p-Adic L-Functions for Unitary Shimura Varieties

I: Construction of the Eisenstein Measure

To John Coates, with admiration

Michael Harris^{1}, Jian-Shu Li^{2}, Christopher M. Skinner^{3}

Received: September 30, 2005 Revised: November 15, 2006

Abstract. We construct the Eisenstein measure in several variables on a quasi-split unitary group, as a first step towards the construction of p-adic L-functions of families of ordinary holomorphic modular forms on unitary groups. The construction is a direct generalization of Katz’ construction of p-adic L-functions for CM fields, and is based on the theory of p-adic modular forms on unitary Shimura varieties developed by Hida, and on the explicit calculation of non-degenerate Fourier coefficients of Eisenstein series.

2000 Mathematics Subject Classification: Primary 11F33, 11R23; Secondary 14G35

Introduction

This is the first of a projected series of papers devoted to studying the relations between p-adic L-functions for GL(n) (and unitary groups), congruences between stable and endoscopic automorphic forms on unitary groups, and Selmer groups for p-adic representations. The goals of these papers are outlined in the survey article [HLS]. The purpose of the present installment is to prepare the ground for the construction of p-adic L-functions in sufficient generality for the purposes of subsequent applications to congruences and Selmer groups.

1Institut de Mathématiques de Jussieu, U.M.R. 7586 du CNRS; Member, Institut Universitaire de France

2Partially supported by RGC-CERG grants HKUST6115/02P, HKUST601805 and a grant from the Chinese Academy of Science

3Partially supported by a grant from the National Science Foundation and a fellowship from the David and Lucile Packard Foundation

Documenta Mathematica · Extra Volume Coates (2006) 393–464

394 Michael Harris, Jian-Shu Li, Christopher M. Skinner

p-adic L-functions for unitary Shimura varieties, I 395

The first general conjectures on the construction of p-adic L-functions for ordinary motives were elaborated by Coates in [Co]. The conjectured p-adic analytic functions of [Co] interpolate the quotients of normalized values of Lfunctions at critical points, in the sense of Deligne. The normalization proceeds in two steps. The critical values are first rendered algebraic, by dividing by their Deligne periods. Next, they are p-stabilized: the Euler factors at p and ∞ are modified according to a complicated but explicit recipe. Coates’ conjecture is that the resulting values are p-adically interpolated by a p-adic analytic function of Iwasawa type, associated to a p-adic measure. In our setting, the Deligne period is generally replaced by a certain Petersson norm or an algebraic multiple thereof; the relation of this Petersson norm to the Deligne period is discussed at length in [H3]. In [Pa], Panchishkin points out that Coates’ recipe can be adapted unchanged for motives satisfying a condition weaker than ordinarity, which he calls admissibility and which Perrin-Riou and Greenberg have called the Panchishkin condition. Although this is somewhat obscured by the automorphic normalization, we work in the generality of Panchishkin’s admissibility condition. Panchishkin also conjectures the existence of more general p-adic L-functions in the absence of admissibility; we do not address this question. We work with automorphic forms on the unitary groups of hermitian vector spaces over a CM field K, with maximal totally real subfield E. We assume every prime of E dividing p splits in K; we also impose a hypothesis (1.1.2) linking primes above p to signatures of the unitary group at real places of E. Unitary groups, unlike GL(n), are directly related to Shimura varieties. We show that the special values of L-functions of automorphic forms on unitary groups satisfy the congruences needed for the construction of p-adic L-functions by appealing to the fact that the corresponding Shimura varieties are moduli spaces for abelian varieties of PEL type. In this our approach is directly modeled on Katz’s construction [K] of p-adic L-functions for Hecke characters of CM fields; indeed, for groups of type U(1) our results reduce to those of Katz. The starting point of Katz’s construction is Damarell’s formula and its generalizations due to Shimura, which relate the values of arithmetic Eisenstein series at CM points to special values of L-functions of arithmetic Hecke characters. A generalization of Damerell’s formula in higher dimensions is the construction of standard L-functions of unitary groups by the doubling method. This was first developed systematically in the article [PSR] of Piatetski-Shapiro and Rallis, though special cases had been discovered independently by Garrett, and a more thorough development in classical language is contained in the books [S97, S00] of Shimura. The local theory for unitary groups was ignored in [PSR] but was worked out in [L2] and [HKS]. Our p-adic L-functions are actually attached to Hida families of nearly ordinary modular forms on a unitary group G = U(V). As in [K], the main step is the construction of an Eisenstein measure on a large unitary group H, attached to the sum of two copies V ⊕ (−V) of V. The hermitian form on −V has been multiplied by −1, so that H is quasi-split and its associated Shimura variety has a point boundary component, stabilized by a maximal parabolic subgroup, the Siegel parabolic. The Eisenstein series attached to the Siegel parabolic are the direct generalizations of the classical Eisenstein series on GL(2). The Eisenstein measure is a p-adic measure on a product T of copies of ℤ_{p}^{×} with values in the algebra of p-adic modular forms on H interpolating such Eisenstein series. The theory of p-adic modular forms on H was developed by Hida in [Hi04, Hi05]. As in [K], these forms belong to the algebra of functions on the Igusa tower, which is a rigid analytic étale covering of the ordinary locus of the Shimura variety attached to H. The existence of the Eisenstein measure relies crucially on the irreducibility of the Igusa tower; this was established in some generality by Hida, though easier arguments due to Chai and Hida himself suffice for the case at hand (cf. [Ch, Hi06]). The Eisenstein measure associates, by integrating over T with respect to this measure, p-adic modular forms to continuous functions on T. The integrals of characters of T of finite order, which determine the measure, are classical holomorphic (Siegel) Eisenstein series on H and as such are associated to explicit functions (“sections”) belonging to degenerate principal series induced from characters of the Siegel parabolic. These sections factor as tensor products of local sections over the primes of E. At almost all finite primes the local sections are unramified and present no difficulty, and we simplify the theory by choosing local sections at ramified primes, other than those dividing p, that are insensitive to p-adic variation of the character of T. With our choice of data, the Fourier coefficients of the Eisenstein series at a chosen point boundary component also factor over primes. All the work in constructing the Eisenstein measure then comes down to choosing local data at primes above p such that the corresponding local coefficients satisfy the necessary Kummer congruences. Our strategy for choosing local data follows [K] in making use of a partial Fourier transform. Unlike in [K], our construction is systematically adelic and isolates the local considerations at p. The Eisenstein measure is designed to pair with Hida families – on G × G, not on G itself – and thus depends on several variables, considerably complicating the calculations. The doubling method was used by Böcherer and Schmidt in [BS] to construct standard p-adic L-functions for Siegel modular forms. They do not use p-adic modular forms; their approach is to construct the p-adic measure directly in terms of normalized special values of complex L-functions. Their approach applies to all critical values, unlike the present paper, which avoids reference to non-holomorphic differential operators (and their p-adic analogues). Presumably their techniques work for quasi-split unitary groups as well. We have not attempted to compare our results where they can be compared, namely in the local analysis at the prime p, since our group is locally isomorphic to GL(2n), in principle much simpler than a symplectic group. As predicted by Coates, the shape of the modified Euler factor at a prime v dividing p depends on the p-adic valuations of the eigenvalues of Frobenius at v. On the other hand, as in [H3], the fact that a critical value of the standard L-function is an algebraic multiple of a period of an arithmetic modular form on the doubled group G×G – in other words, the Petersson norm of an arithmetic modular form on G – can be expressed in terms of Hodge numbers. Then the Panchishkin condition, applied to the standard L-function for GL(n)_{K}, roughly states that, for each v dividing p, the modified Euler factor at v is given by a natural partition of the Frobenius eigenvalues at v that corresponds to the signature of the unitary group at real places assigned to v by Hypothesis (1.1.2). The form of the modified Euler factor at p is thus linked to the real form of G. This is reflected in the fact that the natural embedding of the Shimura variety attached to G × G in that attached to H in general does not define a map of Igusa towers. In order to pair p-adic modular forms on H with p-adic modular forms on G × G, the natural embedding has to be replaced by a padic translation (cf. (2.1.11)), which is exactly what is needed to provide the expected modification of the Euler factor. The main innovation of our construction concerns the zeta integral at p. As in [K], the use of a partial Fourier transform to define local data at p with the appropriate congruence properties to construct the Eisenstein measure is precisely what is needed to obtain the modified Euler factor at p directly as a local zeta integral, up to some volume factors. For U(1), this was proved by Katz by direct computation. In general, we obtain the result as an immediate application of the local functional equation for the Godement-Jacquet integral representation of the standard L-function of GL(n). These calculations are presented in Part II.

Why the present construction is not altogether satisfactory. The first reasons have to do with somewhat arbitrary restrictions on the scope of our result. We have only constructed the p-adic L-function for holomorphic automorphic forms of scalar weight. Moreover, for any fixed scalar weight, we have only studied the p-adic interpolation of the critical values at a fixed point s_{0}, though we allow the inertial characters at p to vary freely. Relaxing these restrictions would require the construction of the p-adic analogues of the classical non-holomorphic weight-raising operators of Maass, as in [K]. There is no doubt that Katz’s constructions can be generalized, but the paper was already quite long without this additional generality, which is not necessary for our intended applications to Selmer groups. Moreover, although Garrett has determined the special values of the archimedean zeta integrals up to rational factors in general, his method does not permit identification up to p-adic units in general.^{4} As mentioned above, our choice of Eisenstein measure is insensitive to p-adic variation at ramified primes not dividing p, and the resulting p-adic L-function is missing its local Euler factors at the corresponding primes. A construction taking ramification away from p into account would probably require at the very least a p-integral version of the Godement-Jacquet theory of local zeta integrals (at primes not dividing p), based on Vignéras’ modular representation theory of GL(n) over local fields. We hope to return to this question in the future. Ignoring a finite number of Euler factors at places prime to p introduces a bounded error in expected applications to Selmer groups. There are also local restrictions at primes dividing p. Working with general r-dimensional Hida families, we expect the values of our p-adic L-functions at algebraic (classical) points to be explicitly related to normalized special values of archimedean L-functions. The normalization involves dividing by a complex period invariant, to which we return momentarily. Our main results assert this to be the case under certain restrictions: at algebraic points corresponding to r-tuples of characters lying in a certain positive cone (the regular case); or when r = 1, where the Hida family is just the family of twists by characters composed with the determinant; or finally when r ≤ 2 but only along an “anticyclotomic” direction. This is sufficient for our intended applications but is certainly less than optimal, and we hope to be able to relax at least the anticyclotomic condition in the final version of Part II. The restrictions allow us to identify the specialization of the Hida family at an algebraic point as an explicit vector in a principal series representation, which can then be used as a test vector in a local zeta integral. The most serious defect of our construction is global. The conjectures of [Co] and [Pa] are expressed in the language of motives, and relate the special values of the p-adic L-function to the special values of the quotient of an archimedean L-function by a complex period invariant attached to the motive. In order for this relation to make sense, one needs to know that this quotient is an algebraic number, and so the statements of the conjectures of [Co] and [Pa] require Deligne’s conjecture on the critical values of motivic L-functions as a preliminary hypothesis.^{5} Our archimedean L-functions are attached to automorphic forms rather than to motives, and the period invariants are defined, as in Shimura’s work, as (suitable algebraic multiples of) Petersson norms of arithmetic holomorphic modular forms on the appropriate Shimura varieties. The conjectural relation of these Petersson norms to Deligne’s motivic periods, up to rational factors, is discussed in [H3], at least when the ground field is ℚ. Partial results in this direction are obtained in [H4, H5], using an elaborate inductive argument, based on the theta correspondence, for establishing period relations between automorphic forms on unitary groups of different signatures. It is not beyond the realm of imagination that such techniques can eventually provide relations between Petersson norms up to integral factors, though it may well be beyond the limits of anyone’s patience. Even the relatively favorable case of Shimura curves, where no products of periods are involved, required extraordinary efforts on the part of Prasanna [Pr]. However, and this is the most important point, even assuming integral period relations for Petersson norms, we still need to compare products of Petersson norms to motivic periods. When n = 2 and the ground field is ℚ, Hida realized long ago that the ratio of the Petersson norm to the motivic period generates the congruence ideal, and is itself the specialization of a p-adic L-function. When n > 2 we do not know how to use the automorphic theory to study the analogous ratios.

4Shimura calculates the archimedean zeta integrals precisely in [S97], but only for forms of scalar weight. His scalar weights, unlike those treated here, are non-constant functions on the set of real primes; thus he is forced to work with Maass operators.

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5The more general conjectures of Perrin-Riou concern non-critical values of motivic Lfunctions, and the normalizing periods are defined by Beilinson’s conjectures; in general, this is far beyond the scope of the automorphic theory as it presently stands.

Documenta Mathematica · Extra Volume Coates (2006) 393–464

Contents of this paper. To keep this first paper in the series to a reasonable length we have decided to break it into two parts. Part I, by recalling the theory of p-adic modular forms on unitary groups and constructing the Eisenstein measure, sets up the ground work for the construction of the p-adic L-functions. More precisely, §1 recalls the theory of modular forms on unitary Shimura varieties, a theory ultimately due to Shimura but presented here in the setting of [H1]. We present the theory of p-adic modular forms on unitary Shimura varieties in §2, following Hida’s generalization of the constructions of Deligne and Katz for GL(2). Most of these results are at least implicitly due to Hida, but we have highlighted some special features adapted to the embedding of Igusa towers mentioned above. The calculation of the local coefficients at p of Eisenstein series occupies the greater part of §3, the rest of which is concerned with the local coefficients at the remaining places, and the relation of local to global coefficients, due essentially to Shimura. We conclude §3 with the construction of the Eisenstein measures. Part II will develop Hida theory for p-adic modular forms on unitary groups G, carry out the related zeta-integral calculations from the doubling method, and complete the construction of p-adic L-functions. It will also establish a dictionary between the motivic and automorphic normalizations, and in particular will verify that the modified Euler factors at p are as predicted in [Co] and [Pa].

Acknowledgments

The authors, working on three separate continents, are grateful to the institutions that have provided us the opportunity to meet occasionally. Harris began working on this project in 2001, while visiting the Centre de Recerca Matemàtica at the Universitat Autònoma de Barcelona, and continued as a guest of the Hong Kong University of Science and Technology in 2001. Li and Skinner were guests of the Institut de Mathématiques de Jussieu in 2002 and 2003, respectively. Harris and Skinner were both invited to Harvard University in the spring of 2004. It remains to express our thanks to the colleagues who have encouraged this project in its various manifestations. We thank Eric Urban, who corrected some of our initial misconceptions and thereby incited us to consider more general p-adic L-functions than we had originally intended to construct. Ching-Li Chai has patiently answered our many questions related to the theory of p-adic modular forms. We are especially grateful to Haruzo Hida, who has generously and unhestitatingly shared his expertise and advice since the beginning of our collaboration, and has been a permanent source of encouragement, while warning us that the project would be with us longer than we might have expected.

Finally, it is a special privilege and pleasure to dedicate this article to John Coates. His insights and taste have shaped our field for a generation; his generosity, especially in supporting young researchers, is unparalleled; and his personal charm is in large part responsible for making number theory a most enviable profession.

0. Notation and Conventions

Let G be a reductive algebraic group over the number field F. If v is a place of F we let G_{v} = G(ℚ_{v}); if v is archimedean we let 𝔤_{v} = Lie(G_{v})_{ℂ}. We let G_{∞} denote ∏_{v|∞} G_{v}, the product being over all archimedean places of F, and let 𝔤_{∞} = ∏_{v|∞} 𝔤_{v}. In practice we will denote by K_{∞} a subgroup of G_{∞} which is maximal compact modulo the center of G. We let \overline{ℚ} denote the algebraic closure of ℚ in ℂ. Thus for any number field L we identify the set Σ_{L} of complex embeddings of L with the set Hom(L, \overline{ℚ}). Let ℂ_{p} denote the completion of an algebraic closure of ℚ_{p}, with integer ring O_{ℂ_{p}}. We choose once and for all an embedding incl_{p} : \overline{ℚ} → ℂ_{p}, and let ℤ̄_{(p)} = incl_{p}^{−1}(O_{ℂ_{p}}), the corresponding valuation ring. When necessary, we denote by incl_{∞} the given inclusion of \overline{ℚ} in ℂ. Via this pair of inclusions, any embedding τ : L → ℂ of a number field L gives rise to an embedding τ_{p} = incl_{p} ∘ τ : L → ℂ_{p}.

(0.1) Unitary groups over CM fields. Let E be a totally real number field of degree d over ℚ and let K be a totally imaginary quadratic extension of E, with ring of integers O. Let c ∈ Gal(K/E) denote the non-trivial automorphism, and ε_{K} the character of the idele classes of E associated to the quadratic extension K. We fix a CM type of K, i.e. a subset Σ ⊂ Σ_{K} such that Σ ∐ Σc = Σ_{K}. Let V be an n-dimensional K-vector space, endowed with a non-degenerate hermitian form < •, • >_{V} relative to the extension K/E. For each σ ∈ Σ_{K}, < •, • >_{V} defines a hermitian form < •, • >_{σ} on the complex space V_{σ} = V ⊗_{K,σ} ℂ. We let (a_{σ}, b_{σ}) denote the signature of the form < •, • >_{σ}. Note that (a_{cσ}, b_{cσ}) = (b_{σ}, a_{σ}) for all σ ∈ Σ_{K}. The hermitian pairing < •, • >_{V} defines an involution c̃ on the algebra End(V) via

(0.1.1)  < a(v), v′ >_{V} =< v, ac̃(v′) >,

and this involution extends to End(V ⊗_{ℚ} R) for any ℚ-algebra R. We define ℚ-algebraic groups U(V) = U(V, < •, • >_{V}) and GU(V) = GU(V, < •, • >_{V}) over ℚ such that, for any ℚ-algebra R, (0.1.2)

\begin{matrix}U(V)(R) = \{g ∈ GL(V ⊗_{ℚ} R) | g · c̃(g) = 1\}; \\ GU(V)(R) = \{g ∈ GL(V ⊗_{ℚ} R) | g · c̃(g) = ν(g) for some ν(g) ∈ R^{×}\}. \end{matrix}

Thus GU(V) admits a homomorphism ν : GU(V) → 𝔾_{m} with kernel U(V). There is an algebraic group U_{E}(V) over E such that U(V) ⟶^{∼} R_{E/ℚ}U_{E}(V), where R_{E/ℚ} denotes Weil’s restriction of scalars functor. This isomorphism identifies automorphic representations of U(V) and U_{E}(V). The groups U(V) (resp. GU(V)) are all inner forms of the same quasi-split unitary group (resp. unitary similitude group), denoted U_{0} (resp. GU_{0}). The group U_{0} is of the form U(D_{0}, χ̃(∗)_{0}) where D_{0} is the matrix algebra and χ̃(∗)_{0} is an appropriate involution. Then U_{0,∞} ≅ U(\frac{n}{2}, \frac{n}{2})^{[E:ℚ]} if n is even, U_{0,∞} ≅ U(\frac{n−1}{2}, \frac{n+1}{2})^{[E:ℚ]} if n is odd.

(0.2) Haar measures. The bulk of this article and its companion, Part II, is devoted to calculations involving Fourier transforms, zeta integrals, and Petersson inner products of automorphic forms on the groups U(V) of (0.1). The integrals are defined with respect to local and adelic Haar measures. The natural adelic Haar measure on G = U_{E}(V) is Tamagawa measure d^{τ}g, associated to an invariant top differential ω rational over E on G. Let δ(E) denote the discriminant of E. The adelic Tamagawa measure d^{τ}g factors up to normalization as a product of local measures

(0.2.1)  d^{τ}g = |δ(E)|^{− \frac{dim G}{2}} L(1, ε_{K})^{−1} ∏_{v} d^{τ}g_{v}

where d^{τ}g_{v} is the measure defined by ω_{v} if v is real and by L_{v}(1, ε_{K})ω_{v} if v is finite. The Tamagawa number τ(G) of G is vol(G(ℚ)\G(A), d^{τ}g) = 2. For finite v the volume of any compact open set with respect to d^{τ}g_{v} is always a rational number. An alternative measure, traditionally used in the calculation of zeta integrals, is dg = ∏_{v} dg_{v} where dg_{v} = d^{τ}g_{v} for archimedean v but dg_{v} is chosen to give volume 1 to a hyperspecial maximal compact subgroup K_{v} at almost all finite primes. Let S_{G} be the set of finite places v of E where ω_{v} is not an O_{E,v} generator of the module of top differentials; in particular, the group G is unramified at v ∉ S_{G} and so G(E_{v}) has hyperspecial maximal compacts. The relation is

(0.2.2)  d^{τ}g_{v} = L_{v}(1, ε_{K}) · A_{v}(n)dg_{v}, A_{v}(n) = (q_{v})^{− dim G} · |G_{v}(k_{v})|

where G_{v} is the smooth reductive group scheme over Spec(O_{E,v}) associated to the hyperspecial subgroup K_{v}. If for v ∈ S_{G} (which includes the finite places where G has no hyperspecial maximal compact) we arbitrarily set d^{∗}g_{v} = dg_{v} for v ∈ S_{G}, then

(0.2.3) vol(G(ℚ)\G(A), dg))/vol(G(ℚ)\G(A), d^{τ}g) =

= |δ(E)|^{\frac{dim G}{2}} · ∏_{v∉S_{G}} A_{v}(n)^{−1} = ∏_{j=1}^{n} L^{S_{G}}(j, ε_{K}^{j})

where L^{S_{G}} denotes the partial L-function with the factors at S_{G} removed.

Given an open compact subgroup K ⊂ G(A_{f}), we let dμ_{K}(g) be the Haar measure that gives each connected component of _{K}S(G) = G(ℚ)\G(A)/K_{∞}K total volume 1, for any maximal compact subgroup K_{∞} ⊂ G(ℝ). When V is totally definite, so G(ℝ) = K_{∞}, dμ_{K}(g) is counting measure on the finite set KS(G). In general,

(0.2.4)  dμ_{K}(g) = \frac{C(G, K)}{2}d^{τ} g

where the class number C(G, K) = |π_{0}(_{K}S(G)| can be determined explicitly.

1. Automorphic forms on unitary groups

(1.1) Ordinary primes for unitary groups.

Let (V, < •, • >_{V}) be a hermitian pairing as in (0.1). Let p be a rational prime which is unramified in K (hence in particular in the associated reflex field E(V)), and such that every divisor of p in E splits completely in K. Choose an inclusion incl_{p} : \overline{ℚ} ↪ ℂ_{p} as above. Composition with incl_{p} defines an identification Σ_{K} ⟶^{∼} Hom(K, ℂ_{p}), hence for every τ ∈ Hom(K, ℂ_{p}) we can define a signature

(1.1.1)  (a_{τ} , b_{τ}) = (a_{σ}, b_{σ}) if τ = incl_{p} ∘ σ.

We assume the triple (Σ, incl_{p}, (a_{σ}, b_{σ})_{σ∈Σ_{K}}) to be ordinary in the following sense:

(1.1.2) Hypothesis. Suppose σ, σ′ ∈ Σ have the property that incl_{p} ∘ σ and incl_{p} ∘ σ′ define the same p-adic valuations. Then a_{σ} = a_{σ′}.

When a_{σ} = n for all σ ∈ Σ – this is the definite case, to be described in detail later – or more generally, when a_{σ} = a for all σ ∈ Σ is constant, this comes down to the following hypothesis, used by Katz in the case n = 1:

(1.1.3) Hypothesis. For σ, σ′ ∈ Σ, the p-adic valuations defined by incl_{p} ∘ σ and incl_{p} ∘ σ′ċ are distinct.

As Katz observes in [K], our hypotheses on p guarantee that Σ’s satisfying (1.1.2) exist. We let Σ_{p} denote the set incl_{p} ∘ σ | σ ∈ Σ} of ℂ_{p}-embeddings of K. Complex conjugation c acts on the set of primes of K dividing p, and the set of all such primes of K is the disjoint union

(1.1.4)  Hom(K, ℂ_{p}) = Σ_{p} ∐ Σ_{p}c.

Hypothesis (1.1.2) was suggested by Fargues, who observed that it is equivalent to the condition that the completion of the reflex field of the Shimura variety attached to G (see §1.2) at the place defined by incl_{p} is ℚ_{p}. This is in turn equivalent, by a criterion of Wedhorn [We], to the condition that the ordinary locus of the completion of the Shimura variety at incl_{p} is non-empty (see (2.1.7), below). We reformulate the elementary condition (1.1.2) in equally elementary terms. We have a canonical isomorphism V_{p} ⟶^{∼} ⊕_{w|p} V_{w} where V_{w} = V ⊗_{K} K_{w}. Let V_{p,Σ_{p}} and V_{p,Σ_{p}c} be, respectively, the preimages of the subspaces ⊕_{w|p,w∈Σ_{p}}V_{w} and ⊕_{w|p,w∈Σ_{p}c}V_{w}, where the notation w ∈ Σ_{p} designates those w such that w is the valuation determined by some σ_{p} ∈ Σ_{p}. In particular,

(1.1.5)  V_{p,Σ_{p}} ⟶^{∼} ⊕_{w|p,w∈Σ_{p}} V_{w}.

The fact that all primes of E above p split in K/E is equivalent to the condition that the ℚ_{p}-vector space V_{p} = R_{K/ℚ}V ⊗_{ℚ} ℚ_{p} decomposes ℚ_{p}-rationally as V_{p} = V_{p,Σ_{p}} ⊕ V_{p,Σ_{p}c}. The decomposition (1.1.5) is tautologically ℚ_{p}-rational. For any w dividing p, let

Σ_{w} = {σ ∈ Σ_{K} | σ_{p} = w}.

Equivalent to (1.1.2) is the hypothesis:

(1.1.6) Hypothesis. There is a ℚ_{p}-rational K ⊗_{ℚ} ℚ_{p}-submodule W (sig) ⊂ V_{p,Σ_{p}} (resp. F^{0}V_{p} ⊂ V_{p}) such that W (sig) = ⊕_{w|p,w∈Σ_{p}}W (sig)_{w} (resp. F^{0}V_{p} = ⊕_{w|p}F^{0}V_{w}) with dim W (sig)_{w} = a_{σ} for any σ ∈ Σ (resp. dim F^{0}V_{w} = a_{σ} for any σ ∈ Σ_{w}).

In the definite case we just have W (sig) = V_{p,Σ_{p}}. Under hypothesis (1.1.2) we write (a_{w}, b_{w}) = (a_{σ}, b_{σ}) for any σ ∈ Σ_{w}.

(1.2) Shimura varieties and automorphic vector bundles.

Let (V, < •, • >_{V}) be an n-dimensional hermitian space over K as above. As in [H4], we let −V denote the space V with hermitian form < •, • >_{−V} = − < •, • >_{V} and 2V denote the doubled hermitian space V ⊕ (−V) with hermitian form the sum of < •, • >_{V} and < •, • >_{−V}. We define U(2V) and GU(2V) as in (1.1); in particular, GU(2V) denotes the rational similitude group. The stabilizer in U(2V) of the direct sum decomposition 2V = V ⊕ (−V) is naturally isomorphic to the product U(V) × U(−V), embedded naturally in U(2V). Similarly, the stabilizer in GU(2V) is isomorphic to the subgroup G(U(V) × U(−V)) ⊂ GU(V) × GU(−V), defined by

(1.2.1) G(U(V) × U(−V)) = {(g, g′) ∈ GU(V) × GU(−V) | ν(g) = ν(g′)}.

Let (W, < •, • >_{W}) be any hermitian space over K. To the group G = GU(W) one can canonically attach a Shimura datum (G, X), and hence a Shimura variety Sh(W) = Sh(G, X), as follows. For each σ ∈ Σ, let (a_{σ}, b_{σ}) denote the signature of the hermitian form induced by < •, • >_{W} on the complex space W_{σ} = W ⊗_{K,σ} ℂ. Let GU(a_{σ}, b_{σ}) = GU(W_{σ}) denote the real unitary similitude group, and let X^{a_{σ},b_{σ}} denote the GU(a_{σ}, b_{σ})(ℝ)-conjugacy class of homomorphisms R_{ℂ/ℝ}𝔾_{m,ℂ} → GU(a_{σ}, b_{σ}) defined in [H4, p. 143]. The product X = X(W) = ∏_{σ∈Σ} X^{a_{σ},b_{σ}} is naturally a G(ℝ)-conjugacy class of homomorphisms R_{ℂ/ℝ}𝔾_{m,ℝ} → G_{ℝ}, and the pair (G, X) satisfies the axioms of [D] defining a Shimura variety – unless W_{σ} is definite for all σ, in which case one can attach a zero-dimensional Shimura variety to (G, X) all the same, as in [H3]. We recall that the complex-valued points of Sh(G, X) are given by

(1.2.2)  Sh(G, X)(ℂ) = \underleftarrow{lim}_{K} G(ℚ)\X × G(A_{f})/K,

where K runs over open compact subgroups of G(A_{f}). We let _{K}Sh(G, X) denote the associated variety whose complex points are given by G(ℚ)\X × G(A_{f})/K. If W′ is a second hermitian space, the above construction applies to groups of the form G(U(W) × U(W′)), defined by analogy with (1.2.1), yielding a Shimura datum (G(U(W) × U(W′)), X(W, W′)). With the above conventions, it is immediate that the natural map G(U(W) × U(W′)) → GU(W ⊕ W′) defines a map of Shimura data (G(U(W) × U(W′)), X(W, W′)) → G(U(W ⊕ W′)), X(W ⊕ W′)), hence a morphism of Shimura varieties

(1.2.3) Sh(W, W′) = Sh((G(U(W) × U(W′)), X(W, W′))) → Sh(W ⊕ W′).

When E = ℚ, this is worked out in detail in [H4]. In particular, we obtain a map

(1.2.4)  Sh(V, −V) → Sh(2V).

The group GU(2V) is always quasi-split; in particular, up to isomorphism, it does not depend on the choice of V of dimension n. The corresponding Shimura variety always has a canonical model over ℚ. The more general Shimura varieties Sh(W), Sh(W, W′) are defined over reflex fields E(W), E(W, W′), respectively, of which one can only say in general that they are contained in the Galois closure of K over ℚ. It is easy to see, however, that E(V, −V) = E(V), and the general theory of canonical models implies that the map (1.2.4) is rational over E(V). If E = ℚ then K is a quadratic imaginary field, and E(V) = K unless V is quasi-split, in which case E(V) = ℚ. When V is a definite hermitian space, E(V) is the reflex field E(K, Σ) of the CM type (K, Σ). We will be working with holomorphic automorphic forms on G, when G is of the form G = GU(W) or GU(W, W′). These are constructed as follows; for details, see [H1]. Let K_{∞} ⊂ G(ℝ) be the stabilizer of a point x ∈ X (= X(W) or X(W, W′)); thus K_{∞} contains a maximal connected compact subgroup of G(ℝ), as well as the real points of the center Z_{G} of G. In fact, K_{∞} is the group of real points of an algebraic subgroup, also denoted K_{∞}, of G, the centralizer of the torus x(R_{ℂ/ℝ}𝔾_{m,ℂ}). Moreover, the derived subgroup of G is simply connected, hence K_{∞} is connected. Hence one can speak of algebraic representations of K_{∞} and their extreme weights. If τ : K_{∞} → GL(W_{τ}) is an algebraic representation, then there exists a holomorphic vector bundle [W_{τ}] on Sh(G, X); more precisely, there exists a canonical holomorphic structure on the C^{∞} vector bundle

(1.2.5)  [W_{τ}] = \underleftarrow{lim}_{K} G(ℚ)\G(ℝ) × W_{τ} × G(A_{f})/K_{∞}K,

where K_{∞} acts on the right on G(ℝ) and on the left on W_{τ}, yielding a natural map to

\underleftarrow{lim}_{K} G(ℚ)\G(ℝ) × G(A_{f})/K_{∞}K = \underleftarrow{lim}_{K} G(ℚ)\X × G(A_{f})/K = Sh(G, X)(ℂ).

A holomorphic automorphic form on G of type τ is a global section f ∈ H^{0}(Sh(G, X), [W_{τ}]); when G contains a rational normal subgroup isogenous to SL(2)_{ℚ} one needs to add a growth condition at infinity. The representation τ is included in the notation for [W_{τ}], but is superfluous; [W_{τ}] can be defined without reference to a choice of K_{∞} (or, equivalently, a choice of p ∈ X), and has a canonical model rational over a number field E(W_{τ}), containing the reflex field E(G, X), and attached canonically to the set of extreme weights of W_{τ}. In particular, the space H^{0}(Sh(G, X), [W_{τ}]) has a canonical rational structure over E(W_{τ}). However, since we have chosen K_{∞}, we can also realize holomorphic automorphic forms of type τ as W_{τ} -valued functions on the adèle group of G via (1.2.5). Let A(G) denote the space of automorphic forms on G(ℚ)\G(A). Then (1.2.6)

H^{0}(Sh(G, X), [W_{τ}]) ⟶^{∼} A_{hol,τ} (G) := {f ∈ (A(G) ⊗ W_{τ})^{K_{∞}} | 𝔭^{−}f = 0},

canonically. Here

(1.2.7)  𝔤_{∞} = Lie(K_{∞})_{ℂ} ⊕ 𝔭^{−} ⊕ 𝔭^{+}

is the Harish-Chandra decomposition, and the choice a base point x ∈ X, and hence K_{∞} and the decomposition (1.2.7), is implicit in the notation A_{hol,τ} (G). We also write the right-hand side of (1.2.6) as

(A(G) ⊗ W_{τ})^{K_{∞}}[𝔭^{−}],

the 𝔭^{−}-torsion in (A(G) ⊗ W_{τ})^{K_{∞}}.

If X = X(V, −V) with V a definite hermitian space, then K_{∞} = GU(V, −V)(ℝ). If X = X(2V), with V again definite, we can take K_{∞} to be GU(V, −V)(ℝ) ⊂ GU(2V). With this choice, the Harish-Chandra decomposition (1.2.7) is rational over E(V, −V) = E(V) = E(K, Σ).

Restricting forms.

Let G = GU(V, −V), X = X(V, −V), G′ = GU(2V), and X′ = X(2V). Pick x ∈ X(V, −V). This determines a base point in X′ and hence K′_{∞} ⊆ G′_{∞} in addition to K_{∞}, with K_{∞} being identified with a subgroup of K′_{∞} via the canonical embedding of G into G′. Suppose τ is a one-dimensional representation of K′_{∞}. This then determines a one-dimensional representation of K_{∞} by restriction, and we obtain holomorphic vector bundles [W τ] and [W′_{τ}] on Sh(V, −V) = Sh(G, X) and Sh(2V) = Sh(G′, X′), respectively, having canonical models over the respective fields E(W_{τ}) and E(W′_{τ}). There is canonical map from the pull-back of [W′_{τ}] under the morphism (1.2.4) to [W_{τ}] and therefore a homomorphism:

(1.2.8)  res_{V,τ} : H^{0}(Sh(2V), [W′_{τ}]) → H^{0}(Sh(V, −V), [W_{τ}]).

This is rational over E(W_{τ}). Over the complex numbers (1.2.8) is compatible in the obvious way with the isomorphisms in (1.2.6) and the restriction of forms in A_{hol,τ} (G′) to G(A), which gives forms in A_{hol,τ} (G).

Connected components.

We let G = GU(V). Let C denote the algebraic group G/G^{der} over ℚ. Let G(ℝ)^{+} denote the identity component of G(ℝ), G(ℚ)^{+} = G(ℚ) ∩ G(ℝ)^{+}. For any open compact subgroup K ⊂ G(A_{f}), the set π_{0}(_{K}Sh(G, X)(ℂ)) of connected components of _{K}Sh(G, X)(ℂ) is given by \overline{G(ℚ)^{+}}\G(A_{f})/K, where \overline{G(ℚ)^{+}} denotes the closure of G(ℚ)^{+} in G(A_{f}). Let C_{K} ⊂ C(A_{f}) denote the image of K under the natural map; let C^{+} ⊂ C(A_{f}) denote the image of G(ℚ)^{+}. Now G^{der} is an inner form of the simply-connected group SL(n), hence satisfies strong approximation. It follows (cf. [D, (2.1.3.1)]) that

(1.2.9)  π_{0}(_{K}Sh(G, X)(ℂ)) = C(K) =^{def.} C(A_{f})/C_{K} · C^{+}.

We can define a Shimura datum (C, X(C)) to be the quotient of (G, X) by G^{der}. The corresponding Shimura variety Sh(C, X(C)) also has a modular interpretation in terms of level structures on certain direct factors of rank one over K of certain tensor powers of the Tate modules of abelian varieties with CM by K. The tensor power in question depends on the signatures (a_{σ}, b_{σ}). The natural map _{K}Sh(G, X)(ℂ) → C(K) = π_{0}(_{K}Sh(G, X)(ℂ)) becomes a morphism of moduli spaces. This interpretation will not be used in the sequel.

(1.3) PEL structures.

Let G = GU(V). Notation is as in the previous section. Write

K ⊗_{ℚ} \overline{ℚ} = ⊕_{σ∈Σ_{K}} \overline{ℚ}σ,

and let e_{σ} ∈ K ⊗_{ℚ} \overline{ℚ} be the corresponding orthogonal idempotents. We decompose V ⊗_{ℚ} \overline{ℚ} as a K ⊗ ℚ\overline{ℚ}-module as V ⊗_{ℚ} \overline{ℚ} = V_{Σ} ⊕ V_{Σc}, where V_{Σ} is the sum of the spaces V_{σ} = e_{σ}(V ⊗_{ℚ} \overline{ℚ}) for σ ∈ Σ, and similarly for V_{Σc}. Inside V ⊗_{ℚ} \overline{ℚ} we consider a variable K ⊗_{ℚ} \overline{ℚ}- submodule F^{0}V satisfying (1.3.0) Property. For any σ ∈ Σ_{K}, the projection F^{0}V_{σ} = e_{σ}F^{0}V of F^{0}V on V_{σ} is of dimension a_{σ}.

Let T be an indeterminate and, for x ∈ K, let P_{0}(x, T) ∈ \overline{ℚ}[T] denote the characteristic polynomial of x, acting on F^{0}V. It follows from the definition of the reflex field E(V) that P_{Σ}(x, T) ∈ E(V)[T], independently of the choice of F^{0}V. Indeed, Shimura defined E(V) to be the field generated by traces of elements of K acting on F^{0}V. Choose a purely imaginary element ℷ ∈ K, i.e. an element such that T r_{K/E}(ℷ) = 0. The form < •, • >_{V,ℷ}= ℷ· < •, • >_{V} is skew-hermitian. When we fix a prime p we will always assume ℷ to be a unit at p. Fix a compact open subgroup K ⊂ GU(V)(A_{f}). We consider the following functor from the category of schemes over E(V) to the category of sets:

(1.3.1)  S ↦ _{K}A_{V} (S) = _{K}A_{V,ℷ}(S) = {(A, λ, ι, α)}

where (1.3.1.1) A is an abelian scheme over S, viewed as an abelian scheme up to

isogeny;

(1.3.1.2) λ : A → Â is a polarization; (1.3.1.3) ι : K → End_{S}(A) ⊗ ℚ is an embedding of ℚ-algebras; (1.3.1.4) α : V (A_{f}) ⟶^{∼} V^{f}(A) is an isomorphism of K-spaces, modulo K.

Here V^{f}(A) = ∏_{ℓ} T_{ℓ}(A) ⊗ ℚ is the adelic Tate module, viewed as a ind-pro-étale sheaf over S; it’s K-structure comes from (1.2.1.3). The level K structure of (1.3.1.4) is understood in the sense of Kottwitz [Ko]. These data satisfy the usual compatibility conditions:

(1.3.1.5) The Rosati involution on End_{S}(A)⊗ℚ defined by λ fixes ι(K) and acts

as complex conjugation;

(1.3.1.6) The isomorphism α identifies the Weil pairing on V^{f}(A) with an A_{f}^{×}- multiple of the skew-symmetric pairing on V (A_{f}) defined by tr_{K/ℚ} < •, • >_{V,ℷ}.

Finally, the action induced by ι on Lie_{A/S} satisfies Shimura’s trace condition, which we state here in the equivalent formulation due to Kottwitz. Let P_{ι}(x, T) ∈ O_{S}[T] denote the characteristic polynomial of x, acting on Lie(A/S). We view E(V)[T] as a subalgebra of O_{S}[T]. The Shimura-Kottwitz condition is

(1.3.1.7)  P_{ι}(x, T) = P_{0}(x, T) ∈ O_{S}[T], ∀x ∈ K.

Two quadruples (A, λ, ι, α) and (A′, λ′, ι′, α′) are identified if and only if there is an isogeny ϕ : A → A′, commuting with ι′, prime to the level K in the obvious sense and taking α to α′, and identifying λ′ with a positive rational multiple of λ.

(1.3.2) Theorem (Shimura). For K sufficiently small, the functor (1.3.1) is representable by a quasi-projective scheme over E(V), and this is precisely the canonical model of _{K}Sh(V). As K varies, the natural maps between these functors induce the natural maps between the various _{K}Sh(V). The action of GU(V)(A_{f}) on the tower _{K}Sh(V) preserves the E(V)-rational structure.

For U ⊂ GU(V)(A_{f}) a closed compact subgroup, we write _{U}Sh(V) = \underleftarrow{lim}_{K⊃U} _{K}Sh(V), as K runs over compact open subgroups of GU(V)(A_{f}). This is simply a shorthand for referring to the full tower of the _{K}Sh(V) for K ⊂ U, and we will not need to worry about the nature of the projective limit. The above theory applies in particular to the Shimura varieties Sh(2V) and Sh(V) × Sh(−V). The Shimura variety Sh(V, −V) is defined as the subvariety of Sh(V) × Sh(−V), which parametrizes pairs of quadruples ((A, λ, ι, α), (A^{−}, λ^{−}, ι^{−}, α^{−})), determined by compatibility of polarizations in the obvious sense. As a subvariety of Sh(2V), Sh(V) × Sh(−V) is then the set of quadruples (B, μ, ι_{2}, β) which decompose as a product

(B, μ, ι_{2}, β) ⟶^{∼} (A × A^{−}, λ × λ^{−}, ι × ι^{−}, α × α^{−}).

In particular, β respects the A_{K,f}-decomposition 2V (A_{f}) = V (A_{f}) ⊕ (−V)(A_{f}). The most important level structures β for our purposes do not, however, respect this decomposition. In other words, in the applications, we will not be working with the Shimura variety Sh(V, −V) via its natural embedding in Sh(2V), but rather with a translate of the latter, cf. (2.1.11). For the remainder of this section, let G = GU(V), X = X(V), Sh = Sh(G, X). We identify

GU(V)(ℚ_{p}) = U(V)(ℚ_{p}) × ℚ_{p}^{×} ⟶^{∼} U_{E}(V)(E ⊗_{ℚ} ℚ_{p}) × ℚ_{p}^{×},

where the map to ℚ_{p}^{×} is the similitude factor and U_{E}(V) is as in (0.1). The ordinarity hypothesis (1.1.2) allows us to define subspaces V_{p,Σ_{p}} and V_{p,c·Σ_{p}} of V_{p} as in (1.1.5). The hermitian pairing

V_{p} × V_{p} → E ⊗ ℚ_{p}

determines, and is determined by, a perfect duality V_{p,Σ_{p}} ⊗V_{p,c·Σ_{p}} → E⊗_{ℚ}(ℚ_{p}) of free E ⊗_{ℚ} ℚ_{p} ⟶^{∼} ∏_{w∈Σ_{p}} K_{w}-modules. There is thus a natural isomorphism

(1.3.3)  G(ℚ_{p}) ⟶^{∼} GL(V_{p,Σ_{p}}) × ℚ_{p}^{×} ⟶^{∼} ∏_{w,Σ_{w}⊂Σ_{p}} GL(n, K_{w}) × ℚ_{p}^{×}

The indexing by w such that Σ_{w} ⊂ Σ_{p} is a reminder of the fact that several elements of Σ_{p} can correspond to the same divisor w of p. This is just a way of saying, somewhat more carefully than usual, that the unitary group at a split place is isomorphic to a general linear group. We identify G_{ℚ_{p}} with the product of algebraic groups G_{0} × GL(1), where G_{0} = GL(V_{p,Σ_{p}}) as algebraic groups and the map to GL(1), as before, is the similitude factor ν. Fix a compact open subgroup K = K_{p}×K^{p} ⊂ G(A_{f}), with K_{p} ⊂ G(ℚ_{p}), K^{p} ⊂ G(A_{f}^{p}), and let _{K}Sh denote the Shimura variety at level K. Our hypotheses imply that G_{ℚ_{p}} is an unramified group over ℚ_{p}, hence that G(ℚ_{p}) contains hyperspecial maximal compact subgroups; we assume that K_{p} is one such. Then K_{p} is the group of ℤ_{p}-points of an extension of G to a smooth group scheme, also denoted G, over Spec(ℤ_{p}). The choice of K_{p} is equivalent to the choice of a self-dual O ⊗ ℤ_{p}-lattice M_{V} ⊂ V_{p}. Let M_{V,Σ_{p}} ⊂ V_{p,Σ_{p}} be the projection of M_{V}. We can extend G_{0} to a group scheme over ℤ_{p} as G_{0} = GL(M_{V,Σ_{p}}). Then there are isomorphisms

(1.3.4) K_{p} = G(ℤ_{p}) ⟶^{∼} G_{0}(ℤ_{p}) × GL(1, ℤ_{p}) = ∏_{w,Σ_{w}⊂Σ_{p}} GL(n, O_{w}) × ℤ_{p}^{×}

compatible with the factorization (1.3.3). We also assume K^{p} is sufficiently small, in a sense we will make precise later. When G = G(2V) ⊃ G(V, −V), we choose K_{p} so that M_{2V} = M_{V} ⊕ M_{−V} with M_{V} ⊂ V ⊗ ℚ_{p} and M_{−V} ⊂ (−V) ⊗ ℚ_{p} self-dual lattices; this is equivalent to the assumption that K_{p} ∩ G(V, −V)(ℚ_{p}) is a hyperspecial maximal compact subgroup of G(V, −V)(ℚ_{p}). In (2.1) we will impose additional conditions on the choice of M_{V} in the general case.

(1.4) Automorphic vector bundles on unitary Shimura varieties, again.

Notation is as in the previous sections: G = GU(2V), resp. GU(V, −V), X = X(2V), resp. X(V, −V), and [W_{τ}] is an automorphic vector bundle on Sh(G, X). In (1.2) we have fixed the stabilizer K_{∞} ⊂ G(ℝ) of a point x ∈ X. Choose a maximal torus T_{∞} ⊂ K_{∞}, an algebraic subgroup over ℝ necessarily containing the image of x. Then T_{∞} is also a maximal torus in G. A specific choice of pair (T_{∞}, K_{∞}) can be obtained as follows. Decompose (V, < •, • >_{V}) as an orthogonal direct sum of one-dimensional hermitian spaces over K:

(1.4.1)  (V, < •, • >_{V}) = ⊕_{i=1}^{n}(V_{i}, < •, • >_{i}).

We assume the V_{i} are numbered so that, for any σ ∈ Σ, V_{i,σ} = V_{i} ⊗_{K,σ} ℂ has signature (1, 0) for i ≤ r_{σ} and signature (0, 1) for i > r_{σ}. Let −V_{i} denote V_{i} with the hermitian form − < •, • >_{i}. Let GU^{⊕_{i}}(V, −V) denote the subgroup of the torus ∏_{i} GU(V_{i})×∏_{i} GU(−V_{i}) defined by equality of similitude factors. We obtain embeddings of Shimura data (1.4.2)

(GU^{⊕_{i}}(V, −V),∏_{i}(X_{i}×X′_{i})) ↪ (GU(V, −V), X(V, −V)) ↪ (GU(2V), X(2V))

where ∏_{i}(X_{i} × X′_{i}) is an appropriate product of point symmetric spaces determined by the signatures of each V_{i} and −V_{i}. We write

Sh^{⊕_{i}}(V, −V) = Sh(GU^{⊕_{i}}(V, −V), ∏_{i}(X_{i} × X′_{i})),

the superscript ^{⊕_{i}} serving as a reminder of the choice of direct sum decomposition above. Define (GU^{⊕_{i}}(V), ∏_{i} X_{i}) ⊂ (GU(V), X(V)), GU^{⊕_{i}}(−V), ∏_{i} X′_{i}) ⊂ (GU(−V), X(−V)) analogously. The groups GU^{⊕_{i}}(V), GU^{⊕_{i}}(−V), and GU^{⊕_{i}}(V, −V), defined over ℚ, are maximal ℝ-elliptic tori in GU(V), GU(−V), and GU(V, −V) or GU(2V), and we take T_{∞} to be the group of real points of one of these tori. We can of course find K_{∞} containing T_{∞}, though K_{∞} will in general not be defined over ℚ. The Shimura data (GU^{⊕_{i}}(V), ∏_{i} X_{i}), etc., define CM points of the corresponding unitary Shimura varieties. The group T_{∞} is a maximal torus in a reductive group of type A, and we parametrize its roots in the usual way. In the case G = GU(V), G_{ℂ} is naturally isomorphic to ∏_{σ∈Σ} GL(n, ℂ) × GL(1, ℂ), the last term coming from the similitude factor. Thus the group X(T_{∞}) of characters λ of the algebraic torus T_{∞} consists of d-tuples (a_{1,σ}, …, a_{n,σ})_{σ∈Σ} of n-tuples of integers, indexed by σ ∈ Σ, together with a single integer a_{0} for the similitude factor. The (a_{j,σ}) are given by the restriction of the character λ to T_{∞} ∩ U(V), whereas a_{0} is given by the restriction of λ to the maximal ℝ-split torus in T_{∞} ∩ Z_{G}: if tI_{n} ∈ G(ℝ) is a real central element then λ(tI_{n}) = t^{a_{0}}. The parameters satisfy the relation

(1.4.3)  a_{0} ≡ ∑_{j,σ} a_{j,σ} (mod 2).

Given an ordering on the roots of the maximal torus T_{∞} ⊂ G, the dominant weights are then the characters parametrized as above, with a_{i,σ} ≥ a_{i+1,σ}, for all σ and i = 1, …, n − 1. We choose a set of positive roots containing the roots in 𝔭^{−}. The n-tuple corresponding to σ will often be written with a semi-colon (a_{1,σ}, …, a_{a_{σ},σ}; −b_{b_{σ},σ}, …, −b_{1,σ}) or occasionally (a_{1,σ}, …, a_{a_{σ},σ}; −b_{b_{σ},σ}, …, −b_{1,σ}; a_{0}) when the term a_{0} needs to be stressed, in such a way that it gives a dominant weight of the σ-factor of K_{∞} ∩ U(V)_{∞}, U(V)_{∞} ≡ ∏_{σ} U(a_{σ}, b_{σ}) if and only if

(1.4.4)  a_{1,σ} ≥ ⋯ ≥ a_{a_{σ},σ}, b_{1,σ} ≥ ⋯ ≥ b_{b_{σ},σ}

The parametrization in G = GU(2V) is the same as above, except that n is replaced by 2n and a_{σ} = b_{σ} = n. For G = GU(V, −V), we place the parameters for GU(V) and GU(−V) side by side. If K is sufficiently small, _{K}Sh(V) carries a universal abelian scheme _{K}A endowed with PEL structure of the appropriate type. Let p_{K} : _{K}A → _{K}Sh(V) denote the structure map and put

ω = ω_{V} = p_{K,∗}Ω_{_{K}A/_{K}Sh(V)}^{1}.

This is a locally-free sheaf on _{K}Sh(V) of rank dn = [E : ℚ] dim_{K} V with a natural action of O_{_{K}Sh(V)} ⊗_{ℚ} K, the K-action coming from (1.3.1.3). If we extend the ground field to contain E, then ω breaks up as ω = ⊕_{σ∈Σ}ω_{σ} corresponding to the canonical decomposition E ⊗_{ℚ} K = ⊕_{σ∈Σ}K. Each ω_{σ} is a locally-free O_{_{K}Sh(V)}-sheaf of rank n. The sheaf ⊗_{σ∈Σ}ω_{σ} is the canonical bundle associated to (0, ..., 0; 1, 0, ..., 0; 1)_{σ∈Σ}. The canonical bundles for other τ’s can be constructed as follows, again assuming the ground field contains E. Let F l(ω_{σ})_{_{K}Sh(V)} be the scheme representing the functor

S ↦ (E_{1} = ω_{σ}_{/S} ⊃ E_{2} ⊃ ⋯ ⊃ E_{n} ⊃ E_{n+1} = 0 ;

ϕ_{i} : E_{i}/E_{i+1} ⟶^{∼} O_{S}, i = 1, ..., n).

There is an obvious action of D_{σ} = 𝔾_{m}^{n}_{/_{K}Sh(V)} on F L(ω_{σ}): d = (d_{1}, ...., d_{n}) acts by multiplying ϕ_{i} by d_{i}. Let π_{σ} : F L(ω_{σ}) → _{K}Sh(V) be the structure map. For each τ_{σ} ∈ X(D_{σ}) we define a locally-free sheaf ρ_{τ_{σ}} on _{K}Sh(V) by H^{0}(U, ρ_{τ_{σ}}) = H^{0}(π_{σ}^{−1}(U), O_{F L(ω_{σ})})[τ_{σ}], where the [τ_{σ}] signifies the submodule on which D_{σ} acts through τ_{σ}. We identify each τ_{σ} with an n-tuple of integers (m_{1,σ}, ...., m_{n,σ}) in the usual way and say that such a τ_{σ} is dominant if m_{1,σ} ≥ ⋯ ≥ m_{n,σ}. Given a d-tuple τ = (τ_{σ})_{σ∈Σ} of dominant characters, let ρ_{τ} = ⊗_{σ∈Σ}ρ_{τ_{σ}}. Then we can naturally identify ρ_{τ} with [W_{τ}], where the character of T_{∞} associated to τ is (m_{1,σ}, ..., m_{a_{σ},σ}; m_{a_{σ}+1,σ}, ..., m_{n,σ}). These identifications respect the maps in (1.2.8) in the obvious way.

(1.5) Fourier expansions of modular forms.

In this section we consider the Shimura datum (GU(2V), X(2V)). The symmetric domain X(2V) is holomorphically isomorphic to the product of [E : ℚ] = |Σ| copies of the irreducible tube domain X_{n,n} of dimension n^{2} attached to the group U(n, n). Let P = P_{Δ} ⊂ G be the maximal parabolic defined in §1.5. The group of real points of P stabilizes the 0-dimensional boundary component of this product of tube domains. Fourier expansion with respect to U(ℝ) defines the q-expansion of a holomorphic automorphic form on X relative to a congruence subgroup of GU(2V, ℚ). By work of Fujiwara [F], extending the results of Chai and Faltings, one can also define q-expansions for sections of the automorphic vector bundles [W_{τ}] over _{K}𝕊 when K_{p} is hyperspecial. In [Hi04, Hi05], Hida defined q-expansions on the closed Igusa tower. We will formulate this theory in an adelic version analogous to the characteristic zero formulation in [H1, §6] and [P]. In [H1, §6] we attach a Shimura datum (G_{P} , X_{P}) to the rational parabolic subgroup P ⊂ GU(2V). The domain X_{P} is a version of the point boundary component mentioned above, and G_{P} is a torus; specifically, G_{P} is contained in the center of the standard Levi component of P. Recall the definition of G_{P}: the standard rational representation of G on R_{K/ℚ}(2V) carries a family of Hodge structures of type (−1, 0) + (0, −1), corresponding to the family of abelian varieties of PEL type over Sh(2V). In a neighborhood of the boundary component corresponding to P, this family degenerates to a mixed Hodge structure of type (0, 0) + (−1, −1). Actually, the formulation in [H1] is not quite correct: in general the boundary Shimura datum should be defined as in [Pink], where X_{P} is a homogeneous space for G_{P} (ℝ) finitely fibered over a G_{P} (ℝ) conjugacy class of homomorphisms R_{ℂ/ℝ} → G_{P,ℝ}. In the present case, G_{P} (ℝ) has two connected components, corresponding to upper and lower hermitian half-spaces, and X_{P} consists of two points. The Shimura variety Sh(G_{P} , X_{P}) is zero-dimensional, and one easily verifies it is of PEL type. Indeed, it parametrizes pairs (α_{Σ}, α_{m}) where α_{Σ} is a complete level structure on the abelian variety with complex multiplication type (K, Σ), and α_{m} is an isomorphism

α_{m} : ∏_{q} ℚ_{q}/ℤ_{q} ⟶^{∼} ∏_{q} μ_{q^{∞}}

Thus as long as one works in finite level K_{P} prime to p, there is no difficulty defining an integral model _{K_{P}} 𝕊(G_{P}) of _{K_{P}} Sh(G_{P} , X_{P}). For general level K_{P}, there is a unique normal integral model, and we define this to be _{K_{P}} 𝕊(G_{P}). We let U_{P} denote the unipotent radical of P, and let U^{∗} = Hom(U_{P} (ℚ), ℚ). This is the vector space denoted 𝔤^{−2}(ℚ)^{∗} in [H1, ]. The space U^{∗} ⊗ ℝ contains a self-adjoint cone, homogeneous under P (ℝ)/U_{P} (ℝ), and denoted C in [H1, 5.1]; we let U^{∗}(C) = U^{∗} ∩ C. Let [W_{τ}] be an automorphic vector bundle over S̄, as above. There is an automorphic vector bundle [W_{τ_{P}}] over Sh(G_{P} , X_{P}), and a map

(1.5.1)  F.J.^{P,0} : Γ(Sh(2V), [W_{τ}]) → ˆ ⨁_{β∈U^{∗}}Γ(Sh(G_{P} , X_{P}), [W_{τ_{P}}])

defined, with slightly different notation, in [H1, (6.3.3)], and in [Pink, §12]. Here ˆ⨁ is understood as the subset (f_{β}) of the direct product over β ∈ U^{∗} such that f_{α} = 0 for all but finitely many β ∉ U^{∗}(C). If F.J.^{P,0}(f) = (f_{β}) for some f ∈ Γ(Sh(2V), [W_{τ}]), then the usual Fourier expansion is written ∑ f_{β}q^{β}. The Koecher principle asserts that, for n > 0, F.J.^{P,0} is supported on U^{∗}(C), and even for n = 0 one takes care only to consider f with that property. Since C is self-adjoint, it can also be viewed as a cone in U_{P} (ℝ). One obtains a more reassuring variant of the q-expansion in the following way. Let N = dim U_{P}, and let

(1.5.2)  Λ = Λ(K^{p}) = U_{P} (ℚ) ∩ K(U, m) ⊂ U_{P} (ℚ).

Note that Λ is a lattice in U_{P} (ℚ) and does not depend on m. We choose a polyhedral cone 𝔠 ⊂ C generated by a basis {λ_{1}, …, λ_{N}} of Λ:

𝔠 = {∑_{i=1}^{N} a_{i}λ_{i} | a_{i} ≥ 0}

and let 𝔠^{∗} ⊃ U^{∗}(C) be the dual cone:

𝔠^{∗} = {v ∈ U^{∗}(C) | v(λ_{i}) ≥ 0, i = 1, … N}.

Let Λ^{∗} = Hom(Λ, ℤ), viewed as a subgroup of U^{∗}(C). Let R be a O_{v}-algebra and M a free R-module. The intersection Λ^{∗} ∩ 𝔠^{∗} is a free monoid on N generators β_{i}, i = i, …, N, and the ring of formal series

(1.5.3)  R[[q^{Λ^{∗}∩𝔠^{∗}}]] = { ∑_{β∈Λ^{∗}∩𝔠^{∗}} f_{β}q^{β}},

with f_{β} ∈ R, and with the usual multiplication rule q^{β} · q^{β′} = q^{β+β′}, is then isomorphic to R[[q^{β_{1}}, … q^{β_{N}}]]. We define the R[[q^{Λ^{∗}∩𝔠^{∗}}]]-module

M[[q^{Λ^{∗}∩𝔠^{∗}}]] = M ⊗_{R} R[[q^{Λ^{∗}∩𝔠^{∗}}]] = { ∑_{β∈Λ^{∗}∩𝔠^{∗}} f_{β}q^{β}}

where now f_{β} ∈ M for all β. Taking

M^{0} = M^{0}([W_{τ_{P}}], K_{P} (m)) = Γ(_{K_{P} (m)}𝕊(G_{P}), [W_{τ_{P}}])

for appropriate m, F.J.^{P,0} can be regarded as a map

(1.5.4) F.J.^{P,0} : Γ(_{K(U,m)}Sh(2V), [W_{τ}]) → M^{0}([W_{τ_{P}}], K_{P} (m))[[q^{Λ^{∗}∩𝔠^{∗}}]].

Letting K^{p} run over a fundamental set of open subgroups of G(A_{f}^{p}) corresponds to letting Λ^{∗} grow to a ℤ_{(p)}-lattice in U_{P} (ℚ), or equivalently to adding nth roots of the generators q^{β_{i}} of R[[q^{Λ^{∗}∩𝔠^{∗}}]] for all n prime to p.

(1.5.5) One-dimensional τ’s.

In the present article we will mainly consider W_{τ} of dimension one. More precisely, [W_{τ_{P}}] is the automorphic vector bundle associated to an algebraic character, say τ_{P}, of the torus G_{P}. Fix a base point x ∈ _{K_{P} (m)}𝕊(G_{P})(ℂ); for instance, we can take x to be the image of the element 1 ∈ GU(2V)(A) under the isomorphism G_{P} (ℚ)\G_{P} (A)/K_{P} (m) ⟶^{∼} _{K_{P} (m)}𝕊(G_{P})(ℂ). Let W_{τ_{P}} be the stalk at x of [W_{τ_{P}}]. Then H^{0}(𝕊(G_{P}), [W_{τ_{P}}]) can be canonically identified with the space M(W_{τ_{P}} (ℂ), K_{P} (m)) of W_{τ_{P}} (ℂ)-valued automorphic forms on G_{P} of infinity type τ_{P}^{−1}; i.e., the space of functions

c : G_{P} (ℚ)\G_{P} (A)/K_{P} (m) → W_{τ_{P}} (ℂ)

such that c(g_{∞}g) = τ_{P} (g_{∞})^{−1}c(g) for all g ∈ G_{P} (A) and all g_{∞} ∈ G_{P} (ℝ). Choosing a basis of W_{τ_{P}} (ℂ) identifies M(W_{τ_{P}} (ℂ), K_{P} (m)) with the space

(1.5.5.1) X_{τ_{P}} (G_{P}; K_{P} (m)) = = {c : G_{P} (ℚ)\G_{P} (A)/K_{P} (m) → ℂ | c(g_{∞}g) = τ_{P} (g_{∞})^{−1}c(g)} spanned by ℂ-valued Hecke characters of the indicated infinity type. This in turn identifies the Fourier expansion of a holomorphic modular form with an element of X_{τ_{P}} (G_{P}; K_{P} (m))[[q^{Λ^{∗}∩𝔠^{∗}}]]. In this notation we can regard ℂ as a O_{v}-algebra or, more prudently, regard both ℂ and O_{v} as algebras over the ring of integers of some number field.

(1.5.6) Comparison with the transcendental theory

Let ψ : A/ℚ → ℂ^{×} be a non-trivial additive character, with local component ψ_{v} at the place v of ℚ, such that ψ_{∞}(x) = e^{2πix}. For any β ∈ U^{∗}(ℚ) we define the character

(1.5.6.1)  ψ_{β} : U(ℚ)\U(A) → ℂ^{×} | ψ_{β}(u) = ψ(β(u)), u ∈ U(A).

A section f ∈ Γ(Sh(2V), [W_{τ}]) can be identified with a W_{τ} (ℂ)-valued automorphic form on GU(2V)(A), belonging to the space on the right-hand side of (1.2.6), which This automorphic form will again be denoted f. We assume we are given an isomorphism of W_{τ} (ℂ) with ℂ, so that f is viewed as a complex-valued automorphic form. The Fourier coefficients of such an f are then defined, classically, as functions on GU(2V, A) by

(1.5.6.2)  f_{β}(h) = ∫_{U(ℚ)\U(A)} f(uh)ψ_{−β}(u)du

For h = (h_{∞}, h_{f}) ∈ GU(2V, A), the holomorphy of f implies a factorization f_{β}(h) = f_{β,∞}(h_{∞})f_{β,f}(h_{f}) where f_{β,∞} depends only on τ and β. Explicitly, if we write h_{∞} = p_{∞}k_{∞} with p_{∞} ∈ P (ℝ) and k_{∞} ∈ K_{∞}, we have

(1.5.6.3)  f_{β,∞}(p_{∞}k_{∞}) = τ(k_{∞})^{−1}e^{2πiβ(Z(p_{∞}))}

where Z(p_{∞}) = p_{∞}(x) ∈ U(ℂ), with x the fixed point of K_{∞} in X(2V) and X(2V) is realized as the tube domain U(ℂ) over the self-adjoint cone C in U(ℝ) and the action of P (ℝ) on the tube domain is the standard one. For more details, see [H1, II]. We write q^{β}(h_{∞}) = f_{β,∞}(h_{∞}). The function f can be recovered from the Fourier coefficients by Fourier inversion, to which we add Koecher’s principle:

(1.5.6.4)  f(h) = ∑_{β∈U^{∗}∩C} f_{β}(h) = ∑_{β∈U^{∗}∩C} q^{β}(h_{∞})f_{β,f}(h_{f}).

It follows that the finite parts f_{β,f} of f_{β}, as β varies, suffice to determine the form f. Suppose f is invariant under the compact open subgroup K ⊂ GU(2V)(A_{f}). Now the derived subgroup GU(2V)^{der} is simply-connected, hence strong approximation is valid, and it follows that the coefficients f_{β,f} are uniquely determined by their values on any subset C′ ⊂ GU(2V)(A_{f}) which maps surjectively onto the quotient C(K) defined as in (1.2). Let L_{P} ⊂ P be the standard Levi component, the centralizer of G_{P}. Then we can take C′ = L_{P} (A_{f}^{p}). It follows that (1.5.6.5) Transcendental q-expansion principle. A form f ∈ A_{hol,τ} (GU(2V)) is determined by the values f_{β,f}(h_{f}) for h_{f} ∈ L_{P} (A_{f}^{p}).

To simplify the comparison of the algebraic and transcendental theories, we introduce the “Shimura variety” Sh(L_{P} , X_{P}) attached to L_{P}:

(1.5.6.6)  Sh(L_{P} , X_{P}) = Sh(G_{P} , X_{P}) ×^{G_{P} (A_{f})} L_{P} (A_{f}).

This can be interpreted as an inductive limit of profinite schemes over E(G_{P} , X_{P}) = ℚ, with natural L_{P} (A_{f})-action. The normal integral model 𝕊(G_{P}) extends similarly to an L_{P} (A_{f})-equivariant normal integral model 𝕊_{P} of Sh(L_{P} , X_{P}). The automorphic vector bundles [W_{τ_{P}}] on Sh(G_{P} , X_{P}) extend trivially to L_{(}A_{f})-equivariant vector bundles on Sh(L_{P} , X_{P}). As in (1.5.5), we can write

M = M([W_{τ_{P}}], K_{L(P)}(m)) = Γ(_{K_{L(P)}(m)}𝕊_{P} , [W_{τ_{P}}])

for an appropriate compact open subgroup K_{L(P)}(m) ⊂ L_{P} (A_{f}), and identify the latter with

(1.5.6.7) X_{τ_{P}} (L_{P}; K_{L_{P}} (m)) = = {c : G_{P} (ℚ)\G_{P} (ℝ) · L_{P} (A_{f})/K_{L(P)}(m) → ℂ | c(g_{∞}h) = τ_{P} (g_{∞})^{−1}c(h)}

where now h ∈ L_{P} (A_{f}) but g_{∞} ∈ G_{P} (ℝ). Ignoring the level structure, the Fourier expansion (1.5.6.4), with h_{f} restricted to L_{P} (A_{f}^{p}), then corresponds to a map

(1.5.6.8)  F.J.^{P} : Γ(Sh(2V), [W_{τ}]) → ˆ ⨁_{β∈U^{∗}}Γ(Sh(L_{P} , X_{P}), [W_{τ_{P}}])

defined over ℚ. By (1.5.6.5), this map is injective.

(1.5.7) Trivializations.

A good choice of basis of W_{τ_{P}} is provided by the theory of degenerating abelian varieties of type _{K(U,m)}A_{2V} (1.3.1); cf. [K, p. 212 ff.], [H1, Lemma 6.6], and [Pink,12.20]. The automorphic vector bundle W_{τ_{P}} is some power, say the kth, of the relative canonical sheaf (bundle of top differentials) on the universal degenerating abelian scheme over the toroidal compactification. Its natural basis is then the product

(1.5.7.1)  ( ⋀_{j=1}^{N} dq^{β_{j}}/q^{β_{j}})^{⊗k} = (2πi)^{Nk}(⋀_{j} dz_{j})^{k},

where the tube domain coordinate z_{j} on X(2V) is defined by q^{β_{j}} = e^{2πiz_{j}}. This basis is defined over ℤ_{(p)} because the coordinates q^{β_{j}} are used to define the toroidal compactification over ℤ_{(p}) in [F]. Thus the trivialization (1.5.7.1) is compatible with the theory of p-adic modular forms, just as in [K], and allows us to identify

(1.5.7.2)  F.J.^{P} (f)_{β}(h) = q^{β}(h_{∞})f_{β,f}(h_{f}), h = (h_{∞}, h_{f}) ∈ L_{P} (A)

where the left-hand term is (1.5.6.8) and the right-hand expression is from (1.5.6.4).

(1.6) p-integral models and p-integral sections.

Let p be a rational prime, and assume hypothesis (1.1.2) is satisfied. Let L′ be a finite extension of ℚ containing E(V) and let O′ be the ring of integers of L′. For simplicity we will assume that L′ also contains E. Fix a sufficiently small compact open subgroup K = K_{p} × K^{p} ⊂ G(A_{f}), as in §1.3. Then it is known (cf. [Ko]) that _{K}Sh(G, X) admits a smooth integral model _{K}𝕊 = _{K}𝕊(G, X) over the valuation ring O′_{(p)} that is a moduli space for abelian varieties with additional structure of PEL type (the moduli problem is that of (1.3.1) but with ι now an embedding O_{(p)} → End_{S}(A) ⊗ ℤ_{(p)}). Moreover, if L′ also contains E(W_{τ}) for every τ (a finite set of τ suffices) then the automorphic vector bundles [W_{τ}] extend naturally to locally free sheaves over_{K}𝕊. In particular, the construction of the ρ_{τ} ’s from §1.4 can be carried out over_{K}𝕊; these then provide integral structures on the various [W_{τ}]’s. Both _{K}𝕊 and the integral structures on the [W_{τ}]’s are functorial with respect to change of the level subgroup K^{p} away from p. In particular, we occasionally drop the notation K in what follows. By our hypotheses on p, and by an elementary approximation argument, the decomposition (1.4.1) can be taken integral over O_{(p)}. We assume that K is so defined so that K_{p} ∩ GU^{⊕_{i}}(V, −V)(ℚ_{p}) is again a maximal compact. Then _{K}Sh^{⊕_{i}}(V, −V) (where the subscript _{K} has the obvious meaning) also has a model over O′_{(p)}, which we denote _{K}𝕊^{⊕_{i}} = _{K}𝕊^{⊕_{i}}(V, −V). The natural map _{K}Sh^{⊕_{i}}(V, −V) → _{K}Sh(G, X) (which is just the inclusion of certain CM points) extend to a map _{K}𝕊^{⊕_{i}} → _{K}𝕊, which can be uses to detect p-integrality of sections of the [W_{τ}]’s, as we now explain. Let A_{Σ} be an abelian variety over \overline{ℚ}, of dimension 2d, with complex multiplication by K of type Σ, and assume End(A_{Σ})⊗_{ℤ} ℤ_{(p)} = O_{(p)}. In other words, O_{(p)} acts on the object “A_{Σ} ⊗_{ℤ} ℤ_{(p)}” defined by A_{Σ} in the category of abelian varieties up to prime-to-p isogeny. One knows A_{Σ} extends to an abelian scheme, also denoted A_{Σ}, over the valuation ring ℤ̄_{(p)}, also with action by O_{(p)} up to prime-to-p isogeny. There is a decomposition

(1.6.1)  H_{DR}^{1}(A_{Σ}/ℤ̄_{(p)}) ⟶^{∼} ⊕_{σ∈Σ_{K}} Ω(Σ)_{σ},

with each Ω(Σ)_{σ} a free ℤ̄_{(p)}-module of rank one. Choose ℤ̄_{(p)}-generators ω_{σ}, σ ∈ Σ_{K} of Ω(Σ)_{σ}. On the other hand, the topological (Betti) homology H_{1}(A_{Σ}(ℂ), ℤ̄_{(p)}) is a free rank one O ⊗_{ℤ} ℤ̄_{(p)}-module, hence admits a decomposition

(1.6.2)  H_{1}(A_{Σ}(ℂ), ℤ̄_{(p)}) ⟶^{∼} ⊕_{σ∈Σ_{K}} (ℤ̄_{(p)})_{σ}

where (ℤ̄_{(p)})_{σ} is the submodule of H_{1}(A_{Σ}(ℂ), ℤ̄_{(p)}), isomorphic to ℤ̄_{(p)}, on which O acts via σ. Choose ℤ̄_{(p)}-generators γ_{σ′} ∈ (ℤ̄_{(p)})_{σ′}, for σ′ ∈ Σ_{K}. The natural pairing (integration)

Int : H_{DR}^{1}(A_{Σ}/ℤ̄_{(p)}) ⊗ H_{1}(A_{Σ}(ℂ), ℤ̄_{(p)}) → ℂ

defines invariants

(1.6.3)  p_{K}(σ, Σ) = Int(ω_{σ}, γ), σ ∈ Σ_{K}, γ ∈ H_{1}(A_{Σ}(ℂ), ℤ̄_{(p)})

where γ is taken to be a free O ⊗_{ℤ} ℤ̄_{(p)} generator of H_{1}(A_{Σ}(ℂ), ℤ̄_{(p)}). It is easy to see that Int(ω_{σ}, γ) depends only on the projection of γ on (ℤ̄_{(p)})_{cσ}, hence that the complex number p_{K}(σ, Σ) is well defined up to multiplication by units in (ℤ̄_{(p)})^{×}. Indeed, both H_{1}(A_{Σ}(ℂ), ℤ̄_{(p)}) and H_{DR}^{1}(A_{Σ}/ℤ̄_{(p)}) are invariants of the prime-to-p isogeny class of A_{Σ}, so the invariants p_{K}(σ, Σ) are independent of the choice of base point in the prime-to-p isogeny class of A_{Σ}, up to (ℤ̄_{(p)})^{×}-multiples. It is well-known that any two choices of A_{Σ} can be related by a prime-to-p isogeny (concretely, any idèle class of K mod K_{∞}^{×} can be represented by an idèle trivial at p). Thus the p_{K}(σ, Σ) can be considered well-defined invariants of Σ, once a base point in the isogeny class is chosen. Now the elements of Σ_{K} generate the character group of the torus R_{K/ℚ}𝔾_{m}, hence their restrictions to the subtorus GU(V_{i}), for any V_{i} as above, generate the character group of the latter. We only consider characters of R_{K/ℚ}𝔾_{m} trivial on the Zariski closure of a sufficiently small congruence subgroup of the units in K. These are characters of the Serre group, and can be identified with the formal linear combinations ∑_{σ∈Σ_{K}} n_{σ}σ with n_{σ} ∈ ℤ such that n_{σ} + n_{σc} is independent of σ. For such characters we define

(1.6.4)  p_{K}(∑_{σ} n_{σ}σ, Σ, V_{i}) = ∏_{σ} p_{K}(σe(i, σ), Σ)^{n_{σ}}

where e(i, σ) = 1 if i ≤ a_{σ} and e(i, σ) = c otherwise. More generally, if κ is a character of ∏_{i} GU(V_{i}) × ∏_{i} GU(−V_{i}), written as an n-tuple of pairs of formal linear combinations

( ∑_{σ∈Σ_{K}} n_{i,σ}σ, ∑_{σ∈Σ_{K}} n_{i,σ}^{−}σ)

we define

(1.6.5) p_{K}(κ, Σ, 2V) = ∏_{i} p_{K}(∑_{σ} n_{i,σ}σ, Σ, V_{i}) · ∏_{i} p_{K}(∑_{σ} n_{i,σ}^{−}σ, Σ, −V_{i}).

Here p_{K}(∑_{σ} n_{i,σ}σ, Σ, −V_{i}) is defined as in (1.6.4) but with a_{σ} replaced by n−a_{σ} The subgroup T = GU^{⊕_{i}}(V, −V)_{∞} ⊂ ∏_{i} GU(V_{i}) × ∏_{i} GU(−V_{i}) is a maximal torus in K_{∞} (maximal compact mod center in G = GU(2V)). The formalism of CM periods implies that the product on the right in (1.6.5) depends only on the restriction of the algebraic character κ to the subgroup T. Indeed, if the restriction of κ to ∏_{i} U(V_{i}) × ∏_{i} U(−V_{i}) is trivial, then in particular n_{i,σ} = n_{i,σc} for all i and all σ. Since n_{i,σ} + n_{i,σc} is independent of σ for each i, it follows that n_{i} = n_{i,σ} is independent of σ for each i, and one can define n_{i}^{−} likewise. One then has

p_{K}(∑_{σ} n_{σ}σ, Σ, V_{i}) = p_{K}(∑_{σ} σ, Σ, V_{i})^{n_{i}} = p_{K}(|| • ||, 1)^{n_{i}} = (2πi)^{−dn_{i}}

as in [H2,Lemma 1.8.3]. If moreover κ|_{T} ≡ 1, then ∑_{i} n_{i} + n_{i}^{−} = 0, and so the product of powers of 2πi is in fact algebraic. Hence the statement of the following Proposition makes sense:

(1.6.6) Proposition. Let G = GU(2V). Let κ be a character of the torus T that extends to a one-dimensional representation of K_{∞}. Let [W_{τ}] be the corresponding automorphic line bundle over _{K}𝕊. Let D ⊂ Sh^{⊕_{i}}(V, −V)(\overline{ℚ}) be a set of points with the following property: the G(A_{f}^{p}) orbit of the image of D under specialization is Zariski dense in the special fiber of _{K}𝕊. Then f ∈ H^{0}(_{K}𝕊, [W_{τ}]) ⊗_{L′} ℂ belongs to H^{0}(_{K}𝕊, [W_{τ}]) ⊗_{O′} ℤ̄_{(p)} if and only if, for all g ∈ G(A_{f}^{p}), the weight κ component f^{g}[κ] of the restriction of the g translate f^{g} of f to D satisfies

(1.6.7)  p_{K}(κ, Σ, 2V)^{−1}f^{g}[κ](x) ∈ ℤ̄_{(p)}

for all x ∈ D. Here the section f^{g} ∈ H^{0}(_{K}𝕊, [W_{τ}]) ⊗_{L′} ℂ is identified with a classical automorphic form on X(2V) × G(A_{f}) via (1.2.6). The same holds with ℂ replaced by ℂ_{p} and ℤ̄_{(p)} replaced by O_{ℂ_{p}}.

Remark. There is an analogous proposition for [W_{τ}] of arbitrary dimension, but we will not be needing it in the present paper.

Proof. Write H = H^{0}(_{K}𝕊, [W_{τ}]), H̄ = H ⊗_{O′_{(p)}} \overline{ℚ}. Our hypothesis on D implies that D · G(A_{f}^{p}) is Zariski dense in the generic fiber _{K}Sh(G, X). Then (1.6.7), with ℤ̄_{(p)} replaced by \overline{ℚ}, is a version of Shimura’s criterion for f to belong to H̄ (cf. ([H1,§5.3], cf. [H3, III, Lemma 3.10.2] for an explicit statement when K is imaginary quadratic). Then there is a number field L, containing L, such that f ∈ H ⊗_{L′} L. Let H_{p} = H ⊗_{O′_{(p)}} O_{L,(p)}. Thus H_{p} is a free O_{L,(p)}-module of finite rank, and H̄ = H_{p} ⊗_{O_{L,(p)}} \overline{ℚ}. Let 𝔭 be a prime of O_{L,(p)}, necessarily dividing p, and let ϖ be a uniformizer of 𝔭. Thus for some positive integer m we have ϖ^{m}f ∈ H_{p}. Write F = ϖ^{m}f. Condition (1.6.7) asserts that

(1.6.8)  p_{K}(κ, Σ, 2V)^{−1}F^{g}[κ](x) ≡ 0 (mod 𝔭^{m}), ∀x ∈ D

The proposition then comes down to showing that any F satisfying (1.6.8) belongs to 𝔭^{m}H_{p}. Since belonging to 𝔭^{m}H_{p} is a local condition on _{K}𝕊 × O_{L,p}, we can replace the latter by an affine open subset U = Spec(A) flat over O_{L,p}, and H_{p} by a free A-module M_{p}; F is an element of M_{p}. By induction we reduce to the case m = 1. Let Ū = Spec(A/pA) denote the special fiber of U; for a geometric point y of Ū let I_{y} ⊂ A denote the maximal ideal at y. Condition (1.6.8) is the condition that F ∈ I_{y} · M_{p} for y in a Zariski dense subset D̄ of Ū; this is essentially the obvious p-integal version of the results of [H3, (3.10)]. By definition, the intersection ⋂D̄ I_{y} = 𝔭 · A. Since M_{p} is free of finite rank over the noetherian ring A, the proposition is clear. A simple continuity argument now provides the proof in the case where ℤ̄_{(p)} is replaced by O_{ℂ_{p}}.

2. p-adic automorphic forms on unitary groups

(2.1) The Igusa tower, I: Definitions.

Notation is as in §1. Recall the ℚ_{p}-rational K ⊗_{ℚ} ℚ_{p}-submodule F^{0}V_{p} ⊂ V_{p}, defined in (1.1.6), and the K ⊗_{ℚ} \overline{ℚ}-submodule subspace F^{0}V ⊂ V ⊗_{ℚ} \overline{ℚ} introduced at the beginning of (1.3). The flag variety X̂ of K-linear subspaces of V satisfying (1.3.0) has a natural E(V)-rational structure. Hypothesis (1.1.6) is equivalent to the condition that the completion E(V)_{w_{0}} of E(V) at the place w_{0} of E(V) corresponding to incl_{p} is isomorphic to ℚ_{p}, and the K-linear subspace F^{0}V_{p} ⊂ V_{p} is indeed a ℚ_{p} = E(V)_{w_{0}}-rational point of X̂. The skew-hermitian pairing tr_{K/ℚ} < •, • >_{V,ℷ} on V ⊗_{ℚ} ℚ_{p} defines a perfect duality

(2.1.1)  V_{p,Σ_{p}} ⊗ V_{p,cΣ_{p}} → ℚ_{p}.

This duality identifies

V_{w}/F^{0}V_{w} ⟶^{∼} Hom(F^{0}V_{c·w}, ℚ_{p})

for any w dividing p. In this way

(2.1.2)  ∏_{w|p} GL(F^{0}V_{w}) ⟶^{∼} ∏_{w,Σ_{w}⊂Σ_{p}} GL(F^{0}V_{w}) × GL(V_{w}/F^{0}V_{w}),

is naturally isomorphic to the Levi quotient L^{0} of the parabolic

P^{0} = ∏_{w,Σ_{w}⊂Σ_{p}} Stab(F^{0}V_{w}).

Here P^{0} is a viewed as a parabolic subgroup of the unitary group G_{0} rather than the unitary similitude group G. Bear in mind that the action of L^{0} on V_{w}/F^{0}V_{w} is dual to that on F^{0}V_{c·w}. We return to the situation of (1.3.1), and let K = K^{p} × K_{p} where K_{p} = G_{0}(ℤ_{p}) × GL(1, ℤ_{p}) is the hyperspecial maximal compact subgroup of (1.3.4), viewed as the group of ℤ_{p}-points of a smooth reductive group scheme 𝕂 over ℤ_{p} with generic fiber G ×_{ℚ} ℚ_{p}. We assume the subspace F^{0}V_{p} and K_{p} are chosen compatibly, in the sense that P^{0} is the ℚ_{p}-points of a parabolic subgroup ℙ^{0} ⊂ 𝕂 (parabolic in the subgroup of 𝕂 corresponding to G_{0}(ℤ_{p})), and we can define 𝕃^{0} to be the Levi quotient of ℙ^{0} so that L^{)} = (L)^{0}(ℚ_{p}). Equivalently, V_{p} and F^{0}V_{p} contain compatible O_{p}-stable lattices M and M^{0}, respectively, with K_{p} the stabilizer of M, and the decomposition F^{0}V_{p} = ⊕_{w|p}F^{0}V_{w} of (1.1.6) is obtained by extension of scalars from a decomposition M^{0} = ⊕_{w|p}M_{w}^{0}; ℙ^{0} is then the stabilizer in 𝕂 of M^{0}. Where necessary, we write M = M(V), M^{0} = M(V)^{0}, etc., to emphasize the relation with the hermitian space V defining the moduli problem. We write

(2.1.3)  M_{Σ_{p}}^{0} = ⊕_{w,Σ_{w}⊂Σ_{p}}M_{w}^{0}, M_{Σ_{p}}^{−1} = ⊕_{w,Σ_{w}⊂Σ_{p}}M_{w}/M_{w}^{0}

As in the preceding paragraph, the skew-hermitian form tr_{O/ℤ} < •, • >_{V,ℷ} can be normalized to define a natural skew-hermitian perfect duality.

(2.1.4)  M^{0} ⊗ M/M^{0} → ℤ_{p}.

There is also a natural isomorphism

(2.1.5)  M^{0} ⟶^{∼} M_{Σ_{p}}^{0} ⊕ Hom^{c}(M_{Σ_{p}}^{−1}, ℤ_{p}),

where

Hom^{c}(M_{Σ_{p}}^{−1}, ℤ_{p}) = O_{p} ⊗_{O_{p},c} Hom(M_{Σ_{p}}^{−1}, ℤ_{p})

i.e. the natural action of O_{p} on Hom(M_{Σ_{p}}^{−1}, ℤ_{p}) is composed with complex conjugation. Let _{K^{p}}A^{p} = _{K^{p}}A_{V,ℷ}^{p} be the functor

S ↦ {(A, λ, ι, α^{p})}

where A is now an abelian scheme over S up to prime-to p-isogeny, λ is a polarization of degree prime to p, ι : O_{(p)} → End_{S}(A) ⊗ ℤ_{(p)} is an embedding of ℤ_{(p)}-algebras, and α^{p} : V (A_{f}^{p}) ⟶^{∼} V^{f,p}(A) is a prime-to-p O_{(p)}-linear level structure modulo K^{p}. The forgetful map _{K}A → _{K^{p}}A^{p} is obviously an isomorphism. The functor _{K^{p}}A^{p} is representable over the integer ring O_{w_{0}} of E(V)_{w_{0}} by a scheme we will denote _{K}𝕊, as in (1.4).

(2.1.6) Igusa Schemes The following constructions are compatible with change of the level subgroup K^{p}, and with passage to the limit over all K^{p}. Hence we drop the subscript K^{p} for the time being. We view A^{p} as a functor on the category of schemes over O_{w_{0}}. Points of A^{p}(S) will be denoted \underline{A}. Define three families of functors above A^{p}, indexed by non-negative integers m:

(2.1.6.1)  Ig_{1,m}(S) = \{(\underline{A}, j^{et})\}, j^{et} : A[p^{m}] ↠ (M/M^{0}) ⊗ ℤ/p^{m}ℤ.

(2.1.6.2) Ig_{2,m}(S) = \{(\underline{A}, j^{o})\}, \underline{A} = (A, λ, ι, α^{p}), j^{o} : M^{0} ⊗ μ_{p^{m}} ↪ A[p^{m}].

(2.1.6.3) Ig_{3,m}(S) = \{(\underline{A}, j^{0}, j^{(−1)})\},

j^{0} : M_{Σ_{p}}^{0} ⊗ μ_{p^{m}} ↪ A[p^{m}]_{Σ_{p}}, j^{(−1)} : A[p^{m}]_{Σ_{p}} ↠ M_{Σ_{p}}^{−1} ⊗ ℤ/p^{m}ℤ.

In each case \underline{A} designates a quadruple (A, λ, ι, α^{p}) ∈ A^{p}(S). The maps j^{0}, j^{et}, j^{0}, and j^{(−1)} are all assumed O/p^{m}O-linear.

(2.1.6.4) Lemma. The functors Ig_{i,m}, i = 1, 2, 3, are all relatively representable over A^{p}, and are canonically isomorphic for all m. These isomorphisms are compatible with the natural forgetful projection maps Ig_{i,m+1} → Ig_{i,m} for all i; moreover, these projection maps are étale for all m.

Proof. Since the polarization λ is assumed of degree prime to p, we can use it to identify Â[p^{m}] ⟶^{∼} A[p^{m}]. The isomorphism Ig_{1,m} ⟶^{∼} Ig_{2,m} is then obtained by combining the duality (2.1.4) with Cartier duality A[p^{m}] × Â[p^{m}] → μ_{p^{m}}. The isomorphism between Ig_{2,m} and Ig_{3,m} is obtained in a similar way from (2.1.5). Compatibility of these isomorphisms with the forgetful projection maps is obvious. Finally, the projection Ig_{1,m+1} → Ig_{1,m} is obviously étale, since it corresponds to lifting a trivialization of the étale quotient of A[p^{m}] to one of the étale quotient of A[p^{m+1}].

Since the isomorphisms in (2.1.6) are canonical, we write Ig_{m} for Ig_{i,m}, i = 1, 2, 3, or Ig(V)_{m} when we need to emphasize V. For any m > 0, the natural forgetful map Ig_{m} → 𝕊 obviously factors through the inclusion of the ordinary locus 𝕊^{ord} ⊂ 𝕊. The limit Ig_{∞} = \underleftarrow{lim}_{m} Ig_{m} is an étale Galois covering of 𝕊^{ord} with covering group

𝕃^{0}(ℤ_{p}) = Aut(M^{0}) ⟶^{∼} Aut(M_{Σ_{p}}^{0}) × Aut(M_{Σ_{p}}^{−1}).

Let 𝔽 denote the algebraic closure of the residue field of O_{w_{0}}, and let S̄ = K𝕊 ×_{O_{w_{0}}} 𝔽 denote the geometric special fiber of the moduli scheme _{K}𝕊. Let S̄^{ord} = 𝕊^{ord} ×_{Spec(O_{w_{0}} )} S̄ ⊂ S̄ denote the ordinary locus of the special fiber. The following theorem is a special case of a result of Wedhorn [We]:

(2.1.7) Theorem. The ordinary locus S̄^{ord} contains an open dense subscheme of every irreducible component of S̄.

(2.1.8) Modular interpretation of the Igusa tower in the limit

In the limit as m tends to infinity we can reformulate the definition of the Igusa tower in terms of abelian varieties up to isogeny. We prefer to use the models Ig_{3,m}. Let T (𝔾_{m}) = \underleftarrow{lim}_{m} μ_{p^{m}} denote the Tate module of the multiplicative group, viewed as a profinite flat group scheme over Spec(ℤ_{p}). For any vector space W over ℚ_{p} we let W (1) = W ⊗_{ℤ_{p}} T (𝔾_{m}). Consider the functor on schemes over O_{w_{0}}: (2.1.8.1)

\begin{matrix}Ig′_{3,∞}(S) & = \{(\underline{A}, j^{0}, j^{−1})\}, \\ j^{0} : F^{0}V_{Σ_{p}}(1) ↪ ℚ_{p} ⊗_{ℤ_{p}} A[p^{∞}]_{Σ_{p}}, & j^{−1} : ℚ_{p} ⊗_{ℤ_{p}} A[p^{∞}]_{Σ_{p}} ↠ V/F^{0}V_{Σ_{p}}. \end{matrix}

Here \underline{A} = (A, λ, ι, α^{p}) as above, but now A is an abelian variety up to isogeny, and A[p^{∞}] is a p-divisible group up to isogeny, or rather quasi-isogeny (cf. [RZ], 2.8). For fixed m we define Ig′_{3,m} by the same functor as Ig′_{3,∞} but with j^{0} and j^{−1} defined only modulo the principal congruence subgroups modulo p^{m} of GL(M_{Σ_{p}}^{0}) and GL(M_{Σ_{p}}^{−1}), respectively. The usual argument shows that

(2.1.8.2) Lemma. There are canonical isomorphisms Ig′_{3,m} ⟶^{∼} Ig_{3,m} for all m, compatible with the forgetful maps from level p^{m+1} to level p^{m} for all m. In particular, the natural action of 𝕃^{0}(ℤ_{p}) on Ig_{∞} extends canonically to an action of L^{0}(ℚ_{p}).

The final assertion is completely analogous to the existence of an action of G(A_{f}^{p}) in the inverse limit over K^{p}.

(2.1.9) Irreducibility of the Igusa tower

We reintroduce the prime-to-p level subgroups K^{p}, and the level subgroup K = K^{p} × K_{p}. The fiber over ℚ_{p} of the ordinary locus _{K}𝕊^{ord} coincides with K𝕊 ×_{ℤ_{p}} ℚ_{p} = _{K}Sh(V)_{ℚ_{p}}; here, as above, we identify ℚ_{p} = E(V)_{w_{0}}. The generic fibers _{K^{p}}Ig_{m,ℚ_{p}} can be identified with Shimura varieties attached to appropriate level subgroups, as follows. Let U ⊂ ℙ^{0} denote the unipotent radical. For any non-negative integer m, let K(U, m)_{p} ⊂ K_{p} denote the inverse image of U(ℤ_{p}/p^{m}Zp)×GL(1, ℤ_{p}) under the natural map K_{p} → 𝕂(ℤ_{p}/p^{m}Zp). Let K(U, m) = K(U, m)_{p} × K^{p}. The variety _{K(U,m)}Sh, as m tends to infinity, parametrizes quadruples (A, λ, ι, α) where α = (α_{p,m}, α^{p}) with α^{p} as above and

α_{p,m}^{0} : M/p^{m}M ⟶^{∼} A[p^{m}] (mod K(U, m))

is an O/p^{m}O-linear injection that identifies the given skew-symmetric pairing on M/p^{m}M with the Weil pairing on A[p^{m}]. This comes down to an inclusion of (M/p^{m}M)^{K(U,m)} = M^{0}/p^{m}M^{0} in A[p^{m}]_{Σ_{p}} and a Cartier dual surjection of A[p^{m}]_{Σ_{p}} onto M^{−1}/p^{m}M^{−1}. It follows that there are natural isomorphisms

(2.1.9.1)  _{K^{p}}Ig_{m,ℚ_{p}} ⟶^{∼} _{K(U,m)}Sh

compatible with the forgetful maps from level m + 1 to level m. Over ℂ, the connected components of _{K(U,m)}Sh(G, X) are in bijection with the class group C(m) = C(K(U, m)), as at the end of (1.2). Consider the normalization _{ K̂}𝕊 of _{K}𝕊 in _{K(U,m)}Sh(G, X). This is an O_{v}-model of _{K(U,m)}Sh(G, X),

though not a very good one. However the non-singular locus (ˆS̄^{o} is étale over K S̄^{ord}, and _{K^{p}}Ig_{m} is naturally isomorphic to an open subscheme of (ˆS̄)^{o}. In particular, there is a map c_{m} :_{K^{p}} Ig_{m} → C(m), which can be given a modular interpretation as in (1.2). A special case of Corollary 8.17 of [Hi04], (cf. also [Hi05, §10]) is that

(2.1.10) Theorem. ([Hida]) The fibers of c_{m} are geometrically irreducible for all m.

This is proved in [loc. cit.] under a hypothesis labeled (ord), which is equivalent to our hypothesis (1.1.2). Lemma 8.10 of [loc. cit.] makes this explicit, but only for imaginary quadratic K.

(2.1.11) Inclusion of Igusa towers for Sh(V, −V) in Sh(2V)

Applying the previous discussion to the hermitian space 2V, we identify Ig(2V)_{m} = Ig(2V)_{2,m} with the moduli space of quintuples

\{\underline{B} = (B, μ, ι_{2}, β^{p}), j_{2V}^{o} : M(2V)^{0} ⊗ μ_{p^{m}} ↪ B[p^{m}]).

Now M(2V)^{0} is a lattice in the K ⊗_{ℚ} ℚ_{p}-submodule F^{0}(2V)_{p} of (2V)_{p}, which we can choose arbitrarily as long as we respect Hypothesis (1.1.6). For example, we can choose

(2.1.11.1)  F^{0}(2V)_{p} = F^{0}V_{p} ⊕ F^{0}(−V)_{p}

where F^{0}(−V)_{p} ⊂ (−V)_{p} is any K ⊗_{ℚ} ℚ_{p}-submodule satisfying (1.1.6), which for −V amounts to the condition that dim F^{0}(−V)_{w} = n − a_{σ} = a_{cσ} for any σ ∈ Σ_{w}. As K ⊗_{ℚ} ℚ_{p}-module −V is isomorphic to V, and it is particularly convenient to choose F^{0}(−V)_{p} ⊂ (−V)_{p} = V_{p} to be a subspace mapping isomorphically to V_{p}/F^{0}V_{p} under the projection, or equivalently such that (2.1.1) restricts to a duality between F^{0}(−V)_{w} and F^{0}V_{cw} for any w dividing p. We define Ig(V, −V)_{m} ⊂ Ig(V)_{m} ×Ig(−V)_{m} as Sh(V, −V) in (1.3) as the subvariety with compatible polarizations. Then, ignoring prime-to p level structures, the reduction modulo p of the natural morphism Sh(V, −V) ⊂ Sh(2V) defines a family of morphisms Ig(V, −V)_{m} → Ig(2V)_{m} whose image, in the version Ig_{2,m}, is the moduli space of quintuples as above where

(B, μ, ι_{2}, β) ⟶^{∼} (A × A^{−}, λ × λ^{−}, ι × ι^{−}, α × α^{−})

as in §1.4 and where

(2.1.11.2) j_{2V}^{o} = j_{V}^{o} ×j_{−V}^{o} : M(V)^{0}⊗μ_{p^{m}} ×M(−V)^{0}⊗μ_{p^{m}} ↪ A[p^{m}]×A^{−}[p^{m}].

We make this more explicit. Fix w dividing p, let (a, b) = (a_{w}, b_{w}), and choose bases (e_{1}, …, e_{n}) and (f_{1}, …, f_{n}) for V_{w} and (−V)_{w}, respectively, with e_{1}, … e_{a} a basis for F^{0}V_{w}, f_{1}, … f_{b} a basis for F^{0}(−V_{w}). We regard the natural identification of V_{w} with (−V)_{w} as an isomorphism between the two halves of 2V, in such a way that e_{i} is taken to f_{b+i} for 1 ≤ i ≤ a and e_{a+j} is taken to f_{j} for 1 ≤ j ≤ b. The 2n × 2n-matrix γ_{1} = γ_{1;a,b}:

(2.1.11.3)  γ_{1} = γ_{1;a,b} = ( \begin{matrix}I_{a} & 0 & 0 & 0 \\ 0 & 0 & 0 & I_{b} \\ 0 & 0 & I_{a} & 0 \\ 0 & I_{b} & 0 & 0 \end{matrix} ) ,

in the basis (e_{1}, …, e_{n}, f_{1}, … f_{n}) of 2V_{w}, takes the subspace V_{w} ⊂ 2V_{w} to the subspace F^{0}(2V)_{w} defined by (2.1.11.1).

(2.2) The Igusa tower, II: p-adic modular forms.

We now recall Hida’s generalization of the Deligne-Katz construction of p-adic modular forms, for the Shimura varieties Sh(G, X). In the present article we will only need p-adic modular forms in order to define a good notion of pintegrality for certain holomorphic Eisenstein series ramified at p, but later we will use them to construct p-adic L-functions and establish their boundedness . So for the moment we let (G, X) = (GU(V), X(V)) or (GU(V, −V), X(V, −V)). We work with a smooth, projective, toroidal compactification _{K}𝕊˜ of _{K}𝕊. The construction of such compactifications in this setting is due to Fujiwara. The choice of _{K}𝕊˜ is not canonical. However, the universal abelian scheme _{K}𝔸 over K𝕊 extends to a semi-abelian scheme over _{K}𝕊˜ . Hence ω, and therefore each ρ_{τ}, also extends. Let v be the prime of K determined by incl_{p}. We begin by choosing a lifting of _{K} S̄^{ord} to an O_{v}-flat open subscheme of _{K}𝕊˜ . (This is possible since under (1.1.2) E(V)_{w_{0}} = ℚ_{p} so our schemes are all defined over O_{v}.) More precisely, K S̄^{ord} is defined by the non-vanishing of the Hasse invariant H, which can be regarded as a section of a certain automorphic line bundle [L] over S̄. The line bundle L is known to be ample, hence for some power κ >> 0 the section H^{κ} lifts to a section H̃ ∈ Γ(_{K}𝕊, [L]^{κ}). We let _{K}𝕊^{ord} ⊂ _{K}𝕊 be the open subscheme defined by non-vanishing of H̃. This is slightly abusive, since it depends on the choice of lifting H̃, but different choices yield isomorphic theories. For all this, see [Hi05, p. 213 ff.] or [SU]. We let W be a finite flat O_{v} algebra, W_{r} = W/p^{r}W, and let S_{m} = _{K}𝕊^{ord} ⊗_{O_{v}} W_{r}. The S_{r} form a sequence of flat W_{r} schemes, with given isomorphisms

S_{r+1} ⊗_{W_{r+1}} W_{r} ⟶^{∼} S_{r}.

For m ≥ 1, let P_{m} = 𝔸[p^{m}]^{et} = 𝔸[p^{m}]/𝔸[p^{m}]^{0} over _{K} S̄^{ord}, the quotient of the _{K} S̄^{ord} group scheme of p^{m}-division points of 𝔸 by its maximal connected subgroup scheme. This is a free étale sheaf in O_{v}/p^{m}O_{v}-modules over _{K} S̄^{ord}, hence lifts canonically, together with its O_{v}-action, to an étale sheaf over S_{r} for all r. Following [Hi04], we define T_{r,m} to be the lifting to S_{r} of the corresponding principal GL(n, O/p^{m}O)-bundle (resp.., GL(n, O/p^{m}O) × GL(n, O/p^{m}O)- bundle) Ig_{m}(V) = Ig_{1,m} (resp., Ig_{m}(V, −V)), defined by (2.1.6.1) (resp., as in (2.1.11)); note that our indices are not the same as Hida’s. Let

V_{r,m} = Γ(T_{r,m}, O_{T_{r,m}}); V_{r,∞} = \underrightarrow{lim}_{m} V_{r,m}; V_{∞,∞} = \underleftarrow{lim}_{r} V_{r,∞}

Note that these carry actions of GL(n, O_{p}) or of GL(n, O_{p}) × GL(n, O), depending on whether G = GU(V) or GU(V, −V). Let U be the upper-triangular unipotent radical of GL(n, O_{p}) or GL(n, O_{p})×GL(n, O_{p}), depending. We then define our space of p-adic modular forms to be

V := V_{∞,∞}^{U}.

We will adopt the convention of adding a superscript V or V, −V when it is necessary to distinguish the groups in question. Hence, V_{V} is the ring of p-adic modular forms for GU(V). It is clear that the construction of the spaces of p-adic modular forms for GU(V, −V) and GU(2V) can be done compatibly, at least when the various prime-to-p level structures are compatible (i.e., there are morphisms KSh(V, −V) → _{K′}Sh(2V)). This gives rise to a restriction map

r_{V} : V_{2V} → V_{(V,−V)}.

The primary goal of this section is to explain why this is a good definition and how it naturally contains all p-adic sections of [W_{τ}] for all τ, and, in the case G = GU(2V), is contained in the power series ring R[[q^{Λ^{∗}∩𝔠^{∗}}]] of (1.5.3) for an appropriate R. For n > 1, the sections of [W_{τ}] are vector-valued functions. To compare them for different τ, we follow Hida and trivialize the [W_{τ}], using the modular definition of T_{r,m}, and then apply the theorem of the highest weight in integral form. The discussion below follows [Hi04,8.1], to which we refer for missing details. Let ω_{r,m} denote the pullback of ω to T_{r,m}. By Cartier duality, the universal surjection (2.1.6.1), with S = T_{1,m}, is equivalent to an isomorphism of group schemes

(2.2.1)  𝔡^{−1} ⊗ (μ_{p^{m}})^{n} ⟶^{∼} 𝔸̂[p^{m}]^{0}.

Here 𝔡^{−1} is the different of K over ℚ, μ_{p^{m}} is the kernel of multiplication by p^{m} in the multiplicative group scheme, 𝔸̂ is the abelian scheme dual to 𝔸, and the superscript ^{0} denotes the maximal connected subgroup scheme. Since (2.2.1) is Cartier dual to an isomorphism of étale group schemes induced by (2.1.6.1), it lifts canonically to T_{r,m}. Since there are canonical isomorphisms

ω_{r,m} ⟶^{∼} Lie(𝔸̂) ⊗ W_{r} ⟶^{∼} Lie(𝔸̂[p^{m}]^{0}) ⊗ W_{r}

we can identify

(2.2.2)  ω_{r,m} ⟶^{∼} 𝔡^{−1} ⊗ Lie(μ_{p^{m}})^{n} ⊗ W_{r} ⟶^{∼} 𝔡^{−1} ⊗ O_{T_{r,m}}^{n}.

as O_{p} ⊗_{ℤ_{p}} W_{r} modules. Since K is unramified at p, 𝔡^{−1} is prime to p, and (2.2.2) reduces to a family of O_{p} ⊗_{ℤ_{p}} W_{r} isomorphisms

(2.2.3)  ω_{r,m} ⟶^{∼} O ⊗_{ℤ_{p}} O_{T_{r,m}}^{n},

compatible as m and r vary. Note that in (2.2.1), (2.2.2), and (2.2.3) the n should be replaced by a 2n if G = GU(V, −V). Suppose that G = GU(V). Now we apply the highest weight formalism as in [Hi05]. Let G_{1} = Res_{O_{p}/ℤ_{p}}GL(n), let B_{1} be the upper-triangular Borel of G_{1}, U_{1} its unipotent radical, and T_{1} the torus of diagonal elements. Let H = G_{1}/U_{1}. Then (2.2.3) yields a family of isomorphisms

(2.2.4)  G_{1}_{/T_{r,m}} ⟶^{∼} GL_{O}(ω_{r,m})

and

(2.2.5)  H_{T_{r,m}} ⟶^{∼} Y_{r,m} =^{def} GL_{O}(ω_{r,m})/U_{can}

where U_{can} is the T_{r,m}-unipotent group scheme corresponding to U_{1} under (2.2.4). The isomorphisms (2.2.5) are compatible with the natural G_{1} × T_{1} actions on the two sides (G_{1} acting on the left and T_{1} on the right) and patch together as r and m vary. Not that for any character κ of T_{1}, taking κequivariant sections (indicated by [κ]) of O_{Y_{r,m}} makes sense. Continuing as in [Hi05, §7], and writing Y = Y_{r,m}, p_{Y} : Y → T_{r,m} the natural map, note that p_{Y,∗}(O_{Y} [κ]) inherits an action of G_{1}(ℤ_{p}), covering the trivial action on T_{r,m}, because p_{Y} is a fibration in G_{1}(ℤ/p^{m}ℤ)-homogeneous spaces. On the other hand, T_{r,m} is a G_{1}(ℤ/p^{m}ℤ)-torsor over S_{r}. We let δ_{m} denote the diagonal action of G_{1}(ℤ/p^{m}ℤ) on p_{Y,∗}(O_{Y} [κ]) over S_{r}. Over S_{r}

(2.2.6)  ρ_{κ} = p_{Y,∗}(O_{Y} [κ])/δ_{m}(G_{1}(ℤ/p^{m}ℤ)),

From the isomorphism (2.2.5) one obtains an isomorphism

ϕ_{m} : H^{0}(S_{m}, ρ_{κ})

⟶^{∼} {f ∈ Mor_{V_{m,m}}(G_{1}_{/V_{m,m}}, G_{a}_{/V_{m,m}}) | f(hgut) = κ(t)h · f(g),

h ∈ G_{1}(ℤ_{p}), u ∈ U_{1}, t ∈ T_{1}}.

These isomorphisms are clearly compatible with varying m. Composing with the evaluation at the identity map yields a map

β_{κ} : H^{0}(S_{m}, ρ_{κ}) → V_{m,m}^{U_{1}}.

Because of the compatibilities as m varies, this also makes sense for m = ∞, in which case we have an injection

(2.2.7)  β_{κ} : H^{0}(S_{∞}, ρ_{κ}) → V = V_{∞,∞}^{U_{1}}.

The image of β_{κ} is naturally contained in V[κ]. From (2.2.7) we obtain an injection (2.2.8) Ig : H^{0}(_{K(U,∞)}Sh(V), [W_{τ}])⊗\overline{ℚ}ℂ_{p} = H^{0}(_{K(U,∞)}Sh(V), ρ_{τ})⊗\overline{ℚ}ℂ_{p} → V ⊗_{O_{v}} ℂ_{p}.

This is defined by restricting a section of H^{0}(_{K(U,∞)}Sh(V), ρ_{τ}) to a formal neighborhood of the Igusa tower in the special fibre of the normalization _{ K̂}𝕊 of K𝕊 in _{K}Sh(V). When G = GU(V, −V) the same arguments apply, but in the definition of G_{1}, GL(n) is replaced by GL(n) × GL(n), and in (2.2.4) GL_{O}(ω_{r,m}) is replaced by the subgroup preserving the splitting of Lie(𝔸̂) coming from the splitting of 𝔸. In particular, when the prime-to-p levels are compatible, there is a commutative diagram (2.2.9)

H^{0}(_{K(U,∞)}Sh(2V), [W_{τ}]) ⊗\overline{ℚ} ℂ_{p} ⟶^{res′} H^{0}(_{K(U,∞)}Sh(V, −V), [W_{τ}]) ⊗\overline{ℚ} ℂ_{p}
Ig_{2V} ↓  ^{Ig_{V,−V}} ↓
V_{2V} ⊗_{O_{v}} ℂ_{p}  ⟶^{r_{V}}  V_{V,−V} ⊗_{O_{v}} ℂ_{p}

where res′ is the map coming from the inclusion of Igusa towers as in (2.1.11).

(2.3) p-adic modular forms and the q-expansion principle.

Now we return to the situation of (1.5), with the Shimura datum (GU(2V), X(2V)). We write Sh(L_{P}) instead of Sh(L_{P} , X_{P}). For simplicity, we again restrict attention to one-dimensional [W_{τ}]. Then the Fourier expansion of (1.5.6.8), applied to

H^{0}(_{K(U,∞)}Sh(2V), [W_{τ}]) := \underrightarrow{lim}_{m} H^{0}(_{K(U,m)}Sh(2V), [W_{τ}]),

takes values in

ˆ⨁_{β∈U^{∗}}H^{0}(_{K_{P} (∞)}Sh(L_{P}), [W_{τ_{P}}]) := ˆ ⨁_{β∈U^{∗}} \underrightarrow{lim}_{m} H^{0}(_{K_{L_{P}} (m)}Sh(L_{P}), [W_{τ_{P}}]).

These can be translated into locally constant functions on L_{P} (A_{f}) as in the discussion following (1.5.6.5), and as indicated there, it suffices to consider values on L_{P} (A_{f}^{p}). In [Hi04, 8.3.2], Hida explains how to fill in the lower horizontal arrow in the following commutative diagram:

(2.3.1)

H^{0}(_{K(U,∞)}Sh(2V), [W_{τ}]) ⊗\overline{ℚ} ℂ_{p} ⟶^{F.J.^{P}} ˆ ⨁_{β∈U^{∗}}H^{0}(_{K_{P} (∞)}Sh(L_{P}), [W_{τ_{P}}]) ⊗\overline{ℚ} ℂ_{p}
Ig↓  ^{=}↓
V ⊗_{O_{v}} ℂ_{p}  ⟶^{(F.J.^{P})_{ℂp}} ˆ ⨁_{β∈U^{∗}}H^{0}(_{K_{P} (∞)}𝕊_{P} , [W_{τ_{P}}]) ⊗_{O_{v}} ℂ_{p}

More precisely, and more usefully, Hida explains how to construct an integral map

(2.3.2)  V ⟶^{F.J.^{P}} ˆ ⨁_{β∈U^{∗}}H^{0}(_{K_{P} (∞)}𝕊_{P} , O_{𝕊_{P}})

which yields the bottom line of (2.3.1) upon tensoring with ℂ_{p}.^{6} Now we can state

Theorem 2.3.3 (q-expansion principle, [Hi04]).

(a) The map F.J.^{P} of (2.3.2) is injective and its cokernel has no p-torsion.

(b) Let f ∈ H^{0}(_{K(U,∞)}Sh(2V), [W_{τ}]) and suppose f is defined over \overline{ℚ}, viewed as a subfield of ℂ or of ℂ_{p}. Then the expansions F.J.^{P} (f), defined via (2.3.2) or (1.5.6.8), coincide, and the following are equivalent:

(i) Ig(f) ∈ H^{0}(S_{∞}, ρ_{κ}) ⊗ O_{ℂ_{p}} (ii) F.J.^{P} (f) has coefficients in O_{ℂ_{p}}.

Here, as in (1.5), the coefficients of F.J.^{P} (f) can be viewed as functions on L_{P} (A_{f}), and to test their integrality it suffices to consider their values on L_{P} (A_{f}).

When n = 1 and E = ℚ, this theorem, or rather the corrected version of this theorem incorporating a growth condition at the cusps, is essentially due to Katz; for general E, still with n = 1, it is due to Ribet. The principal ingredient in the proof is the irreducibility theorem 2.1.10.

(2.4) The case of definite groups.

We end our discussion of p-adic modular forms with a naive description when V is definite. The comparison of this naive description, which is useful, for calculations, and the geometric description of the previous section is made in (2.4.7). We will need it to understand how the restriction of a p-adic modular form on U(2V) to U(V, −V) can be described in the naive sense. Throughout this section we assume that < •, • >_{σ} is positive definite for all σ ∈ Σ (so a_{σ} = n for all σ).

6Actually Hida only considered the case of level prime to p; the general case is treated in [SU].

Documenta Mathematica · Extra Volume Coates (2006) 393–464

(2.4.1) Spaces of forms and rational structures.

For applications to definite unitary groups, we can avoid similitude factors, so for the moment we let G denote U(V) or U(−V) (since these are canonically identified, the distinction is made primarily for ease of subsequent notation). In what follows, we consider only compact subgroups K ⊂ G(A^{f}) of the form K = ∏_{v} K_{v}, the product being over finite places of ℚ, with K_{v} a subgroup of G_{v}. We fix a rational prime p such that all places of E dividing p split in K and let K^{p} = K ∩ G(A^{f,p}) ≅ ∏_{v≠p} K_{v}. Let ρ be a complex algebraic character of G. Via the fixed isomorphism ℂ_{p} ≅ ℂ we view ρ as an algebraic character over ℂ_{p}. Then ρ has a model over some finite extension F of ℚ_{p}. We fix such an F. For each finite place v of ℚ let 𝔰_{v} : K_{v} → GL(𝔏_{v}) be a finite-dimensional F -representation of K_{v} factoring through a finite quotient of K_{v} and such that 𝔰_{v} and 𝔏_{v} are trivial for almost all v and for v = p. Let 𝔰 = ⊗_{v,F} 𝔰_{v} and 𝔏 = ⊗_{v,F} 𝔏_{v}. The product G_{∞} × K acts on ℂ ⊗_{F} 𝔏 via ρ ⊗ 𝔰. For a finite set S of places of ℚ and a finite-dimensional complex vector space H let C^{∞}(G(A^{S}), H) denote the space of functions from G(A^{S}) to W_{σ}(ℂ) that are smooth as functions of the infinite component of G(A^{S}) and locally constant as functions of the finite component. If S contains ∞, G′ ⊂ G(A^{S}) is an open subgroup, and M is any set, then we write C^{∞}(G′, M) for the set of locally constant functions from G′ to M. Let

A_{0}(G, K, ρ, 𝔰) = = {f ∈ C^{∞}(G(A), ℂ ⊗_{F} 𝔏)) | f(γg · g_{∞}k) = (ρ ⊗ 𝔰)(g_{∞} × k)^{−1}f(g)},

where γ ∈ G(ℚ), g ∈ G(A), g_{∞} ∈ G_{∞}, and k ∈ K. For any F -algebra R let

A_{f}(G, K, ρ, 𝔰)(R) =

= {f ∈ C^{∞}(G(A^{f}), R ⊗_{F} 𝔏) | f(γ · gk) = (ρ ⊗ 𝔰)(γ × k^{−1})f(g)},

where γ ∈ G(ℚ), g ∈ G(A^{f}), and k ∈ K. Note that there is a canonical isomorphism

(2.4.1.1)  A_{f}(G, K, ρ, 𝔰)(R) = A_{f}(G, K, ρ, 𝔰)(F) ⊗_{F} R.

Restriction to G(A^{f}) defines a natural isomorphism

(2.4.1.2)  res : A_{0}(G, K, ρ, 𝔰) ⟶^{∼} A_{f}(G, K, ρ, 𝔰)(ℂ),

and hence, by (2.4.1.1), A_{f}(G, K, ρ, 𝔰)(F) defines an F -structure on A_{0}(G, K, ρ, 𝔰). When ρ or 𝔰 is the trivial one-dimensional representation, we drop it from our notation.

(2.4.2) Integral structures.

Let R be a commutative ring. For any R[K]-module M let

A(G, K, M) = = {f ∈ C^{∞}(G(A^{f}), M) | f(γgk) = k^{−1} · f(g), γ ∈ G(ℚ), k ∈ K}.

If K′ ⊆ K is an open subgroup then A(G, K, M) ⊆ A(G, K′, M) and there is a trace map tr_{K′,K} : A(G, K′, M) → A(G, K, M) defined by tr_{K′,K}f(x) = ∑_{y∈K/K′} yf(xy). These maps are clearly functorial in M and R and they satisfy

(2.4.2.1)  tr_{K′′,K} = tr_{K′,K} ∘ tr_{K′′,K′}, K′′ ⊆ K′ ⊆ K.

Let A be the ring of integers of F. We choose a K_{v}-stable A-lattice Λ_{𝔰_{v}} in each 𝔏_{v} and let Λ_{𝔰} = ⊗_{v,A}Λ_{𝔰_{v}}. Clearly A(G, K, Λ_{𝔰}) is an A-lattice in A_{f}(G, K, 𝔰)(F). Let Γ_{K} = G(ℚ) ∩ K_{p}. For χ an R^{×}-valued character of Γ_{K} and M an R[K]- module let

A_{f}(G, K, χ, M) = {f ∈ C^{∞}(G(A^{f,p}) × K_{p}, M |
f(γgk) = χ(γ^{−1}) · k^{−1}f(g), γ ∈ Γ_{K}, k ∈ K}.

Weak approximation shows that restriction to G(A^{f,p}) × K_{p} yields an isomorphism

(2.4.2.2)  A(G, K, M) ⟶^{∼} A_{f}(G, K, 1, M).

Similarly, when R is an F -algebra, restriction to G(A^{f,p}) × K_{p} yields an isomorphism

(2.4.2.3)  A_{f}(G, K, ρ, 𝔰)(R) ⟶^{∼} A_{f}(G, K, ρ, 𝔏 ⊗_{F} R).

It follows from (2.4.2.3) that to define an A-lattice in A_{f}(G, K, ρ, 𝔰)(F) it suffices to define an A-lattice in 𝔏. In particular, A_{f}(G, K, ρ, Λ_{𝔰}) defines an Alattice in A_{f}(G, K, ρ, 𝔰). For K′ ⊆ K we define a trace map tr_{K′,K} : A_{f}(G, K′, χ, M) → A_{f}(G, K, χ, M) just as we did above. These maps also satisfy (2.4.2.1) and are functorial in M, and R and agree with our previous definitions via (2.4.2.2) when χ = 1.

(2.4.3) p-adic forms For a topological space X and a group G′ = H × H′ with H ⊆ G(ℚ_{p}) and H′ ⊆ G(A^{f,p}) open sets, we write C^{p}(G′, X) for the space of maps from G′ to X that are continuous on H (for the p-adic topology) and locally constant on H′. Let G_{1} denote the group scheme R_{O_{E,p}/ℤ_{p}}GL(n) over ℤ_{p} and fix an identification of G with G_{1} over ℚ_{p}. Let B ⊆ G_{1} be its upper-triangular Borel. Let P ⊇ B be a standard parabolic of G_{1}. Let L be its standard Levi subgroup and U_{P} its unipotent radical. Upon fixing an identification O_{E,p} = ∏_{w|p} O_{E,w} we have G_{1}(ℤ_{p}) = ∏_{w|p} GL(n, O_{E,w}), P (ℤ_{p}) = ∏_{w|p} P_{w}(O_{E,w}) where P_{w} ⊆ GL(n) is a standard parabolic corresponding to a partition p_{w} : n = n_{1,w} + ⋯ + n_{l_{w},w} of n, and L(ℤ_{p}) = ∏_{w|p} L_{w} where L_{w} is the set of block diagonal matrices diag(A_{1}, ..., A_{l_{w}}) with A_{i} ∈ GL(n_{i,w}, O_{E,w}). Let L_{1} ⊆ L(ℤ_{p}) be the subgroup ∏_{w|p} L_{w,1} where L_{w,1} is the subgroup defined by det(A_{i}) = 1. Let P_{1} = L_{1}U_{P} (ℤ_{p}). For m ≥ 0 let U_{P,m} = {x ∈ G_{1}(ℤ_{p}) | x mod p^{m} ∈ (P_{1} mod p^{m})}. So ∩U_{P,m} = P_{1}. Let I_{P,m} = {x ∈ G_{1}(ℤ_{p}) | x mod p^{m} ∈ P (ℤ_{p}/p^{m})}. Assume that K = G_{1}(ℤ_{p})×K^{p}. Let K_{P,m} = U_{P,m}×K^{p} and let K_{P} = P_{1}×K^{p}. Then ∩K_{P,m} = K_{P}. Let R be a p-adic ring and M any finite R-module that is also an R[K]-module on which K_{p} acts trivially. Let

A_{p}(G, K_{P} , M)={f ∈ C^{p}(G(A^{f}), M) | f(γgk) = k^{−1}·f(g), γ ∈ G(ℚ), k ∈ K_{P} }.

Since M/p^{r}M is discrete, the canonical projections M ↠ M/p^{r}M together with (2.4.2.2) induce a canonical isomorphism

(2.4.3.1)  A_{p}(G, K_{P} , M) ⟶^{∼} \underleftarrow{lim}_{r} \underrightarrow{lim}_{m} A_{f}(G, K_{P,m}, M/p^{r}M).

Let A and Λ_{𝔰} be as in (2.4.2) and take R = A. Then Λ_{𝔰} provides an important example of an M as above. We call A_{p}(G, K_{P} , Λ_{𝔰}) the space of (Λ_{𝔰}-valued) p-adic modular forms on G relative to P (and K). When P is understood then we just call this the space of p-adic modular forms.

(2.4.4) Characters

The group L(ℤ_{p}) normalizes each K_{P,m}, m > 0, and so acts on A_{p}(G, K_{P} , M) via right translation, determining an action of

Z_{P} = L(ℤ_{p})/L_{1} = P (ℤ_{p})/P_{1} ⟶^{∼} \underleftarrow{lim}_{m} I_{P,m}/U_{P,m}.

For any R^{×}-valued character χ of Z_{P} we define A_{p}(G, K_{P} , M)[χ] to be the submodule on which Z_{P} acts via χ. Note that

(2.4.4.1) Z_{P} ⟶^{∼} ∏_{w|p}(O_{E,w}^{×})^{l_{w}}, diag(A_{1}, ..., A_{l_{w}})) ↦ (det(A_{1}), ..., det(A_{l_{w}}).

By an arithmetic character of Z(ℤ_{p}) we will mean a character χ such that χ = χ_{0}ρ with χ_{0} a finite-order character and ρ arising from the restriction of an algebraic character of G as in (2.4.1). For an arithmetic character χ let m_{χ} be the smallest integer such that χ_{0} is trivial on I_{P,m_{χ}}/U_{P,m_{χ}}. For m ≥ m_{χ} we can extend χ to a character of I_{P,m} by setting χ(x) = χ(z) where z ∈ Z_{p} is such that z has the same image as x in I_{P,m}/U_{P,m}. We also extend χ to a character of the center of L(ℚ_{p}) as follows. We fix a uniformizer ξ_{w} of O_{E,w} for each w|p. Then we put

χ(diag(ξ_{w}^{r_{1}}1_{n_{1,w}}, ...., ξ_{w}^{r_{lw}} 1_{n_{lw,w}})) = ρ(diag(ξ_{w}^{r_{1}}1_{n_{1,w}}, ...., ξ_{w}^{r_{lw}} 1_{n_{lw,w}})).

Since any element of the center of L(ℚ_{p}) can be uniquely written as a product of a diagonal element as above and an element in L(ℤ_{p}) this is enough to define the desired extension. For any R^{×}-valued arithmetic character χ = χ_{0}ρ of Z_{P} we have injective maps

(2.4.4.2)  \begin{matrix}r_{χ} : A_{f}(G, K_{P,m}^{0}, χ, M) ↪ A_{p}(G, K_{P} , M)[χ], m ≥ m_{χ}, \\ r_{χ}(f)(g) = χ(x_{p})f(x), \\ g = γx, γ ∈ G(ℚ), x ∈ G(A^{f,p}) × I_{P,m}, \end{matrix}

where K_{P,m}^{0} = K^{p} × I_{P,m}. A product decomposition of g as in (2.4.4.2) always exists by weak approximation. An important observation is that the r_{χ}’s induce an isomorphism

(2.4.4.3)  \underrightarrow{lim}_{m≥n} A_{f}(G, K_{P,m}^{0}, χ, M/p^{r}M) ⟶^{∼} A_{p}(G, K_{P} , M/p^{r}M)[χ].

For the surjectivity we note that for any f ∈ A_{p}(G, K_{P} , M/p^{r}M)[χ] if m is sufficiently large then f belongs to A(G, K_{P,m}, M/p^{r}). For g ∈ G(A^{f,p}) × I_{P,m} let s_{χ}(f)(g) = χ(g_{p}^{−1})f(g). Then s_{χ}(f) ∈ A_{f}(G, K_{P,m}^{0}, χ, M/p^{r}) and r_{χ}(s_{χ}(f)) = f.

(2.4.5) Hecke actions.

Let K be an open compact subgroup of G(A^{f}). Suppose H ⊆ G(A) is a subgroup containing K and M is a ℤ[K]-module on which K_{v} acts trivially for all v not in some finite set Σ_{M}. For an open subgroup K′ ⊆ K let C(H, K′, M) be the space of functions f : H → M such that f(gk) = k^{−1}f(g) for all k ∈ K′. Then for any g ∈ H ∩ G(A^{f}) such that g_{v} = 1 if v ∈ Σ_{M} and any two open subgroups K′, K′′ ⊆ K, the double coset K′gK′′ determines a map from C(H, K′, M) to C(H, K′′, M) by

(2.4.5.1)  [K′gK′′]f(x) = ∑_{i} f(xg_{i}^{−1}), K′gK′′ = ⊔K′g_{i}.

This map is obviously functorial in M. It is easy to see that from (2.4.5.1) we get actions of double cosets on the various modules of functions defined in the preceding sections; one need only observe that these actions preserve the requisite topological properties. These actions are compatible with all the various comparisons and isomorphisms described so far. One important observation is that if g is such that the K_{P,m}gK_{P,m} have the same left-coset representatives for all m, then from (2.4.3.1) we get an action of T (g) = \underrightarrow{lim}_{m}[K_{P,m}gK_{P,m}] on A_{p}(G, K_{P} , M). If we further assume that g_{p} is in the center of L(ℚ_{p}), then T (g) commutes with the action of P (ℤ_{p}) and hence stabilizes each A_{p}(G, K_{P} , M)[χ], χ a character of Z_{P}. Let C_{P} ⊂ G_{1}(ℚ_{p}) be those elements g in the center of L(ℚ_{p}) such that

(2.4.5.2)  g^{−1}U_{P} (ℤ_{p})g ⊆ U_{P} (ℤ_{p}).

For such g we also have (2.4.5.3)

I_{P,m}gI_{P,m} = ⊔I_{P,m}gu_{i} and U_{P,m}gU_{P,m} = ⊔U_{P,m}gu_{i}, u_{i} ∈ U_{P} (ℤ_{p}).

Also, for g, g′ ∈ C_{P},

(2.4.5.4)  \begin{matrix}I_{P,m}gI_{P,m} · I_{P,m}g′I_{P,m} = I_{P,m}gg′I_{P,m} \\ U_{P,m}gU_{P,m} · U_{P,m}g′U_{P,m} = U_{P,m}gg′U_{P,m} \end{matrix} ,

where the multiplications are the usual double-coset multiplications. Let M_{1} = R_{O_{E,p}/ℤ_{p}}M_{n×n}. Suppose K_{p} ⊆ G_{1}(ℤ_{p}) and let Δ_{K} be the semigroup in M_{1}(ℚ_{p}) generated by K_{p} and those g such that g^{−1} ∈ C_{P}. Let M be an A[K]-module for which there exists a finite set of places Σ_{M}, p ∉ Σ_{M}, such that K_{v} acts trivially on M if v ∉ Σ_{M}. Let g ∈ G(A^{f}) be such that g_{v} = 1 for all v ∈ Σ_{M}, g_{p} ∈ C_{P}, and suppose that

(2.4.5.5)  KgK = ⊔Kg_{i}, g_{i,p}^{−1} ∈ Δ_{K}.

Under this assumption we define an action of KgK on A_{f}(G, K, χ, M) by

(2.4.5.6)  \begin{matrix}(KgK)f(x) = ∑_{i} χ(γ_{i})f(x_{i}), \\ γ_{i} ∈ G(ℚ), γ_{i}xg_{i}^{−1} = x_{i} ∈ G(A^{f,p}) × K_{p}; \end{matrix}

the assumption (2.4.5.5) ensures that γ_{i}^{−1} ∈ Δ_{K}. Let χ be an A^{×}-valued arithmetic character of Z_{P}. If g_{p} ∈ C_{P} and g_{v} = 1 for v ∈ Σ_{M} then (2.4.5.3) implies that (2.4.5.5) holds with K replaced by K_{P,m}^{0} for any m ≥ m_{χ}. In particular, (2.4.5.6) defines an action of T̃(g) = (K_{P,m}^{0}gK_{P,m}^{0}) on A_{f}(G, K_{P,m}^{0}, χ, M), m ≥ m_{χ}, which is multiplicative in such g by (2.4.5.4). Moreover, viewing A_{f}(G, K_{P,m}^{0}, χ, Λ_{𝔰}) as an A-submodule of A_{f}(G, K_{P,m}, ρ, σ)(F) we find that

(2.4.5.6)  T̃(g) = χ^{−1}(g_{p})[K_{P,m}gK_{P,m}].

Additionally, it is clear from the definitions that

(2.4.5.7)  r_{χ} ∘ T̃(g) = T (g) ∘ r_{χ},

where r_{χ} is as in (2.4.4.2).

(2.4.6) Pairings.

For K ⊆ G(A^{f}) an open compact subgroup, let

KS(G) = G(ℚ)\G(A^{f})/K.

This is a finite set. Let R be a commutative ring and let M, M′ be R[K]- modules on which K_{p} acts trivially. Suppose (•, •) : M × M′ → R is a Kequivariant R-pairing. Given an R^{×}-valued character χ of Γ_{K} we define an R-pairing

< •, • >_{K}: A_{f}(G, K, χ, M) × A_{f}(G, K, χ^{−1}, M′) → R,

(2.4.6.1)  < f, g >_{K}= ∑_{[x]∈_{K}S(G)}(f(x), g(x)), x ∈ G(A^{f,p}) × K_{p}.

These pairings (integration with respect to the measure dμ_{K}(g) of (0.2.4)) are clearly functorial in R, M, M′. The following lemma records some basic but important properties of these pairings. For simplicity we will assume that

(2.4.6.2)  γxk = x, γ ∈ G(ℚ), x ∈ G(A^{f}), k ∈ K ⟹ k = 1.

This holds for sufficiently small K.

(2.4.6.3) Lemma. Assume (2.4.6.2).

(i) If (•, •) is a perfect pairing, then so is < •, • >_{K}.

(ii) Let K′ ⊆ K be an open subgroup. Then

(2.4.6.4)  \begin{matrix}< f, tr_{K′,K}(h) >_{K}=< f, h >_{K′}, \\ f ∈ A_{f}(G, K, χ, M), h ∈ A_{f}(G, K′, χ^{−1}, M′). \end{matrix}

(iii) Suppose there exists a finite set of places Σ_{M} such that K_{v} acts trivially on M if v ∉ Σ_{M}. Let K′, K′′ ⊆ K be open subgroups and let g ∈ G(A^{f}) be such that g_{p} = 1 and g_{v} = 1 for all v ∈ Σ_{M}. Then

(2.4.6.5)  \begin{matrix}< [K′′gK′]f, h >_{K′}=< f, [K′g^{−1}K′′]h >_{K′′}, \\ f ∈ A_{f}(G, K′′, χ, M), h ∈ A_{f}(G, K′, χ^{−1}, M′). \end{matrix}

Part (i) holds because A_{f}(G, K, χ, M) is spanned by the functions δ_{x,m}, x ∈ G(A^{f,p}) × K_{p}, m ∈ M, defined by

δ_{x,m}(y) = \{ \begin{matrix}χ(γ^{−1}) · k^{−1}m & y = γxk, γ ∈ Γ_{K}, k ∈ K, \\ 0 & otherwise. \end{matrix}

The assumption (2.4.6.2) ensures that these functions are well-defined. Part (ii) is also clear from (2.4.6.2). Part (iii) follows from part (ii) and the observation that

[K′′gK′](f(x)) = tr_{K′∩g^{−1}K′′g,K′}(f(xg^{−1})).

For our purposes, the most important situation to which we will apply Lemma (2.4.6.3) is when R is the integer ring of some finite extension of F, χ comes from an arithmetic character of Z_{P}, and M = Λ_{𝔰} ⊗_{A} R. In this case we let M′ = Hom_{A}(Λ_{𝔰}, R), the latter being an R[K]-module with the usual action, and let (•, •) be the canonical pairing between M and M′. Let

(2.4.6.6)  < •, • >_{m,χ,σ}=< •, • >_{K_{P,m}^{0}}, m ≥ m_{χ},

where the right-hand side is defined by (2.4.6.1) with our current choices of M, M′, etc. Assuming that (2.4.6.2) holds for K_{P,m}^{0}, then all the conclusions of Lemma (2.4.6.3) hold for < •, • >_{m,χ,σ}.

(2.4.7) Comparison with the geometric picture.

Previously, we defined spaces of p-adic modular form for GU(V) from a geometric perspective. We now compare these to the spaces in (2.4.3). For simplicity we will assume that the similitude character maps K onto ℤ̂^{×}. In the definite situation the geometric constructions of (2.2) are simple. The varieties _{K(U,m)}Sh(V) clearly all have models over O_{v}; the base change to O_{v}/p^{r} is just T_{r,m}. From this it is easily deduced that V_{r,m}^{U}Γ(T_{r,m}, O_{T_{r,m}}) is naturally identified with the set of O_{v}/p^{r}-valued functions on_{K(U,m)}Sh(V) and so, under our hypotheses on K, with A_{f}(G, K(U, m), O_{v}/p^{m}) (in particular, these identifications are compatible with varying r and m. Thus we have that

(2.4.7.1)  V = \underleftarrow{lim}_{r} \underrightarrow{lim}_{m} V_{r,m}^{U} = \underleftarrow{lim}_{r} \underrightarrow{lim}_{m} A_{f}(G, K(U, m), O_{v}/p^{m}).

Then (2.4.3.1) identifies V with A_{p}(G, K_{B}, O_{v}). The spaces of p-adic modular forms for other parabolics are obtained by taking U_{P} -invariants. The restriction on K can be dropped; then V is identified with a direct sum of copies of A_{p}(G, K_{B}, O_{v}).

3. Fourier coefficients of Siegel Eisenstein series on unitary groups

(3.0) Conventions for automorphic forms on unitary groups. We let Σ_{E} denote the set of archimedean places of E. Let W be any hermitian space over K of dimension n, and define −W and 2W = W ⊕ (−W) as in §1. Set

W^{d} = {(v, v) | v ∈ W }, W_{d} = {(v, −v) | v ∈ W }

These are totally isotropic subspaces of 2W. Let P be the stablizer of W^{d} in U(2W). As a Levi component of P we take the subgroup M ⊂ U(2W) which is stablizer of both W^{d} and W_{d}. Then M ≃ GL(W^{d}). We let U denote the unipotent radical of P. The decomposition 2W = W^{d} ⊕ W_{d} is a complete polarization. Choose a basis (u_{1}, …, u_{m}) for W, so that (u_{i}, u_{i}) is a basis for W^{d}. Let (−v_{j}, v_{j}), j = 1, …, m, be the dual basis of W_{d}. For any A ∈ GL(n)_{K}, we define

m(A) to be the element of U(2W) with matrix ( \begin{matrix}A & 0 \\ 0 & ^{t}Ā^{−1} \end{matrix} ) in the basis

{(u_{i}, u_{i})} ∪ {(−v_{j}, v_{j})}, where Ā is the image of A under the non-trivial Galois automorphism of K/E. We will let

w = ( \begin{matrix}0 & 1_{n} \\ −1_{n} & 0 \end{matrix} )

in the same basis; then P \P wP is the big cell in the Bruhat decomposition of P \U(2W). All automorphic forms will be assumed K_{∞}-finite, where K_{∞} will be a maximal compact modulo center subgroup of either U(2W)(ℝ) or U(W)(ℝ), as appropriate. Conventions are as in §1.5; in particular K_{∞} will be associated to a CM point, except where otherwise indicated. We let GU(2W) be the group of rational similitudes, as in §1. Let GP ⊂ GU(2W) denote the stabilizer of W^{d}, and let GM be the normalizer of M in GP. We can identify GM ⟶^{∼} M × 𝔾_{m} where M acts as GL(W^{d}) and 𝔾_{m} acts via the center of GL(W_{d}). Here and below 𝔾_{m} designates 𝔾_{m,ℚ}. In other words, writing GP in standard form:

(3.0.1)  GP = \{( \begin{matrix}A & B \\ 0 & D \end{matrix} )\}

with D = d ·^{t}c(A)^{−1} for some scalar d, we can identify the factor 𝔾_{m} ⊂ GM with the group of matrices

(3.0.2)  \{d(t) = ( \begin{matrix}1_{n} & 0 \\ 0 & t1_{n} \end{matrix} )\} ⊂ GU(2W).

Let v be any place of E, | · |_{v} the corresponding absolute value on ℚ_{v}, and let

(3.0.3)  δ_{v}(p) = |N_{K/E} ∘ det(A(p))|v^{\frac{n}{2}} |ν(p)|^{− \frac{1}{2} n^{2}}, p ∈ GP (E_{v}).

This is the local modulus character of GP (E_{v}). The adelic modulus character of GP (A), defined analogously, is denoted δ_{A}. Let χ be a Hecke character of K. We view χ as a character of M(𝔸_{E}) ⟶^{∼} GL(W^{d}) via composition with det. For any complex number s, define

δ_{P,A}^{0}(p, χ, s) = χ(det(A(p))) · |N_{K/E} ∘ det(A(p))|_{v}^{s}|ν(p)|^{−ns}

δ_{A}(p, χ, s) = δ_{A}(p)δ_{P,A}^{0}(p, χ, s) =

= χ(det(A(p))) · |N_{K/E} ∘ det(A(p))|_{v}^{\frac{n}{2} +s}|ν(p)|^{− \frac{1}{2} n^{2}−ns}.

The local characters δ_{P,v}(·, χ, s) and δ_{P,v}^{0}(·, χ, s) are defined analogously. The restrictions to M of the characters δ_{P,v}, δ_{P,v}^{0}, and so on are denoted by the same notation. As in (2.2), the symmetric domain X(2W) is isomorphic to the X_{n,n}^{d} of tube domains. Let τ_{0} ∈ X(2W) be a fixed point of K_{∞}, X^{+} the connected component of X(2W) containing τ_{0}, GU(n, n)^{+} ⊂ GU(2W)(ℝ) the stabilizer of X^{+}. Thus X^{+} ⟶^{∼} ∏_{σ∈Σ_{E}} X_{n,n;σ}^{+} with X_{n,n;σ}^{+} the symmetric space associated to U(n, n) = U(E_{σ}). Let GK_{∞} ⊂ GU(n, n)^{+} be the stabilizer of τ_{0}; thus GK_{∞} contains K_{∞} as well as the center of GU(n, n). In the tube domain realization, the canonical holomorphic automorphy factor associated to GP and GK_{∞} is given as follows. Let τ = (τ_{σ})_{σ∈Σ_{E}} ∈ X^{+} and

h = ( ( \begin{matrix}A_{σ} & B_{σ} \\ C_{σ} & D_{σ} \end{matrix} ) )_{σ∈Σ_{E}} ∈ GU(n, n)^{+}. Then the triple

(3.0.4) J(h, τ) = (C_{σ}τ_{σ} + D_{σ})_{σ∈Σ_{E}}, J′(h, τ) = (C̄_{σ}^{t}τ_{σ} + D̄_{σ})_{σ∈Σ_{E}}, ν(h)

defines a canonical automorphy factor with values in (GL(n, ℂ)×GL(n, ℂ))^{d} × GL(1, ℝ) (note the misprint in [H3, 3.3]). Write J(h) = J(h, τ_{0}) = (J_{σ}(h))_{σ∈Σ_{E}} and define J′(h) and J′_{σ}(h) analogously. Given a pair of integers (μ, κ), we define a complex valued function on GU(n, n)^{+}:

(3.0.5) J_{μ,κ}((h_{σ})_{σ∈Σ_{E}}) = ∏_{σ∈Σ_{E}} det J_{σ}(h)^{−μ} · det(J′_{σ}(h))^{−μ−κ} · ν(h)^{n(μ+κ)}

For purposes of calculation, we let τ_{0} = (σ(ℷ))_{σ∈Σ}, where ℷ is the trace zero element of K chosen in (1.4). We also write ℷ_{σ} = σ(ℷ). Then the stabilizer GK_{∞} is rational over the reflex field E(GU(2W), X(2W)) = E(K, Σ), and the map h ↦ J(h) is a rational function on the algebraic group GU(2W) with values in GK_{∞}, rational over E(K, Σ).

(3.1) The Siegel Eisenstein series and doubling.

In this section we let G denote U(W), H = U(2W), viewed alternatively as groups over E or, by restriction of scalars, as groups over ℚ. Identifying G with U(−W), we obtain a natural embedding G × G ⊂ H. We choose maximal compact subgroups K_{∞,G} = ∏_{v∈Σ_{E}} K_{v,G} ⊂ G(ℝ) and K_{∞} = ∏_{v∈Σ_{E}} K_{v} ⊂ H(ℝ) – as at the end of the previous subsection – such that

K_{∞} ∩ (G × G)(ℝ) = K_{∞,G} × K_{∞,G}.

We will be more precise about these choices in (4.3).

(3.1.1) Formulas for the Eisenstein series

Let χ be a unitary Hecke character of K. We view χ as a character of M(𝔸_{E}) ⟶^{∼} GL(W^{d}) via composition with det. Consider the induced representation

(3.1.1.1)  I(χ, s) = Ind(χ| · |_{K}^{s}) ⟶^{∼} ⊗_{v} I_{v}(χ_{v}| · |_{v}^{s}),

the induction being normalized; the local factors I_{v}, as v runs over places of E, are likewise defined by normalized induction. At archimedean places we assume our sections to be K_{∞}-finite. For a section f(h; χ, s) ∈ I(χ, s) (cf. [H4, I.1]) we form the Eisenstein series

(3.1.1.2)  E_{f}(h; χ, s) = ∑_{γ∈P (k)\U(2V)(k)} f(γh; χ, s)

This series is convergent for Re(s) > n/2, and it can be continued to a meromorphic function on the entire plane. We now fix an integer m ≥ n and assume

(3.1.1.3)  χ|_{A} = ε_{K}^{m}

Then the main result of [T] states that the possible poles of E_{f}(g; χ, s) are all simple, and can only occur at the points in the set

(3.1.1.4)  \frac{n − δ − 2r}{2} , r = 0, …, [\frac{n − δ − 1}{2}],

where δ = 0 if m is even and δ = 1 if m is odd.

(3.1.2) The standard L-function via doubling. Let (π, H_{π}) be a cuspidal automorphic representation of G, (π^{∨}, H_{π^{∨}}) its contragredient, which we assume given with compatible isomorphisms of G(A)-modules

(3.1.2.1)  π ⟶^{∼} ⊗_{v} π_{v}, π^{∨} ⟶^{∼} ⊗_{v} π_{v}^{∨}.

The tensor products in (3.1.2.1) are taken over places v of the totally real field E, and at archimedean places π_{v} is a admissible (𝔤_{v}, K_{v,G})-module, which we assume to be of cohomological type, with lowest K_{v,G}-type (cf., e.g., [L1]) τ_{v}. For each v we let (•, •)_{π_{v}} denote the canonical bilinear pairing π_{v} ⊗ π_{v}^{∨} → ℂ.

Let f(h; χ, s) be a section, as above, φ ∈ H_{π}, φ′ ∈ Hπ̌, and let φ′_{χ}(g) = φ′(g)χ^{−1}(det g′). We define the zeta integral: (3.1.2.2)

Z(s, φ, φ′, f, χ) = ∫_{G×G)(ℚ)\(G×G)(A)} E_{f}((g, g′); χ, s)φ(g)φ′_{χ}(g′)dgdg′.

The Haar measures dg = dg′ on G(A) are normalized as in (0.2.2). The relation to the integral in terms of Tamagawa measure is determined by (0.2.3). The theory of this function, due to Piatetski-Shapiro and Rallis [PSR], was worked out (for trivial χ) by Li [L2] and more generally in [HKS,§6]. We make the following hypotheses: (3.1.2.4) Hypotheses

(a) There is a finite set of finite places S_{f} of E such that, for any nonarchimedean v ∉ S_{f}, the representations π_{v}, the characters χ_{v}, and the fields K_{w}, for w dividing v, are all unramified;

(b) The section f admits a factorization f = ⊗_{v}f_{v} with respect to (3.1.1.1).

(c) The functions φ, φ′ admit factorizations φ = φ_{S_{f}} ⊗ ⊗_{v∉S_{f}} φ_{v}, φ′ = φ_{S_{f}} ⊗ ⊗_{v∉S_{f}} φ′_{v}, with respect to (3.1.2.1)

(d) For v ∉ S_{f} non-archimedean, the local vectors f_{v}, φ_{v}, and φ′_{v}, are the normalized spherical vectors in their respective representations, with (φ_{v}, φ′_{v})_{π_{v}} = 1.

(e) For v archimedean, the vector φ_{v} (resp. φ′_{v}) is a non-zero highest (resp.

lowest) weight vector in τ_{v} (resp. in τ_{v}^{∨}), such that (φ_{v}, φ′_{v})_{π_{v}} = 1. We let S = Σ_{E} ∪ S_{f}. Define

(3.1.2.5)  d_{n}(s, χ) = ∏_{r=0}^{n−1} L(2s + n − r, ε_{K}^{n−1+r}) = ∏_{v} d_{n,v}(s, χ),

the Euler product on the right being taken only over finite places;

(3.1.2.6)  Q_{W}^{0} (φ, φ′) = ∫_{G(ℚ)\G(A)} φ(g)φ′(g)dg;

(3.1.2.7) Z_{S}(s, φ, φ′, f, χ) = ∫_{∏_{v∈S} G(E_{v})} f_{v}((g_{v}, 1); χ, s)(π_{v}(g_{v})φ, φ′)dg_{v};

Z̃_{S}(s, φ, φ′, f, χ) = ∏_{v∈S}[d_{n,v}(s, χ)]Z_{S}(s, φ, φ′, f, χ).

The integral in (3.1.2.7) converges absolutely in a right halfplane and admits a meromorphic continuation to all s.^{7} We have the following identity of meromorphic functions on ℂ:

7For non-archimedean places this is worked out in detail in [HKS]. There is no published reference for unitary groups at archimedean places in general. Shimura [S97] calculates the archimedean integrals explicitly for holomorphic automorphic forms of scalar weight. For general π_{∞} meromorphic continuation is established by Kudla and Rallis [KR] for symplectic groups by reduction to principal series. The same technique applies to unitary groups, bearing in mind that not all unitary groups are quasi-split. For the special values we have in mind we appeal to the explicit calculations of Garrett [G].

Documenta Mathematica · Extra Volume Coates (2006) 393–464

(3.1.2.8) Basic Identity of Piatetski-Shapiro and Rallis.

d_{n}(s, χ)Z(s, φ, φ′, f, χ) = Z̃_{S}(s, φ, φ′, f, χ)L^{S}(s + \frac{1}{2}, π, χ, St).

Here L^{S}(s+ \frac{1}{2}, π, χ, St) = ∏_{v∉S} L_{v}(s+ \frac{1}{2}, π_{v}, χ_{v}, St), where L_{v}(s+ \frac{1}{2}, π_{v}, χ_{v}, St) is the local Langlands Euler factor attached to the unramified representations π_{v} and χ_{v} and the standard representation of the L-group of G × R_{K/ℚ}𝔾_{m,K}.^{8}

For any place v ∉ S, there is a formal (unramified) base change from π_{v} to a representation BC(π_{v}) of G(K ⊗_{E} E_{v}) ⟶^{∼} GL(m, K ⊗_{E} E_{v}), and

L_{v}(s, π_{v}, χ_{v}, St) = L(s, BC(π_{v}) ⊗ χ_{v} ∘ det),

where the right-hand term is the standard Godement-Jacquet Euler factor (cf. [H4,I.1] for a further discussion). If we assume φ_{S_{f}} and φ′_{S_{f}} to be factorizable over the v ∈ S_{f}, with respect to the isomorphisms (3.1.2.1), then the integral Z_{S} also breaks up as a product of local integrals multiplied by the factor Q_{W}^{0} , as in [H3,H4], as well as [PSR,Li92]. To treat congruences it seems preferable not to impose factorizability at this stage. However, under special hypotheses on the local data we can obtain a factorization, as follows. Write G_{v} = G(E_{v}), and let K_{v} ⊂ G(E_{v}) be a compact open subgroup fixing φ. The natural map G × G → P \H defines an isomorphism between G × 1 and the open G × G orbit in the flag variety P \H [PSR, p. 4]. In particular, P · (G × 1) is open in H and P ∩ (G × 1) = {1}. It follows that, if Y is any locally constant compactly supported function on G_{v}, there is a unique section f_{Y} (h, χ, s) ∈ I_{v}(χ_{v}, s) such that f_{Y} ((g, 1), χ, s) = Y (g) for all g ∈ G_{v}, s ∈ ℂ. Let f_{K_{v}}(h, χ, s) = f_{Y (K_{v})}(h, χ, s), where Y (K_{v}) is the characteristic function of the open compact subgroup K_{v} chosen above. With this choice, we have

(3.1.2.9)  ∫_{G_{v}} f_{K_{v}}((g_{v}, 1); χ, s)(π_{v}(g_{v})φ, φ′)dg_{v} = vol(K_{v})

for any s. If we choose f_{v} = f_{K_{v}} for all v ∈ S_{f}, the basic identity becomes

d_{n}(s, χ)Z(s, φ, φ′, f, χ) =

= d_{n,S}(s, χ) · vol(K_{S_{f}})Z_{∞}(s, φ, φ′, f, χ)L^{S}(s + \frac{1}{2}, π, χ, St),

where d_{n,S}(s, χ) = ∏_{v∈S_{f}} [d_{n,v}(s, χ)], K_{S_{f}} = ∏ K_{v}, and

(3.1.2.10) Z_{∞}(s, φ, φ′, f, χ) = ∫_{∏_{v∈Σ_{E}} G(E_{v})} f_{v}((g_{v}, 1); χ, s)(π_{v}(g_{v})φ, φ′)dg_{v}.

8As in the previous footnote, there is no published reference for the meromorphic continuation and functional equation of standard L-functions of unitary groups, although the results of Kudla and Rallis for symplectic groups adapt to the case of unitary groups. In the applications we will restrict attention to π admitting base change to automorphic representations of GL(n, K), which immediately implies the analytic continuation of the standard L-functions.

Documenta Mathematica · Extra Volume Coates (2006) 393–464

The integrals in (3.1.2.10) are purely local in the following sense. For any archimedean place v we can define a local analogue of (3.1.2.7) by

(3.1.2.11)  Z_{v}(s, φ_{v}, f_{v}, χ_{v}) = ∫_{G(E_{v})} f_{v}((g_{v}, 1); χ_{v}, s)π_{v}(g_{v})φ_{v}dg_{v}.

This is a function of s with values in the K_{∞,v}-finite vectors of π_{v}, absolutely convergent and holomorphic in a right half-plane, and admitting a meromorphic continuation to ℂ (see note 5). Let τ_{v}^{+} ⊂ τ_{v} denote the line spanned by the highest weight vector φ_{v}, let p_{v}^{+} : π_{v} → τ_{v}^{+} denote orthogonal projection. Define the meromorphic function Z_{v}(s, f_{v}, χ_{v}) by

p_{v}^{+}(Z_{v}(s, φ_{v}, f_{v}, χ_{v})) = Z_{v}(s, f_{v}, χ_{v}) · φ_{v}.

This is well-defined, because τ_{v}^{+} is a line, and does not depend on the choice of φ_{v} because both sides are linear functions of φ_{v}. Let Z_{∞}(s, f, χ) = ∏_{v∈Σ_{E}} Z_{v}(s, f_{v}, χ_{v}) It then follows that

(3.1.2.12)  Z_{∞}(s, φ, φ′, f, χ) = Z_{∞}(s, f, χ)Q_{W}^{0} (φ, φ′),

hence that

(3.1.2.13) d_{n}(s, χ)Z(s, φ, φ′, f, χ)

= d_{m,S}(s, χ) · vol(K_{S_{f}})Z_{∞}(s, f, χ)L^{S}(s + \frac{1}{2}, π, χ, St)Q_{W}^{0} (φ, φ′)

We note the following consequence of the basic identity in the form (3.1.2.13). Let K_{f} = K_{S_{f}} × K^{S}, where K^{S} = ∏_{w∉S} K_{w} is a product of hyperspecial maximal compact subgroups fixing φ and φ′.

(3.1.2.14) Hypothesis. We assume f, s = s_{0}, and χ can be chosen so that

d_{n,S}(s_{0}, χ)Z_{∞}(s_{0}, f, χ) ≠ 0.

Thus we are staying away from poles of the local Euler factors in d_{n,S}(s, χ)) and the global Euler products d_{n}(s, χ) and L^{S}(s+ \frac{1}{2}, π, χ, St) have neither zeros nor poles at s = s_{0}. This hypothesis is easy to verify in practice, e.g. in the situation of [H3]; the only subtle point is the non-vanishing of Z_{∞}(s_{0}, f, χ) when ϕ_{v} is holomorphic and the Eisenstein series defined by f_{v} is nearly holomorphic, and in this case the non-vanishing follows from the arguments of Garrett [G]. Let A_{0}(π, S), resp. A_{0}(π^{∨}, S) denote the space spanned by K_{f}-invariant cusp forms on G, that generate irreducible automorphic representations whose vcomponent is isomorphic to π_{v} (resp. to π_{v}^{∨}) for all v ∉ S_{f}, and belonging to the highest weight subspace τ_{v}^{+} of τ_{v} (resp. to the lowest weight subspace of τ_{v}^{∨} for all v ∈ Σ_{E}. Then (3.1.2.12) asserts that the bilinear forms Z(s_{0}, φ, φ′, f, χ) and Q_{W}^{0} on A_{0}(π, S) are proportional. (If π occurs with multiplicity one in A_{0}(G), then this is automatic.) This simplifies the arguments of §3 of [H3], proving, when E = ℚ, that critical values of L(s, π, χ, St) are K-multiples of a basic period equal to an elementary expression multiplied by a square norm of the form Q_{W}^{0} (φ, φ′), where φ and φ′ are arithmetic holomorphic modular forms of the given type.^{9} In particular, this gives a somewhat more natural proof of Corollary 3.5.12 of [H3], to the effect that, under the hypotheses of loc. cit. (existence of sufficiently many critical values) Q_{W}^{0} (φ, φ′) depends up to arithmetic factors only on the abstract representation π^{S}.

(3.1.2.15) Remark. Local Euler factors L_{v}(s, π_{v}, χ_{v}, St) are defined in [HKS] for all finite places, by the method of Piatetski-Shapiro and Rallis. It should not be difficult to prove by global methods that these factors coincide with L(s, BC(π_{v}) ⊗ χ_{v} ∘ det), at least when π_{v} is a local component of an automorphic cuspidal representation for a definite unitary group. A complete proof would require local functional equations at archimedean primes. When n = 2 the unitary group can be compared simply to the multiplicative group of a quaternion algebra, and the result can be proved easily in that case directly.

(3.1.3) Eisenstein series and zeta integrals on similitude groups.

We now return to the situation of (3.1). Let GH = GU(2W), and consider the subgroup GU(W, −W) = G(U(W) × U(−W) ⊂ GH. The induced representation I(χ, s) and the Eisenstein series E_{f}((g, g′); χ, s) can be extended in various ways to automorphic forms on GH. Let GP ⊂ GH denote the Siegel parabolic defined in (3.2.5). Global characters of GM = M ×𝔾_{m} are given by pairs (χ, υ) where χ is a Hecke character of M^{ab} = R_{K/ℚ}𝔾_{m,ℚ}, lifted to a character of M by composition with the determinant, and υ is a Hecke character of A^{×}/ℚ^{×}. Let

(3.1.3.1)  I(χ, υ, s) = Ind_{GP}^{GH}((χ| · |_{K}^{s}) ∘ det ·υ ∘ ν).

For any section f(h; χ, υ, s) ∈ I(χ, υ, s) we form the Eisenstein series E_{f}(h, χ, υ, s) by the analogue of the formula (3.1.1.2). The character υ factors through a character of GH and does not affect convergence. Let π, π′ be automorphic representations of GU(W), with central characters ξ, ξ′, respectively. Let φ ∈ π, φ′ ∈ π′, and consider φ ⊗ φ′ by restriction as an automorphic form on GU(W, −W). Let Z be the identity component of the center of GU(W, −W), which we may also view as a central subgroup of GH, or (via projection) as a central subgroup of GU(W). We assume

(3.1.3.2)  ξ · ξ′ · ξ_{χ,υ} = 1;

9In [H3] only values of s in the absolutely convergent range are considered, but the argument remains valid in general under hypothesis (3.1.2.14). See [H5] for a more extended discussion of this point.

Documenta Mathematica · Extra Volume Coates (2006) 393–464

here ξ_{χ,υ} is the central character of I(χ, υ, s). We can then define the zeta integral

(3.1.3.3) Z(s, φ, φ′, f, χ, υ) =

∫_{Z(A)GU(W,−W))(ℚ)\(GU(W,−W)(A)} E_{f}((g, g′); χ, s)φ(g)φ′_{χ}(g′)dgdg′.

The basic identity (3.1.2.8) then takes the following form (cf. [H3,(3.2.4)]): (3.1.3.4)

d_{n}(s, χ)Z(s, φ, φ′, f, χ, υ) = Q_{W} (φ, φ′)Z̃_{S}(s, φ, φ′, f, χ)L^{S}(s + \frac{1}{2}, π, χ, St).

where

(3.1.3.5)  Q_{W} (φ, φ′) = ∫_{Z(A)GU(W)(ℚ)\GU(W)(A)} φ(g)φ′(g)ξ_{χ,υ}^{−1}dg

and the remaining terms are as in (3.1.2). The period Q_{W} (φ, φ′) is slightly more natural from the standpoint of Shimura varieties.

(3.1.4) Holomorphic Eisenstein series.

Fix (μ, κ) as in (3.0.1). Define

χ^{∗} = χ · |N_{K/E}|^{\frac{κ}{2}} .

Suppose the character χ has the property that

(3.1.4.1)  χ_{σ}^{∗}(z) = z^{κ}, χ_{cσ}^{∗}(z) = 1 ∀σ ∈ Σ_{E}

Then the function J_{μ,κ}, defined in (3.0.5), belongs to

(3.1.4.2)  I_{n}(μ − \frac{n}{2}, χ^{∗})_{∞} = I_{n}(μ + \frac{κ − n}{2}, χ)_{∞} ⊗ |ν|∞^{\frac{nκ}{2}}

(cf. [H3,(3.3.1)]). More generally, define

(3.1.4.3) J_{μ,κ}(h, s + μ − \frac{n}{2}) = J_{μ,κ}(h)| det(J(h) · J′(h))|^{−s} ∈ I_{n}(s, χ^{∗})_{∞}

When E = ℚ, these formulas just reduce to the formulas in [H3]. Let f_{∞}(h, χ, s) = J_{μ,κ}(h, s+μ−\frac{n}{2}), and suppose the Eisenstein series E_{f}(h; χ, s) is holomorphic at s = 0. The local section J(μ, κ) is a holomorphic vector in the corresponding induced representation, and in what follows we will extend it to a global section f so that E_{f}(h; χ, 0) is a holomorphic automorphic form. This is always the case if χ/|χ| is a character of U(1) and if f is a Siegel-Weil section, as we will be assuming in later articles. It is also the case for the specific sections f considered in (3.2) and (3.3), where holomorphy is verified by explicit calculation of Fourier coefficients (see especially (3.3.3.2), (3.2.2.3), and (3.3.4.8)). As in [H3, (3.3.4)] we can identify E_{f}(h; χ, 0) with an element of H^{0}(Sh(2W), E_{μ,κ}) where E_{μ,κ} is the automorphic vector bundle defined in [H3,(3.3)]. The identification is as in (1.3.6) and depends on a choice of canonical trivialization of the fiber of E_{μ,κ} at τ_{0}. The center of symmetry s = \frac{1}{2} for L(s, π, χ, St) in the unitary normalization corresponds via (3.1.2.8) to a zeta integral with the Eisenstein series at s = s_{0} = 0. Since χ is by (3.1.1.3) a unitary character, this corresponds in turn to the relation to s_{0} = μ + \frac{κ−n}{2} = 0. More generally, the value of the motivically normalized L-function

L^{mot}(s, π, χ∗, St) =^{def} L(s − \frac{n − κ − 1}{2}, π, χ, St)

at s = s_{0} + \frac{n−κ}{2} corresponds as above to the Eisenstein series at s_{0} = μ+ \frac{κ−n}{2}), i.e. at s = μ, as in [H3] (where μ was called m). It follows from (3.1.4.1) that we can choose m in (3.1.1.3) so that

(3.1.4.4)  m = n + 2s_{0} = 2μ + κ;

the assumption m ≥ n translates to s_{0} ≥ 0, so the Eisenstein series is always to the right of the center of symmetry.

(3.2) Fourier coefficients of Eisenstein series:

General considerations.

(3.2.1) Notation and preliminaries.

We let V, 2V = V ⊕ −V, and H = U(2V) be as in (3.1) with n = dim V. Let ℷ be as in (1.4). We fix an orthogonal basis u_{1}, ⋯, u_{n} of V, and set

(3.2.1.1)  e_{j} = (u_{j}, u_{j}), f_{j} = δ_{j} · (−u_{j}, u_{j})

where

(3.2.1.2)  δ_{j} = \frac{1}{2 ℷ < u_{j}, u_{j} >_{V}}

With respect to this basis, the matrix of the skew-hermitian form <, >_{2V,ℷ} is given by

( \begin{matrix}0 & 1_{n} \\ −1_{n} & 0 \end{matrix} )

Let g ∈ GL(V). When no confusion is possible, we use the same letter g to denote the n × n matrix (g_{ij}) given by

g(u_{i}) = ∑_{j=1}^{n} g_{ji}u_{j}

Write δ = diag(δ_{1}, ⋯, δ_{n}). Then g ∈ U(V) if and only if ^{t}ḡδ^{−1}g = δ^{−1}, or equivalently gδ^{t}ḡ = δ. With respect to the basis {e_{i}, f_{j}}, the matrix corresponding to (g, 1) ∈ U(V) × U(V) ⊆ U(2V) is

(3.2.1.3)  (g, 1) = ( \begin{matrix}\frac{1}{2}(1_{n} + g) & \frac{1}{2}(1_{n} − g)δ \\ \frac{1}{2}δ^{−1}(1_{n} − g) & \frac{1}{2}δ^{−1}(1_{n} + g)δ \end{matrix} )

We let

w′ = diag(−1_{V} , 1_{V}) = ( \begin{matrix}−1_{n} & 0 \\ 0 & 1_{n} \end{matrix} ) with respect to 2V = V ⊕ −V

Then with respect to the basis e_{i}, f_{j} we have

(3.2.1.4)  w′ = ( \begin{matrix}δ & 0 \\ 0 & ^{t}δ̄^{−1} \end{matrix} ) · w = m(δ) · w, w = ( \begin{matrix}0 & 1_{n} \\ −1_{n} & 0 \end{matrix} ) ,

This amounts to taking g = −1_{n} in (3.2.1.3). In other words, the coset P · (−1_{n}, 1_{n}) ⊂ P · G × {1} belongs to the big cell P wP, and indeed

(3.2.1.5)  P · (−1_{n}, 1_{n}) = P w · 1

More generally, for any positive integer r ≤ n let V_{r} be the subspace of V spanned by u_{1}, ⋯, u_{r}. Let V_{r}^{⊥} be the othogonal complement of V_{r} in V. We define

(3.2.1.6)  w′_{r} = diag(−1_{V_{r}}, 1_{V_{r}^{⊥}}, 1_{V_{r}}, 1_{V_{r}^{⊥}}) ∈ U(2V)

Then w′ = w′_{n}. With respect to the basis e_{i}, f_{j} we have

(3.2.1.7)  w′_{r} = ( \begin{matrix}_{r}δ & 0 & 0 & 0 \\ 0 & 1_{n−r} & 0 & 0 \\ 0 & 0 & _{r}^{t}δ̄^{−1} & 0 \\ 0 & 0 & 0 & 1_{n−r} \end{matrix} ) · w_{r},

with _{r}δ = diag(δ_{1}, ⋯, δ_{r}), where

(3.2.1.8)  w_{r} = ( \begin{matrix}0 & 0 & 1_{r} & 0 \\ 0 & 1_{n−r} & 0 & 0 \\ −1_{r} & 0 & 0 & 0 \\ 0 & 0 & 0 & 1_{n−r} \end{matrix} ) .

By means of the basis {e_{i}, f_{j}} we identify elements of H as 2n × 2n matrices. Then if v is any finite place of E we define H(O_{v}) to be the subgroup of H(E_{v}) consisting of matrices whose entries are in O_{K} ⊗ O_{v}. Let B be the stablizer of the flag

[e_{1}] ⊂ [e_{1}, e_{2}] ⊂ ⋯ ⊂ [e_{1}, ⋯, e_{n}]

where [e_{1}, ⋯, e_{r}] denotes the linear span of e_{1}, ⋯, e_{r}. Then B is a Borel subgroup, and we have the Iwasawa decomposition H(E_{v}) = B(E_{v})H(O_{v}). In what follows we fix a non-trivial character ψ = ∏ ψ_{v} of A/E, as follows. Let e^{0} = ∏ e_{p}^{0} be the unique character of A_{ℚ}/ℚ such that

e_{∞}^{0}(x) = e^{2πix} (x ∈ ℝ),

and that e_{p}^{0} has conductor ℤ_{p} for every finite p. Let e = ∏ e_{v} be the character of A/E defined by

(3.2.1.9)  e(x) = e^{0}(Tr_{E/Q}(x)) (x ∈ A)

Alternatively, we may characterize e as the unique character of A/E such that for every archimedean place v we have

e_{v}(x) = e^{2π\sqrt{−1}x} (x ∈ E_{v} = ℝ).

An arbitrary character of A/E is given by x ↦ e(ax), with a some element of E. We let ψ be one such character with a ∈ E totally positive, fixed henceforward. We can and will always assume a to be a unit at all primes dividing p. Thus

(3.2.1.10)  ψ(x) = e(ax) = e^{0}(Tr_{E/Q}(ax)) (x ∈ A)

In paricular, for every archimedean place v we have

ψ_{v}(x) = e^{2πa\sqrt{−1}x} (x ∈ E_{v} = ℝ).

(3.2.2) Formulas for Fourier coefficients.

We start with a general Siegel Eisenstein series F = E^{f}(h, χ, s) with f ∈ I(χ, s). Here we have written E^{f} instead of E_{f}, in order to leave space for a subscript to denote Fourier coefficients. Let Her_{n} be the space of all n × n hermitian matrices. For β ∈ Her_{n}(E) we define the character ψ_{β} of U(ℚ)\U(A) by

ψ_{β}(n(b)) = ψ(tr(βb))

Note that we have tr(ββ′) ∈ E for any β, β′ ∈ Her_{n}(E). We now fix a Haar measure dx on U(A) ≃ Her_{n}(A) as follows. First we take counting measure on the discrete subgroup Her_{n}(E) ⊆ Her_{n}(A). We choose dx, so that the quotient mesaure on U(E)\U(A) = Her_{n}(E)\Her_{n}(A) is normalized, with total volume 1. Consider the lattice Λ ⊆ Her_{n}(E) consisting of all hermitian matrices with entries in O_{K}. We shall also need the dual lattice Λ^{∗}, defined by

Λ^{∗} = {β ∈ Her_{n}(E) | tr(βξ) ∈ O_{E} ∀ ξ ∈ Λ}

For each finite place v of E we set

Λ_{v} = Her_{n}(O_{v}) = Λ ⊗ O_{v},

Define Λ_{v}^{∗} similarly. Then Λ_{v} = Λ_{v}^{∗} unless v ramifies in K. Let dx_{v} be the Haar measure of Her_{n}(E_{v}) normalized by ∫_{Λ_{v}} dx_{v} = 1. For any archimedean place v we set

dx_{v} = | ⋀_{j=1}^{n} dx_{jj} ⋀_{j<k}(2^{−1}dx_{jk} ∧ dx̄_{jk})|

where x_{jk} is the (j, k)-entry of x_{v}. There is a constant c(n, E, K) so that

dx = c(n, E, K) · ∏_{v} dx_{v}.

Since Her_{n}(A) is the product of n copies of A and n(n − 1)/2 copies of A_{K}, we obtain (say from [Tate])

(3.2.2.1)  c(n, E, K) = 2^{n(n−1)[E:ℚ]/2}|δ(E)|^{−n/2}|δ(K)|^{−n(n−1)/4},

where δ(E) and δ(K) are the discriminants of E and K. This is the same as [S97], p. 153. For β ∈ Her_{n}(E) we define the β-th Fourier coefficient

F_{β}(h) = ∫_{U(ℚ)\U(A)} F (uh)ψ_{−β}(u)du

as in (1.5.6). We now assume that f is factorizable, and write f = ⊗f_{v}. If β has full rank n then a familiar calculation gives (3.2.2.2)

E_{β}^{f}(h, χ, s) = c(n, E, K)·∏_{v} ∫_{U(E_{v})} f_{v}(wn_{v}h_{v}, χ_{v}, s)ψ_{−β}(n_{v})dn_{v}, (det β ≠ 0)

the product being over all places of E. Here w is the Weyl group element given by (3.2.1.4). Remark 3.2.2.3. Suppose that for at least one place v the function f_{v}(•, χ_{v}, s) is supported on the big cell P (E_{v})wP (E_{v}). Then (3.2.2.2) is valid for h ∈ P (𝔸) and any β. Indeed for h ∈ P (𝔸) we have

f(γh, χ, s) ≠ 0⟹γ ∈ P (E)wP (E) = P (E)wU(E)

So that

E^{f}(h, χ, s) = ∑_{δ∈U(E)} f(wδh, χ, s)

and (3.2.2.1) follows immediately for any β, not necessarily of full rank.

Write

(3.2.2.4)  W_{β,v}(h_{v}, f_{v}, s) = ∫_{U(E_{v})} f_{v}(wn_{v}h_{v}, χ_{v}, s)ψ_{−β}(n_{v})dn_{v}.

This function satisfy a transformation law as follows. Suppose

m = m(A) = ( \begin{matrix}A & 0 \\ 0 & ^{t}Ā^{−1} \end{matrix} ) ∈ M(E_{v})

Then

(3.2.2.5)

W_{β,v}(mh_{v}, f_{v}, s) = |N ∘ det A|_{v}^{n/2−s}χ_{v}(det A) · W_{^{t}ĀβA,v}(h_{v}, f_{v}, s)
= |N ∘ det A|_{v}^{\frac{n−κ}{2} −s}χ_{v}^{∗}(det A) · W_{^{t}ĀβA,v}(h_{v}, f_{v}, s)

where N = N_{K/E}. We now recall a calculation of Shimura. In what follows, a is the totally positive element of E, prime to p, fixed in (3.2.1.10).

(3.2.2.6) Lemma. ([S97], 19.2) Suppose β is of full rank n. Let v be a finite place of E. Let f_{v}(•, χ_{v}, s) be the unique section which is invariant under H(O_{v}), and such that f_{v}(1, χ_{v}, s) = 1. Let m = m(A) ∈ M(E_{v}). Then W_{β,v}(m, f_{v}, s) = 0 unless ^{t}ĀβA ∈ a^{−1}𝔇(E/ℚ)_{v}^{−1}Λ_{v}^{∗}, where 𝔇(E/ℚ)_{v} is the different of E_{v} relative to ℚ_{p} (p being the rational prime lying below v). In this case, one has

W_{β,v}(m, f_{v}, s) =

= |N ∘ det A|_{v}^{n/2−s}χ_{v}(det A)g_{β,m,v}(χ(ϖ_{v})q_{v}^{−2s−n}) · ∏_{j=1}^{n} L_{v}(2s + j, χε_{K/E}^{n−j})^{−1}.

Here L_{v}(•, •) is the local abelian L-factor at v, with χ viewed as a character for A_{E}^{×} by restriction, and g_{β,m,v} is a polynomial with constant term 1 and coefficients in ℤ. Let 𝔇(E/ℚ)_{v} = δ_{v}O_{v} for some δ_{v} ∈ E_{v}. If v is unramified in K, and

det(aδ_{v}^{t}ĀβA) ∈ O_{v}^{×},

then g_{β,m,v}(t) ≡ 1.

Example. Let n = 1. Then β ∈ E_{v}^{×} and A is a scalar. Let r ≥ 0 be the integer determined by

|aδ_{v}ĀAβ|_{v} = q_{v}^{−r}

Then

g_{β,m,v}(t) = \frac{(1 − t)[1 − (qt)^{r+1}]}{1 − qt}

Documenta Mathematica · Extra Volume Coates (2006) 393–464

(3.2.2.7) Corollary. For any finite place v we let T_{v} be the characteristic function of 𝔇(E/ℚ)_{v}^{−1}Λ_{v}^{∗}. Suppose that β ∈ Her_{n}(E) is of full rank n. Let S be a finite set of places including all the archimedean ones and all places ramified in K, and large enough so that the conditions of Lemma 3.2.2.6 are satisfied at any place v ∉ S. Let m = m(A) ∈ M(A). Then

(3.2.2.8) E_{β}^{f}(m, χ, s) = c(n, E, K)·| det A|_{K}^{n−s}χ(det A)·(∏_{v∈S} W_{^{t}ĀβA,v}(1, f_{v}, s))

· ∏_{v∉S}[T_{v}(a^{t}ĀβA)g_{β,m,v}(χ(ϖ_{v})q_{v}^{−2s−n})] · ∏_{j=1}^{n} L^{S}(2s + j, χε_{K/E}^{n−j})^{−1}

Here L^{S}(•, •) is the partial L-function, with χ viewed as a character for A_{E}^{×} by restriction.

(3.2.2.9) Remarks.

(i) In the subsequent sections we will always assume S contains all primes of residue characteristic p. Suppose this is the case and v ∉ S. Then the local factor T_{v}(a^{t}ĀβA)g_{β,m,v}(χ(ϖ_{v})q_{v}^{−2s_{0}−n}) is p-adically integral for any half-integer s_{0}. In particular, the p-adic denominators of the Fourier coefficients E_{β}^{f}(m, χ, s_{0}), normalized by the product of the partial Lfunctions, are determined by the local factors at v ∈ S and by the global factors.

(ii) Let β ∈ Her_{n}(E) be of full rank n. We say β is S-primitive if det(aβ) ∈ O_{v}^{×} for all v ∉ S. The condition depends implicitly on a. Since S contains the ramified primes, the local different factors can be ignored. It follows from (3.2.2.7) that for S-primitive β, the product of local coefficients satisfies

∏_{j=1}^{n} L^{S}(2s_{0} + j, χε_{K/E}^{n−j}) · ∏_{v∉S} W_{β,v}(1, f_{v}, s_{0}) = 1

and in particular is a p-adic unit. (iii) On the other hand, the factors g_{β,m,v}(χ(ϖ_{v})q_{v}^{−2s−n}) are p-units at half-

integer values of s, provided v is prime to p. Our local data at primes v dividing p will guarantee the vanishing of coefficients W_{β,v} unless det(aδ_{v}^{t}ĀβA) ∈ O_{v}^{×}, and we will only evaluate the coefficients at points m = m(A) with A_{v} ∈ GL(n, O_{v})). Thus we will always have the local factors g_{β,m,v}(t) ≡ 1 for v dividing p, and the product

(3.2.2.10)  T^{0}(β, m(A), s) = ∏_{v∉S}[T_{v}(a^{t}ĀβA)g_{β,m,v}(χ(ϖ_{v})q_{v}^{−2s−n})]

will always be a p-adic unit when s ∈ \frac{1}{2}ℤ. (iv) In other words, the p-adic behavior of the Eisenstein series is completely

determined by the global normalizing factor ∏_{j=1}^{n} L^{S}(2s + j, χε_{K/E}^{n−j})^{−1} and by the local factors at v ∈ S. Calculation of the local factors will occupy most of the rest of this section.

(3.3) Local coefficients of holomorphic Eisenstein series.

In this section we consider a finite set S of places as in (3.2.2.7), containing all archimedean places, all places ramified in K/E, all places dividing p, and all places at which the character χ_{v} is ramified. We also include in S a collection of finite places where, to guarantee non-vanishing of local zeta integrals for ramified π_{v}, f_{v} cannot be the unramified vector f_{v}^{unr} ∈ I(χ, s), i.e., the vector invariant under H(O_{v}). At the archimedean places we will take specific local data. Otherwise the data will vary according to circumstances to be defined later. The resulting calculation (3.3.1.5, 3.3.2.1) of the local Fourier coefficients at ramified finite primes is less precise than at unramified places. We treat non-split places, split places, and archimedean places separately.

(3.3.1) Finite non-split places.

Let v be a finite place in S. Suppose first that v does not split in K. We let w be the unique place of K dividing v. We define a special section in I(χ_{v}, s) as follows. Let u_{v} be a Schwartz function on Her_{n}(E_{v}). Define a section f_{v}(h; χ_{v}, s) =^{def} f_{u_{v}}(h; χ_{v}, s) ∈ I(χ_{v}, s) by the condition that it is supported in the big cell P (E_{v})wP (E_{v}), and

(3.3.1.2)  f_{v}(wn(b); χ_{v}, s) = u_{v}(b) (b ∈ Her_{n}(E_{v}))

It is easy to see that W_{β,v}(1, f_{v}, s) = û_{v}(β). Together with the transformation law (3.2.2.5), we find that

(3.3.1.3)  W_{β,v}(m(A)f̃_{v}, s) = | det A|_{v}^{n/2−s}χ_{v}(det A) · û_{v}(^{t}ĀβA)

We now choose a lattice L_{v} ⊂ Her_{n}(E_{v}), and make the following assumption:

(3.3.1.4) Hypothesis. u_{v} is the characteristic function of L_{v}.

Let L_{v}^{∨} be the dual lattice defined by

L_{v}^{∨} = {β | ψ(trβx) = 1 for all x ∈ L_{v}}

Then we have

(3.3.1.5) W_{β,v}(m(A), f_{u_{v}}, s) = T_{v}(^{t}ĀβA)| det A|_{v}^{n/2−s}χ_{v}(det A) · vol(L_{v}),

(3.3.2)Finite split places.

Next we consider the case where v is finite and splits in K, of residue characteristic different from p. Let u_{v} be a Schwartz function on Her_{n}(E_{v}) ≃ M_{n,n}(E_{v}) (n × n matrices with entries in E_{v}). Then there is a section f_{u_{v}}(h; χ_{v}, s) such that f_{v}(•; χ_{v}, 0) has support in P (E_{v})wP (E_{v}), and f_{v}(wn(b); χ_{v}, 0) = u_{v}(b). Formula (3.3.1.3) remains valid for all β. If u_{v} is chosen as in (3.3.1.4), then we write f_{L_{v}} instead of f_{u_{v}}. In what follows, A ∈ GL(n, K_{v}) can be written as a pair (A_{v}, B_{v}) with A_{v}, B_{v} ∈ GL(n, E_{v}), and | det(A)|_{v} = | det(A_{v} · B_{v}^{−1})|_{v}, with conventions as in (3.3.4) below.

(3.3.2.1) Lemma. With f_{v} = f_{L_{v}}, formula (3.3.1.5) is valid for all β.

At split places other choices might be more convenient. For example, let U_{v} ⊂ GL(n, E_{v}) be a compact open subgroup and τ_{v} a finite-dimensional irreducible representation of U_{v}. Let u_{v} be a matrix coefficient of τ_{v}, viewed as a function on U_{v} ⊂ GL(n, E_{v}) and extended by zero to M(n, E_{v}). Then u_{v} takes values in the integers of some cyclotomic field. It then follows immediately from (3.3.1.3) that:

(3.3.2.2) Lemma. The functions û_{v} and W_{β,v}(m(A), f_{v}, 0) are locally constant, compactly supported, not identically zero, and takes values in ℚ^{ab} with denominators bounded p-adically independently of τ_{v}.

Indeed, the integral defining û_{v} is a finite sum of terms, each of which is an algebraic integer multiplied by a volume. The volume lies in ℚ and the denominators are bounded in terms of the orders of finite subgroups of GL(n, E_{v}), independently of τ_{v}. The remaining factors in (3.3.1.5) are p-units.

(3.3.2.3) Remark 3.3.2.3 Alternatively, we can let u_{v} be a matrix valued function, namely the function τ_{v}, with values in End(τ_{v}), extended to zero off K_{v}. The Eisenstein series and its Fourier coefficients will then have values in End(τ_{v}). This will allow us to pair the Eisenstein series with forms taking values in the space of τ_{v} and its dual. The local zeta integral will be essentially a volume.

(3.3.3) Archimedean places.

Let v ∈ S_{∞} be a real place of E. We shall regard elements of H(E_{v}) ≃ U(n, n) as 2n × 2n matrices by means of the basis {e_{i}, f_{j}} chosen in (3.2.1). Let j = σ_{v}(ℷ). We let K_{v} ⊂ H(E_{v}) be the maximal compact subgroup consisting of those matrices k with ^{t}k̄ diag(j^{2}I_{n}, −I_{n})k = diag(j^{2}I_{n}, −I_{n}), where I_{n} denote the identity matrix of size n. Then K_{v} ≃ U(n) × U(n). We make this isomorphism explicit as follows. Set

γ = ( \begin{matrix}1_{n} & 1_{n} \\ j^{−1}1_{n} & −j^{−1}1_{n} \end{matrix} ) ∈ GU(n, n)

Then for any A, B ∈ U(n) one has

k(A, B) = γ ( \begin{matrix}A & 0 \\ 0 & B \end{matrix} ) γ^{−1} ∈ K_{v}

The map (A, B) ↦ k(A, B) is an isomorphism from U(n) × U(n) onto K_{v}. Let x ∈ Her_{n}(ℝ). One easily checks that the Iwasawa decomposition of wn(x) is given by

wn(x) =

=( \begin{matrix}1_{n} & −\frac{x}{−j^{2}+x^{2}} \\ 0 & 1_{n} \end{matrix} )( \begin{matrix}\frac{1}{\sqrt{−j^{2}+x^{2}}} & 0 \\ 0 & \sqrt{−j^{2} + x^{2}} \end{matrix} )k(− \frac{x + j1_{n}}{\sqrt{−j^{2} + x^{2}}} , −\frac{x − j1_{n}}{\sqrt{−j^{2} + x^{2}}} )

Let m = 2μ + κ as in (3.1.4.4), so that χ_{v}(−1) = (−1)^{m} (3.1.1.3). We follow Shimura [S82] and take f_{v} to be (up to sign) the v component of the canonical automorphy factor denoted J_{μ,κ}(h, s − \frac{κ}{2}) in (3.1.4.3); thus f_{v} is holomorphic for s = s_{0} = \frac{m−n}{2}. More precisely,

(3.3.3.1) \begin{aligned}f_{v}(wn(x), χ_{v}, s) & = det(−j^{2} + x^{2})^{−s−n/2} det(\frac{−j1_{n} − x}{\sqrt{−j^{2} + x^{2}}} )^{m} \\ & = (−1)^{mn} · δ(x − j1_{n})^{−s− \frac{m+n}{2}} δ(x + j1_{n})^{−s+ \frac{m−n}{2}} \end{aligned}

In subsequent articles we will identify f_{v} with a Siegel-Weil section for the theta lift of the trivial representation of U(m). Continuing the calculation, and making the simple change of variables, x ↦ x/α, where α = −j/i > 0, we find

W_{β,v}(1, f_{v}, s) =

= (−1)^{mn}(−j/i)^{−2ns} ∫_{Her_{n}(ℝ)} δ(x+i1_{n})^{−s− \frac{m+n}{2}} δ(x−i1_{n})^{−s+ \frac{m−n}{2}} e^{−2πitr(βx)}dx

= (−1)^{mn}(−j/i)^{−2ns}ξ(1_{n}, β; s + \frac{n + m}{2}, s + \frac{n − m}{2})

([S82], p. 274, (1.25)). By ([S82], p. 275, (1.29)), this is equal to

(−i)^{mn}2^{n}π^{n^{2}}(−j/i)^{−2ns}Γ_{n}(s + \frac{n + m}{2})^{−1}Γ_{n}(s + \frac{n − m}{2})^{−1} ×

η(21_{n}, πβ; s + \frac{n + m}{2}, s + \frac{n − m}{2})

Choose A ∈ GL(n, ℂ) with AA^{∗} = πβ, where A^{∗} = ^{t}Ā. By ([S82], p.280-281), we have

η(21_{n}, πβ; s+ \frac{n + m}{2}, s+ \frac{n − m}{2}) = δ(πβ)^{2s} ·η(2A^{∗}A, 1_{n}; s+ \frac{n + m}{2}, s+ \frac{n − m}{2})

= (2π)^{2ns}δ(β)^{2s}e^{−2πtr(β)}ζ(4A^{∗}A; s + \frac{n + m}{2}, s + \frac{n − m}{2})

Thus

W_{β,v}(1, f_{v}, s) =(−i)^{mn}2^{n(m−n+1)}π^{ns+n(m+n)/2}(−j/i)^{−}2nsδ(β)^{s− \frac{n−m}{2}} e^{−2πtr(β)}

Γ_{n}(s + \frac{n + m}{2})^{−1}ω(4A^{∗}A; s + \frac{n + m}{2}, s + \frac{n − m}{2})

The function ω(z; μ, λ) is analytic in μ, λ and satisfies the functional equation

ω(z; n − λ, n − μ) = ω(z; μ, λ)

By (3.15) of Shimura we know ω(z; μ, 0) = 1. So at s = (m − n)/2 we obtain (3.3.3.2)

W_{β,v}(1, f_{v}, \frac{m − n}{2}) =

= (ℷ_{v})^{−mn+n^{2}}(−i)^{−n^{2}}2^{n(m−n+1)}π^{mn} det(β)^{m−n}e^{−2πtr(β)}Γ_{n}(m)^{−1}

= \frac{(ℷ_{v})^{−mn+n^{2}}(−i)^{−n^{2}}2^{n(m−n+1)}π^{mn−n(n−1)/2} det(β)^{m−n}}{∏_{j=1}^{n}(m − j)!} · e^{−2πtr(β)}.

The factor e^{−2πtr(β)} at the end is the value at h_{∞} = 1 of the function denoted q^{β} in §(1.5.6); more precisely, q^{β} factors over the archimedean primes, and e^{−2πtr(β)} is the factor at v. The coefficient preceding this factor is the local contribution at v to the Fourier coefficient f_{β}.

(3.3.4) Local results at primes dividing p (choice of special functions at p)

First we fix some notation.

(3.3.4.1) Notation. Let v be a place of E dividing p. Then v splits in K according to our assumptions. Throughout we shall identify E_{v} with K_{w}, where w is the divisor of v with w ∈ Σ_{p} (see (1.1.4)). We denote by O_{v} the ring of integers of E_{v}, and by 𝔭_{v} the prime ideal in O_{v}. For any pair of positive integers a, b we denote by M_{a,b} or M(a, b) the space of a × b matrices. Let dZ be the normalized Haar measure on M_{n,n}(E_{v}) that assigns measure 1 to M_{n,n}(O_{v}). We write d^{×}Z = dZ/| det Z|^{n}. Let dg_{v} be the normalized Haar measure on GL(n, E_{v}) that assigns measure 1 to GL(n, O_{v}). Then d^{×}Z = A(n) · dg_{v}, where

A(n) = ∫_{GL(n,O_{v})} d^{×}Z = ∏_{j=1}^{n}(1 − q^{−j}) = q^{−n^{2}}#GL(n, 𝔽_{q})

This is just the right hand side of (0.2.2). Thus we may assume that d^{×}Z = L_{v}(1, ε_{K})^{−1}d^{τ} g_{v} in the notation of (0.2). Let χ be the character of 𝔸_{K}^{×} that goes into the definition of our Siegel Eisenstein series. At the place v which splits in K, χ is given by the pair of characters (χ_{1v}, χ_{2v}^{−1}). For the rest of section (3.3.4) we drop the subscript v from our notation, writing χ_{1} for χ_{1v}, etc. On H(E_{v}) ≃ GL(2n, E_{v}), the inducing character is

( \begin{matrix}A & ∗ \\ 0 & B \end{matrix} ) ↦ χ_{1}(det A)χ_{2}(det B) · | det(AB^{−1})|^{s+ρ},

with ρ = n/2. Fix a partition

n = n_{1} + ⋯ + n_{l}

Let P = LU be the standard parabolic subgroup of GL(n) corresponding to the above partition. Let I ⊆ GL(n, O_{v}) be the paraholic subgroup corresponding to P. Thus I consists of matrices Z = (Z_{ij}) (written in blocks with respect to the above partition of n), such that

• Z_{jj} ∈ GL(n_{j}, O_{v}) for 1 ≤ j ≤ l.

• Z_{ij} has entries in O_{v} for 1 ≤ i < j ≤ l.

• Z_{ij} has entries in 𝔭_{v} for i > j. Note that I is an open set in the space M(n, n) of all n × n matrices with entries in E_{v}. Consider l characters ν = (ν_{1}, ⋯, ν_{l}) of E_{v}^{×}. We define our Schwartz function ϕ_{ν} by the formula

(3.3.4.2)  ϕ_{ν}(Z) = \{ \begin{matrix}ν_{1}(det Z_{11}) ⋯ ν_{l}(det Z_{ll}), & Z ∈ I \\ 0, & otherwise \end{matrix}

We use the same letter ν to denote the character of L(O) given by

ν(diag(A_{1}, ⋯, A_{l})) = ν_{1}(det A_{1}) ⋯ ν_{l}(det A_{l})

It is easy to see that the function ϕ_{ν} satisfies the relation

ϕ_{ν}(mZ) = ϕ_{ν}(Zm) = ν(m)ϕ_{ν}(Z) (m ∈ L(O), any Z)

Define Fourier transform by

(3.3.4.3)  F(ϕ)(x) = ∫ ϕ(z)\overline{ψ_{v}(tr(z^{t}x))}dz

The function F(ϕ_{ν}) satisfies the (obvious) condition

(3.3.4.4) F(ϕ_{ν})(mx) = F(ϕ_{ν})(xm) = ν^{−1}(m)F(ϕ_{ν})(x) (m ∈ L(O), any x)

The explicit formula for F(ϕ_{ν}) is given in Part II, Appendix B. Consider another l-tuple of characters μ = (μ_{1}, ⋯, μ_{l}). We can define ϕ_{μ} as above. Take any integer t which is large enough — say larger than the conductors of all the characters μ_{j}. Let

Γ = Γ(𝔭^{t}) ⊆ GL(n, O)

be the subgroup of GL(n, O) consisting of matrices whose off diagonal blocks are divisible by 𝔭^{t}. Note that the restriction of ϕ_{μ} to Γ(𝔭^{t}) is a character. We have

(3.3.4.5)  ϕ_{μ}(γx) = ϕ_{μ}(xγ) = ϕ_{μ}(γ)ϕ_{μ}(x)  (γ ∈ Γ(𝔭^{t}), any x)

Define a related function ϕ̃_{μ} by

ϕ̃_{μ}(x) = \{ \begin{matrix}Vol(Γ(𝔭^{t}); d^{×}Z)^{−1} · ϕ_{μ}(x), & if x ∈ Γ(𝔭^{t}) \\ 0, & otherwise \end{matrix}

Here Vol(Γ(𝔭^{t}); d^{×}Z) is the volume of Γ(𝔭^{t}) with respect to the measure d^{×}Z. We have

Vol(Γ(𝔭^{t}); d^{×}Z)^{−1}= A(n)^{−1}[GL(n, O) : Γ(𝔭^{t})] = (∏_{j=1}^{l} A(n_{j})^{−1})( ∏_{1≤i<j≤l}q^{2tn_{i}n_{j}})

Later on, we shall identify various spaces with M_{n,n}, and ϕ_{μ}, etc, will be viewed as a function on these spaces. We define a Schwartz function Φ_{1} on M(n, n) by

(3.3.4.6)  Φ_{1}(u, v) = ϕ̃_{μ}(\frac{u − v}{2} ) · F(ϕ_{ν})(u + v)

Recall that we have identified U(2V)(E_{v}) with GL(2n, E_{v}). Thus it acts on M(n, 2n) by right multiplications. We take a global section

f(h; χ, s) = ⊗f_{u}(h; χ, s) ∈ Ind(χ| · |^{s})

with u running through all places of E. At the place v we choose the local section by the following formula:

(3.3.4.7)  f_{v}(h; χ, s) = f_{v,μ}(h; χ, s) =^{def} χ_{1}(det h) · | det h|^{s+ρ}
· ∫_{GL(n,E_{v})} Φ_{1}((Z, Z)h)χ_{1}χ_{2}^{−1}(det Z)| det Z|^{2(s+ρ)}d^{×}Z.

Recall that we have the decomposition

2V = V^{d} ⊕ V_{d}

of the doubled space 2V into totally isotropic subspaces. We now define a Weyl element w_{n} that interchanges the two summands above. To make it precise wewrite matrices in blocks corresponding to the decomposition

2V = V ⊕ (−V)

Then we take

w = w_{n} = ( \begin{matrix}1_{n} & 0 \\ 0 & −1_{n} \end{matrix} )

For each index j with 1 ≤ j ≤ n we also define

w_{j} = ( \begin{matrix}1_{j} & 0 & 0 & 0 \\ 0 & 1_{n−j} & 0 & 0 \\ 0 & 0 & −1_{j} & 0 \\ 0 & 0 & 0 & 1_{n−j} \end{matrix} )

(Really, 1_{n} is the identity on V. But the definition of 1_{j} for 0 < j < n implies an implicit choice of an orthogonal basis for V).

(3.3.4.8) Lemma. Let P = P^{d} be the stabilizer of V^{d} in U(2V). Then as a function of h the local section f_{v}(h; χ, s) is supported on the “big cell” P (E_{v})w_{n}P (E_{v}).

Proof. We know that U(2V) is the disjoint union of the double cosets P w_{j}P. Since f_{v} is a section, it suffices to show that

f_{v}(w_{j}p; χ, s) = 0, for any p ∈ P (E_{v}), j < n

As remarked above, the definition of w_{j} involves an implicit choice of a basis, and therefore a decomposition

V = V_{j} ⊕ V^{j}

where V_{j} is of dimension j. Recall that U(2V)(E_{v}) ≃ GL(2n, E_{v}). Under this identification, a typical element of P (E_{v}), written in blocks with respect to the decomposition 2V = V ⊕ −V, is of the form

p = ( \begin{matrix}A & B \\ C & D \end{matrix} )

where A, B, C, D are n × n matrices, and

A + C = B + D

In accordance with the decomposition V = V_{j} ⊕ V^{j}, we may write an n × n matrix as Z = (X, Y) where X is n × j and Y is n × (n − j). Then we find

(Z, Z)w_{j}p = (u, v)

with

u = (X, Y)A + (−X, Y)C, v = ((X, Y)B + (−X, Y)D

Consequently

u − v = (X, 0)(A + D − B − C)

(Here we have used the condition A + C = B + D). The right hand side is a singular matrix unless j = n. Since

Φ_{1}(u, v) = ϕ̃_{μ}(\frac{u − v}{2} )F(ϕ_{ν})(u + v)

and ϕ̃_{μ} is supported on invertible matrices, we find

Φ_{1}((Z, Z)w_{j}p) = 0 for all Z

if j < n. Hence f_{v}(w_{j}p; χ, s) = 0 for j < n and p ∈ P (E_{v}).

We define the Eisenstein series E^{f}(h; χ, s) = E^{f_{μ}}(h; χ, s) and its Fourier coefficients as before. Let P = MN be a Levi decomposition. We assume that M is normalized by w. We will calculate the v-component of E_{β}^{f}(h; χ, s) under the condition that h_{v} ∈ P (E_{v}). In view of the above lemma and Remark 3.2.2.3, we know that the factorization (3.2.2.2) is valid for any β (full rank or otherwise), provided h_{v} ∈ P (E_{v}). However, in (3.3.4.9) we will see that our choice of local data at primes dividing p forces E_{β}^{f}(h, χ, s) = 0 for rank(β) < n, provided h_{v} ∈ P (E_{v}) for at least one place v dividing p. For the remainder of this section we shall calculate

W_{β,v}(h_{v}, f_{v}, s) = ∫_{N(E_{v})} f_{v}(wn_{v}h_{v}; χ_{v}, s)ψ_{−β}(n_{v})dn_{v}

The group N can be identified with the space Herm_{n} of n × n hermitian matrices. We write this isomorphism as

Herm_{n}⟶N, R ↦ n(R)

If R ∈ Herm_{n}(A) then

ψ_{β}(n(R)) = ψ(tr(β^{t}R))

where tr denotes trace of the matrix, followed by tr_{K/E}. We need to explain what this means at the split place v. We have the isomorphism

K ⊗ E_{v} ≃ E_{v} ⊕ E_{v}

where the first summand E_{v} is identified with K_{w}, with w the place of K dividing v, such that w ∈ Σ_{p}. The second summand is then identified with K_{w^{c}}. Also, on the right hand side the trace map is identified with the summation of the two coordinates. This gives rise to

2V ⊗ E_{v} = (2V)_{1} ⊕ (2V)_{2}

etc. Now any R ∈ Herm_{n}(E_{v}) is identified with an arbitrary n × n matrix with coefficients in E_{v}, as follows. We consider

M_{n,n}(K) ⊂ M_{n,n}(K_{w}) = M_{n,n}(E_{v})

Then the embedding

M_{n,n}(K)⟶M_{n,n}(E_{v}) ⊕ M_{n,n}(E_{v}), γ ↦ (γ, γ̄)

extends to an isomorphism

M_{n,n}(K) ⊗ E_{v}⟶M_{n,n}(E_{v}) ⊕ M_{n,n}(E_{v})

Since γ̄ = ^{t}γ for γ ∈ Herm_{n}(E), we see that the image of

Herm_{n}(E_{v}) = Herm_{n}(E) ⊗ E_{v} ⊂ M_{n,n}(K) ⊗ E_{v}

under the above isomorphism is precisely

{(R,^{t}R) | R ∈ M_{n,n}(E_{v})}

Thus we get the identification Herm_{n}(E_{v}) = M_{n,n}(E_{v}) by the map (R,^{t}R) ↦ R. Now if a matrix R ∈ M_{n,n}(E_{v}) is identified with an element of Herm_{n}(E_{v}) as above then a simple calculation gives

ψ_{β}(n(R)) = ψ(2 · tr_{E}(β^{t}R))

This time, on the right hand side tr_{E}(β^{t}R) is the trace of β^{t}R viewed as a matrix with coefficients in E_{v}. We may assume h_{v} ∈ M(E_{v}). Then h_{v} preserves both the diagonal and the anti-diagonal. So there are n × n invertible matrices A and B such that

(Z, −Z)h_{v} = (ZA, −ZA), (Z, Z)h_{v} = (ZB, ZB)

for any Z. Suppose n = n(R). A simple calculation gives

(Z, Z)wnh_{v} = (Z(RB + A), Z(RB − A))

Recalling the definition of Φ_{1} we obtain

(3.3.4.9)  Φ_{1}((Z, Z)wnh_{v}) = ϕ̃_{μ}(ZA)F(ϕ_{ν})(2ZRB)

We already know that f_{v} is supported on the big cell. In the integral expression for f_{v} given by (3.3.4.7) we may translate the variable Z by any element of L(O) and then integrate over L(O) ⊆ GL(n, E_{v}). By formula (3.3.4.9) and the transformation properties of ϕ_{μ} and F(ϕ_{ν}) given by (3.3.4.4)-(3.3.4.5), we see immediately that f_{v} would be identically 0 unless the following conditions are satisfied:

(3.3.4.10)  μ_{j} = ν_{j}χ_{2}χ_{1}^{−1} on O_{v}^{×}, for 1 ≤ j ≤ l

We assume this from now on. Then

f_{v}(wnh_{v}; χ, s) = χ_{1}(det B)χ_{2}(det A)| det BA^{−1}|^{s+ρ}F(ϕ_{ν})(2A^{−1}RB)

By Fourier inversion we obtain (3.3.4.11) Lemma. For h_{v} ∈ M(E_{v}) as above, the v-component of the β-th Fourier coefficient E_{β}^{f}(h, χ, s) = E_{β}^{f_{μ}}(h, χ, s) is given by (3.3.4.12)

W_{β,v}(h_{v}, f_{v}, s) = χ_{1}(det B)χ_{2}(det A)| det AB^{−1}|^{−s+ρ}ϕ_{ν}(^{t}Aβ^{t}B^{−1}),

where ν is defined in terms of μ and χ by (3.3.4.10). In particular, the β-th Fourier coefficient vanishes unless β is of full rank.

The last assertion of the lemma follows from the fact that ϕ_{ν} is supported on I.

(3.3.5) Summary.

Recall that m = n + 2s_{0}. Define

C_{∞}(n, m, K) = ∏_{v∈Σ}(ℷ_{v})^{−mn+n^{2}} · (\frac{(−i)^{−n^{2}}2^{n(m−n+1)}π^{mn−n(n−1)/2}}{∏_{j=1}^{n}(m − j)!} )^{[E:ℚ]} ,

(3.3.5.1)  C^{S}(n, m, K) = c(n, E, K) ∏_{j=0}^{n−1} L^{S}(m + j, χε^{j})^{−1}C_{∞}(n, m, K);

We choose a global section

(3.3.5.2) f = f_{μ}(h, χ, s) = ⨂_{v∉S} f_{v}^{unr} ⊗ ⨂_{v∈S_{∞}} f_{v} ⊗ ⨂_{v∈S_{f}^{p}} f_{u_{v}} ⊗_{v|p} f_{v}(h; χ, s)

in accordance with the preceding sections. The functions f_{v} for v | ∞, resp. v | p, are defined by (3.3.3.1), resp. (3.3.4.7), the characters μ_{j} being determined by ν_{j} and χ by (3.3.4.10). Finally, for v ∉ S, f_{v}^{unr} is the unramified vector in I(s, χ) normalized to take value 1 at 1. Let

𝔼(h, χ, m, f) = 𝔼(h, χ, m, f_{μ}) =^{def} C^{S}(n, K)^{−1}E^{f}(h, χ, s_{0}).

We define the factor T^{0}(β, m(A), s_{0}) by (3.2.2.10). When h = m(A) ∈ M(A_{f}), we write m(A) = m(A^{p}) · ∏_{v|p} h_{v}, and let A_{v} be the local component of A at v for v prime to p. The preceding calculations show that the β-Fourier coefficient of 𝔼(m(A), χ, m, f) equals zero if rank(β) < n. Otherwise, the Fourier coefficient is given by the following formula, in which χ has been replaced by the (motivic) Hecke character χ^{∗} = χ · N_{K/E}^{κ/2} and where for split v in S_{f}^{p} we write χ_{v}^{∗}(det(A_{v})) as an abbreviation for χ_{v}^{∗}(det(A_{v} · B_{v}^{−1}) as in (3.3.2):

(3.3.5.3) 𝔼_{β}(m(A), χ, m, f) = 𝔼_{β}(m(A), χ, m, f_{μ}) =

= T^{0}(β, m(A), s_{0}) det(β)^{(m−n)[E:ℚ]}| det A|_{A}^{\frac{n−κ}{2} −s_{0}}×

× ∏_{v∈Σ_{p}} χ_{1}^{∗}(det B(h_{v}))χ_{2}^{∗}(det A(h_{v}))ϕ_{ν}(^{t}A(h_{v})βB(h_{v})^{−1})×

× ∏_{v∈S_{f}^{p}} χ_{v}^{∗}(det(A_{v}))û_{v}(^{t}Ā_{v}βA_{v})

We have dropped the term q^{β} of (1.5.6). The complete arithmetic Fourier expansion is (3.3.5.4)

𝔼(h_{∞}m(A), χ, m, f) = 𝔼(h_{∞}m(A), χ, m, f_{μ}) = ∑_{β} 𝔼_{β}(m(A), χ, m, f)q^{β}

with m(A) ∈ M(A_{f}) as before.

Remarks (3.3.5.5) By (3.1.4.4) the exponent in the absolute value factor | det A|_{A}^{\frac{n−κ}{2} −s_{0}}

is an integer. Thus these factors are always integers, and in fact are p-units under our standing hypothesis that A(h_{v}) and B(h_{v}) are in GL(n, O_{v}) for all v dividing p. Similarly, since m ≥ n, the factor det(β)^{m−n} is p-adically integral provided β is, and this is guaranteed by our hypothesis on A(h_{v}) and B(h_{v}) and the definition of T^{0}(β, m(A)).

(3.3.5.6) With u_{v} chosen as in (3.3.1) and (3.3.2) at places in S_{f}^{p}, the coefficients are then p-adic integers, and in fact are p-adic units where they are non-zero. Better control of the local theta correspondence at places in S will require different choices of f_{v} at S_{f}^{p}.

(3.3.5.7) In applications to the zeta function we will want to work with finite sums of Siegel-Weil Eisenstein series attached to hermitian spaces V′ that differ locally at non-split primes in S, since at such primes we are forced to take the local sections denoted f̃_{v} of (3.3.1.2), which are not generally Siegel-Weil sections. These Fourier coefficients of these sums remain p-adically integral and since the different V′ represent different β, they are also p-adically primitive.

(3.4) Review of abstract p-adic distributions and measures.

Let T be a torus over ℤ_{p}, and let R be a complete ℤ_{p}-algebra, assumed ℤ_{p}- flat and compact, R[\frac{1}{p}] = R ⊗_{ℤ_{p}} ℚ_{p}. For instance, we can take R = O_{ℂ_{p}}, so that R[\frac{1}{p}] = ℂ_{p}. Let B denote an R[\frac{1}{p}]-Banach space, M ⊂ B the unit ball of elements of norm ≤ 1. If A = R, R[\frac{1}{p}], B, or M, let C(T (ℤ_{p}), A) denote the R-module of continuous A-valued functions on T (ℤ_{p}). Since T (ℤ_{p}) is compact, C(T (ℤ_{p}), B) = C(T (ℤ_{p}), M)⊗_{R} R[\frac{1}{p}]), and this is true in particular for M = R itself. The sup norm makes C(T (ℤ_{p}, R[\frac{1}{p}] into an R[\frac{1}{p}]-Banach space. The locally constant functions in C(T (ℤ_{p}), A) are denoted C^{∞}(T (ℤ_{p}), A). A p-adic distribution on T (ℤ_{p}) with values in an R[\frac{1}{p}]-vector space V is a homomorphism of R-modules

λ : C^{∞}(T (ℤ_{p}, R) → V.

To define a distribution V need not be a Banach space. A B-valued p-adic measure on T (ℤ_{p}) is a continuous homomorphism of R[\frac{1}{p}]-Banach spaces

μ : C(T (ℤ_{p}), R[\frac{1}{p}]) → B.

Let X_{fin}(T) denote the set of characters of finite order of T (ℤ_{p}), viewed as a subset of C^{∞}(T (ℤ_{p}), R) for any sufficiently large p-adic ring R, e.g. R = O_{ℂ_{p}}. The set X_{fin}(T) forms a basis for the R[\frac{1}{p}]-vector space C^{∞}(T (ℤ_{p}), R[\frac{1}{p}]), hence any function χ ↦ v_{χ} from X_{fin}(T) to V determines a V-valued distribution on T (ℤ_{p}) by linearity.

(3.4.1) Lemma. Let χ ↦ m_{χ} be a function from X_{fin}(T) to M, and let λ(m) denote the corresponding B-valued distribution. Then λ(m) extends to a p-adic measure if and only if, for every integer n and for any finite sum ∑_{j} α_{j}χ_{j} with α_{j} ∈ R[\frac{1}{p}] and χ_{j} ∈ X_{fin}(T) such that ∑_{j} α_{j}χ_{j}(t) ∈ p^{n}R for all t ∈ T (ℤ_{p}), we have

(3.4.2)  ∑_{j} α_{j}m_{χ_{j}} ∈ p^{n}M.

This is a version of the abstract Kummer congruences stated as Proposition 5.0.6 of [K].

(3.4.3) Corollary. In Lemma (3.4.1) above, it actually suffices to check (3.4.2) with n = 0.

Indeed, the the condition for n=0 implies the condition for general n: a bounded distribution is a measure. In the next section we will be constructing measures with values in the Banach space of p-adic modular forms on the Shimura variety Sh(2V). Let R = O_{ℂ_{p}}, so that R[\frac{1}{p}] = ℂ_{p}. Let V denote the algebra of p-adic modular forms, as in (2.2.9), and let B = V ⊗_{O_{ℂp}} ℂ_{p}. Let M denote the right-hand side

ˆ ⨁α∈U_{∗}H^{0}(_{K_{P} (∞)}S(G_{P} , X_{P}), O_{𝕊_{P}}) of (2.3.2), and let Q = M ⊗_{O_{ℂp}} ℂ_{p}. The ℂ_{p}-vector space B is a Banach space via the sup norm, whereas Q can be viewed as a ring of formal series over the Banach space H^{0}(_{K_{P} (∞)}S(G_{P} , X_{P}), O_{𝕊_{P}}), hence again becomes a Banach space via the sup norm. The q-expansion map F.J.^{P} (2.3.2) is a continuous homomorphism of Banach spaces. The following proposition follows from the q-expansion principle, as in [DR] or [K], and represents the primary application of the q-expansion principle to our project:

(3.4.4) Proposition. Let T be a torus over ℤ_{p}, and let μ be a p-adic measure on T (ℤ_{p}) with values in Q. Suppose that μ(χ) = ∫_{T (ℤ_{p})} χdμ lies in the image of F.J.^{P} for all χ ∈ X_{fin}(T). Then μ is the image, under F.J.^{P}, of a measure with values in B.

(3.5) Construction of Eisenstein measures. Let ℓ be a positive integer and let T (ℓ)^{0} denote the torus over ℤ_{p} given by (R_{O_{E}/ℤ_{p}}𝔾_{m,O_{E}})^{ℓ}. Thus T (ℓ)^{0}(ℤ_{p}) is canonically isomorphic to ∏_{w|p} O_{w}^{×,ℓ}, where w runs through places of E. This can also be identified with the product of ℓ copies of ∏_{v∈Σ_{p}} O_{v}^{×}, where now v are places of K. The latter form will be the most useful for us. For brevity we write O_{Σ_{p}}^{×} for ∏_{v∈Σ_{p}} O_{v}^{×}. We let

T (ℓ) = T (ℓ)^{0} × (R_{O_{K}/ℤ_{p}}𝔾_{m,O_{K}}).

Then the set X_{fin}(T (ℓ)) of finite order characters of T (ℓ) can be parametrized by (ℓ + 1)-tuples (ν_{1}, …, ν_{ℓ}, χ), where each ν_{i} is a character of finite order of O_{Σ_{p}}^{×}, and χ is a character of finite order of ∏_{v|p} O_{v}^{×} where now v runs over all places of K dividing p. We will further write χ = (χ_{1}, χ_{2}), where χ_{1} is the restriction of χ to ∏_{v∈Σ_{p}} O_{v}^{×} and χ_{2} is a second character of the same group ∏_{v∈Σ_{p}} O_{v}^{×} obtained by restricting χ^{−1} to ∏_{v∈cΣ_{p}} O_{v}^{×} and then composing with c. So in the end, X_{fin}(T) can be viewed as the set of (ℓ+2)-tuples of characters of O_{Σ_{p}}^{×}. The character χ will in practice be the restriction to O_{K,p}^{×} of a character of K_{p}^{×} = ∏_{v|p} K_{v}^{×}, which in turn will most commonly be the p-adic component of a global Hecke character. We introduce additional notation: for j = 1, …, ℓ, we let μ_{j} = ν_{j} · χ_{2} · χ_{1}^{−1}. Let m, n, and s_{0} be as in (3.3.5). Let n = n_{1} + ⋯ + n_{ℓ} be a partition of n and Q the corresponding standard parabolic subgroup of GL(n).

(3.5.1) Theorem. There is a B-valued measure λ_{Q}^{m} on T (ℓ) with the property that, for any ℓ+2-tuple (μ, χ) = (μ_{1}, …, μ_{ℓ}, χ_{1}, χ_{2}) of characters of finite order of O_{Σ_{p}}^{×}.

(3.5.2)  F.J.^{P} ∘ ∫_{T (ℓ)}(μ_{1}, …, μ_{ℓ}, χ)dλ_{Q}^{m} = 𝔼(•, χ, m, f_{μ})

where the right hand side is the q-expansion of (3.3.5.4).

Proof. The right-hand side of (3.5.2) defines the value at (μ, χ) of a Q-valued distribution on T (t). To show that this distribution is in fact a Q-valued padic measure, it suffices, by Corollary (3.4.3), to show that the right-hand side of (3.5.2) satisfies the abstract Kummer congruences (3.4.2) for n = 0. In other words, for any β ∈ U^{∗} ∩ C, the Fourier coefficients 𝔼_{β}(m(A), χ, m, f_{μ}) as (ν, χ) vary, satisfy the abstract Kummer congruences as functions of m(A) ∈ L_{P} (A_{f}), with the coefficients A_{v} ∈ GL(n, O_{v}) for v | p. Bearing in mind the relation (3.3.4.10) between ν and μ, this follows immediately from (3.3.5.3) and Remarks (3.3.5.5) and (3.3.5.6). Now the theorem follows from Proposition 3.4.4 and from the fact that 𝔼(•, χ, m, f_{μ}) is a classical modular form for (μ, χ) ∈ X_{fin}(T (ℓ)).

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Michael Harris UFR de Mathématiques Université Paris 7 2 Pl. Jussieu 75251 Paris cedex 05 FRANCE harris@math.jussieu.fr Jian-Shu Li Department of Mathematics HKUST and Zhejiang University Clear Water Bay Hong Kong matom@ust.hk

Christopher M. Skinner Department of Mathematics Princeton University Princeton, NJ USA cmcls@Math.Princeton.EDU