) ⊗ χ),

which implies

H(φ̂) = H(φ)H(φη)\frac{h(M′′)}{h(M)},

where M′′ is the third (and unique) CM quadratic extension of F inside M′ = MF′.

If χ = η^{1−c} for a Hecke character η of M, χψ is again anti-cyclotomic. We have shown in [H05d] Corollary 5.5:

(h(M)/h(F))L^{−}(ψ)|H(φ) and (h(M)/h(F))L^{−}(ψη)|H(φη),

which is enough to conclude the equality for each (by Nakayama’s lemma):

(h(M)/h(F))L^{−}(ψ) = H(φ) and (h(M)/h(F))L^{−}(ψη) = H(φη)

from (h(M′)/h(F′))L^{−}(ψ̂) = H(φ̂) we have already proven.

We now prove the anticyclotomy of χ: χ = η^{c−1}. Let χ : M_{𝔸}^{×}/M^{×} → {±1} be the quadratic idele character corresponding to M′/M. We want to have a finite order Hecke character η : M_{𝔸}^{×} → μ_{𝔑} such that η^{c−1} = χ, where η^{c}(x) = η(c(x)) for x ∈ M_{𝔸}^{×}.

Let k be a number field. By class field theory, any continuous character of Gal(\overline{ℚ}/k) can be regarded as a continuous idele character: C_{k} = k_{𝔸}^{×}/k^{×} → 𝕋, where

𝕋 = {z ∈ ℂ||z| = 1} .

A given continuous character of C_{k} is of finite order if and only if it is trivial on the identity component of the infinite part k_{∞}^{×} of k_{𝔸}^{×} (cf. [MFG] Proposition 2.2). By Artin reciprocity, any continuous character of C_{k} trivial on the identity component of k_{∞}^{×} ⊂ k_{𝔸}^{×} can be viewed as a (finite order) character of Gal(\overline{ℚ}/k) canonically.

Looking at the exact sequence:

1 → M^{×} → M_{𝔸}^{×} → C_{M} → 1,

by Hilbert’s theorem 90 applied to M^{×} and Gal(M/F) = ⟨c⟩, we find

H^{0}(Gal(M/F), C_{M}) = C_{F} ,

and the kernel of c−1 : x ↦ x^{c−1} is given by C_{F}. A character ϕ : C_{M} → 𝕋 is of the form ϕ = η^{c−1} if and only if ϕ is trivial on C_{F}. Since Gal(M′/F) ≅ (ℤ/2ℤ)^{2}, we find a quadratic character α of C_{F} such that χ = α ∘ N_{M/F}. This shows that χ(x) = α(xx^{c}) = α(x^{2}) = 1 for x ∈ C_{F}. Thus we can write χ = η^{c−1} for a character η : C_{M} → 𝕋.

To have η factor through the Galois group of the maximal abelian extension of M, we need to show that η can be chosen so that its restriction to M_{∞}^{×} is trivial. Since χ = η^{c−1} is trivial on M_{∞}^{×}, η is trivial on (M_{∞}^{×})^{c−1} = Ker(N_{M/F} : M_{∞}^{×} → F_{∞}^{×}). Thus η|_{M_{∞}^{×}} factors through N_{M/F} : M_{∞}^{×} → F_{∞+}^{×}. Replacing η by η(ξ ∘ N_{M/F}) for a Hecke character ξ of F, we may assume that η is trivial on M_{∞}^{×}. This finishes the proof for even degree field.

We now assume that F has odd degree. The above trick of taking totally real quadratic extensions F′/F reduces the proof to the even degree case of M′/F′; so, we get the theorem. □

As we have seen that ψ = φ^{−} if and only if ψ is trivial on C_{F}. If ψ is anticyclotomic, then ψ(x^{c}) = ψ(x^{−1}) (⇔ ψ = 1 on N_{M/F} (M_{𝔸}^{×})). Thus ψ|_{C_{F}} is either the character of M/F or trivial. Since ψ is a Hecke character of M_{𝔸}^{×} of finite order, its infinity type is trivial; so, ψ has to be trivial on C_{F}. This shows

(7.18) If ψ is anticyclotomic, then ψ = φ^{−} for a Hecke character φ of M.

We leave the reader to show that we can take φ to be of finite order (see [HMI] Lemma 5.31).

References

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[PAF] H. Hida, p–Adic Automorphic Forms on Shimura Varieties, Springer Monographs in Mathematics. Springer, New York, 2004 (a list of errata

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[H04] H. Hida, Non-vanishing modulo p of Hecke L–values, in: “Geometric Aspects of Dwork’s Theory, II” (edited by Alan Adolphson, Francesco Baldassarri, Pierre Berthelot, Nicholas Katz, and Francois Loeser), Walter de Gruyter, 2004, pp. 735–784 (preprint downloadable at

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[H05a] H. Hida, p–Adic automorphic forms on reductive groups, Astérisque 298 (2005), 147–254 (preprint downloadable at

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[H05b] H. Hida, The integral basis problem of Eichler, IMRN 34 (2005) 2101–

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[H05c] H. Hida, The Iwasawa μ–invariant of p–adic Hecke L–functions, preprint, 2004 (preprint downloadable at www.math.ucla.edu/~hida)

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Haruzo Hida

Department of Mathematics UCLA Los Angeles CA 90095-1555 USA hida@math.ucla.edu