THE RALLIS INNER PRODUCT

FORMULA AND p-ADIC L-FUNCTIONS

To Steve Rallis, the oracle

Michael Harris^{1}

UFR de Mathématique, Université Paris 7, 2 Pl. Jussieu 75251 Paris cedex 05, FRANCE

Jian-Shu Li^{2}

Department of Mathematics, HKUST and Zhejiang University, Clear Water Bay, Hong Kong

Christopher M. Skinner^{3}

Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA

1. Introduction

This article is a report on work in progress, whose goal is to show in certain cases that congruences satisfied by normalized special values of L-functions of geometric automorphic forms on unitary groups give rise to non-trivial elements of certain Galois cohomology groups. More precisely, letting p be an odd prime number, the Galois cohomology groups, or generalized Selmer groups, can be viewed as modules over appropriate Iwasawa algebras, and, possibly up to p-power torsion, the associated p-adic L-functions divide the characteristic power series, in the sense of Iwasawa theory, of the corresponding Selmer groups. Divisibility in both directions is a version of the main conjecture of Iwasawa theory.

Divisibility results of this kind were first obtained by Ribet for class groups of cyclotomic fields. Ribet observed that divisibility of normalized values of Dirichlet L-functions by p gave rise to congruences modulo p between holomorphic Eisenstein series and cusp forms on GL(2). Here two holomorphic modular forms f and g are considered congruent if their Fourier expansions at the cusp at infinity have algebraic integer coefficients a_{n}(f) ≡ a_{n}(g) (mod p) for all n ≥ 0. Ribet’s technique was vastly generalized by Mazur and Wiles, who obtained the first proof of Iwasawa’s main conjecture for cyclotomic fields, and Wiles extended these arguments to abelian extensions of totally real fields. More recently, this technique has been applied successfully by Urban to congruences of Siegel modular forms,

Typeset by AMS-TEX

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

2Partially supported by NNSFC Grant No. 19928103, RGC-CERG grants HKUST6126/00P, HKUST6115/02P, and the Cheung Kong Scholars Programme

3Partially supported by a grant from the National Science Foundation and fellowships from the Sloane Foundation and the David and Lucile Packard Foundation.

2 THE RALLIS INNER PRODUCT FORMULA AND P -ADIC L-FUNCTIONS

obtaining the first proof of an Iwasawa-type main conjecture for p-adic L-functions of degree greater than 1 – specifically, the symmetric square p-adic L-functions for elliptic modular forms. Admitting some expected consequences of the stable trace formula for certain unitary groups, Skinner and Urban have used similar techniques to prove the main conjecture for the standard p-adic L-function associated to an elliptic modular form for a large class of such forms. The principle underlying these constructions is the following: the normalized value of the L-function in question appears as the constant term of a certain holomorphic Eisenstein series. In very favorable cases, vanishing of the constant term modulo p^{r} implies congruence of the Eisenstein series with a cusp form modulo p^{r}. This in turn implies isomorphism of the semisimplified reductions modulo p^{r} of the Galois representations associated to the cusp form and the Eisenstein series, and following Ribet this isomorphism gives rise to the non-trivial Galois cohomology class, as will be explained in §8. Very favorable cases are unfortunately very rare, however, and the greatest challenge in this approach, as well as in the one described below, is to make the argument work in relatively unfavorable cases.

Our approach is based on a different principle. We work with the standard L-function L(s, π) of an automorphic representation π of the unitary group of a hermitian vector space V′ of dimension n over a CM field K. These L-functions can and will be viewed as the L-functions of the corresponding automorphic representation BC(π) of GL(n, K) obtained by stable base change. Suppose there is an n + 1-dimensional positive-definite hermitian vector space V over K such that the theta lift Θ(π) of π to U(V) is non-trivial. Then the Rallis inner product formula identifies the value at s = 1 of L(s, π), or more precisely of the twist L(s, π, χ) where χ belongs to a certain family of Hecke characters of K, as the ratio of the square norm of the theta lift < θ(f), θ(f) > of a vector f ∈ π by the square norm < f, f > of f, multiplied by a finite product of bad local zeta integrals and explicit correction factors. By an endoscopic automorphic representation of U(V) (more precisely, an endoscopic representation of type (n, 1)), we will understand an automorphic representation of the form Θ(π); stable automorphic representations of U(V) will be automorphic representations not of this form. A principle originating in work of Hida affirms that p-divisibility of < f, f > (resp. < θ(f), θ(f) >) corresponds to congruences modulo p between forms in π (resp. Θ(π)) and forms in other representations π′ of U(V′) (resp. U(V)). Thus p-divisibility of the ratio \frac{<θ(f),θ(f)>}{<f,f>} corresponds, in very favorable cases, to congruences modulo p between forms in Θ(π) and forms in stable automorphic representations of U(V). Applying Ribet’s construction to these congruences one then obtains Galois cohomology classes as before.

THE RALLIS INNER PRODUCT FORMULA AND p-ADIC L-FUNCTIONS 3

Much work is required in order to sort out what we mean by a very favorable case, not to mention making the technique work more generally. We actually need to study congruences in the Iwasawa algebra, which can be viewed as a power series ring over a p-adic integer ring in up to n-variables. This means we need to construct the p-adic L-functions for continuous families – Hida families – of p-adic automorphic forms on U(V′) and U(V). We also need to obtain a version of the Rallis inner product formula adapted to these Hida families. These results are nearly complete, at least for the one- and two-dimensional families that suffice for our purposes, though some questions remain concerning the behavior of bad local zeta integrals. The most serious problem has to do with the p-adic properties of the theta lift. Roughly speaking, p-adic divisibility of the inner product < θ(f), θ(f) > is only useful if we can guarantee that θ(f) is p-adically integral but not itself divisible by p. This problem can be resolved when n = 2. It is likely that we can obtain conditional results for general n, but this is not yet completely clear.

Finally, the normalization of the p-adic L-function depends on the choice of a good integral structure on the cohomology of the Shimura variety attached to the unitary group U(V′). This is well understood when V′ is positive-definite, or when n = 2, but otherwise is completely mysterious. We will restrict our attention to the case of positive-definite V′, with some remarks about the case n = 2. When K is an imaginary quadratic field, which we assume henceforward, the case n = 2 includes the case of p-adic L-functions of elliptic modular forms over ℚ, base changed to K.

The idea of studying the congruences between stable and endoscopic forms on unitary groups was first proposed by Richard Taylor in a message to one of the authors in 1993. The connection to the theta correspondence, and specifically to the Rallis inner product formula, was first observed in 2000 or 2001. Throughout the project we have benefited from the advice and suggestions of many mathematicians, whom we will acknowledge in due course, as our articles are completed. However, we must immediately express our gratitude to Haruzo Hida, without whose advice and encouragement this work would have been simply impossible.

It gives us great pleasure to dedicate this article to Steve Rallis. Steve’s generosity is exceptional and his insight has been a constant source of inspiration. The “Rallis inner product formula” of the title originated in Steve’s work on the Howe duality correspondence and incorporates his results obtained with Piatetski-Shapiro on the standard L-functions of classical groups, on the one hand, and the results of his collaboration with Kudla on the extended Siegel-Weil formula, on the other hand, in its version for unitary groups due to Ichino. The applications of these results to arithmetic are just beginning to be explored. Yet all this work is at least ten years old. Since then Steve has, if anything, been even busier. The possibility is real that arithmetic will never catch up.

2. L-functions of unitary groups

For simplicity, we will work with unitary groups with base field ℚ, though everything goes over with little change to general totally real fields. Thus K is an imaginary quadratic field, given with a fixed embedding as a subfield of ℂ, V′ and V are hermitian spaces of dimension n and n + 1, respectively, over K. We take V to be positive-definite. In the applications V′ will usually be positive-definite as well, but in the present report we assume the signature of the hermitian form on V′ to be (a, b), with a + b = n. We let H = U(V), G = U(V′), viewed as algebraic groups over ℚ, and let K_{∞} ≃ U(a) × U(b) be a maximal compact subgroup of G(ℝ). Let ε_{K} be the quadratic character attached to the extension K/ℚ and c the non-trivial automorphism of K (complex conjugation). We fix a prime p, assumed to split in K.

For the moment we keep V and H in reserve and restrict our attention to automorphic representations π of G unramified at p. The representation π_{∞} will be assumed to belong to the discrete series, and to have cohomology with values in a one-dimensional representation (τ, W) of G(ℝ) = U(a, b):

(2.1)  H^{ab}(Lie(G)(ℝ), K_{∞}; π_{∞} ⊗ W) ≠ 0

(since π_{∞} is in the discrete series, all the cohomology is concentrated in the middle dimension ab). There are more general coefficient systems but we will ignore these.

Via twisted (unitary) stable base change, π can be identified – unconditionally for n ≤ 3 [Rog], conditionally^{4} for general n [HL] – with an automorphic representation BC(π) = π_{K} of GL(n, K), unramified at p, which is cohomological at the unique archimedean place and which satisfies

(polarization)  BC(π)^{∨} ⟶^{∼} BC(π)^{c}.

Arithmetically, this corresponds – again, under some conditions for n > 3 – to a continuous representation σ : Gal(\overline{ℚ}/K) → GL(A) on an n-dimensional \overline{ℚ}_{p}-vector space A which is unramified outside a finite set and is crystalline at p. This can be viewed either as the Galois representation attached to BC(π) or to π; there is no difference. But the former has an easier interpretation: if BC(π) = ⊗_{v}BC(π)_{v}, where v runs over places of K, prime to p, then it follows from [HT] that σ = σ(BC(π)) is characterized by

(2.2)  σ |_{Γ_{v}} ⟶^{∼} L(BC(π)_{v})

up to semisimplification, where v is any finite prime of K not dividing p, Γ_{v} is a decomposition group at v and L is the local Langlands correspondence, appropriately normalized, taking irreducible representations of GL(n, K_{v}) to n-dimensional representations of Γ_{v}.^{5}

The Galois representation σ is realized, up to an explicit abelian twist, on the cohomology of some Shimura variety, not necessarily attached to the group G.^{6} Suppose for the moment that the coefficient representation τ is trivial. Then the weight of the p-adic representation σ, determined by Frobenius eigenvalues at unramified primes, is n − 1, the Hodge-Tate weights are {0, 1, …, n − 1}, and the representation is regular in the following sense: each Hodge-Tate weight has multiplicity one (over the coefficient field). This regularity hypothesis, which remains valid for general τ (the list of Hodge-Tate weights depends on τ) is the principal restriction imposed by the use of Shimura varieties.

We continue to assume τ trivial. The polarization condition has one degree of freedom, namely twisting by a Hecke character χ of K which is anticyclotomic in the sense that χ factors through the map a ↦ a/a^{c}. To χ we can associate a weight k (basically χ(z) = (z/z^{c})^{−k}) and a p-adic incarnation σ_{p}(χ), so that the p-adic representation σ ⊗ σ_{p}(χ) is pure of weight w = n − 1 + k. The L-function with arithmetic normalization:

4One has to assume π is supercuspidal or Steinberg at two places of ℚ that split in K.

5We also need information at primes dividing p, specifically that the p-adic absolute values of the normalized Satake parameters coincide the Newton polygon of the Frobenius operator on crystalline cohomology. At least in the situations considered in [HL] the Katz-Messing theorem applies to provide what we need. For intersection cohomology of general Shimura varieties this should follow from work in progress of A. Nair [N].

6In general one may only obtain the restrictions of σ to the Galois groups of a sufficiently general family of quadratic extensions of K directly on the cohomology of Shimura varieties; cf.

[HT], Theorem VII.1.9.

(2.3)

L(s − \frac{n − 1 + k}{2} , BC(π), χ) (GL(n) version)

= L(s − \frac{n − 1 + k}{2} , π, χ) (U(n) version)
= L(s, σ ⊗ σ_{p}(χ)) (Galois version )

then has critical special values, in the sense of Deligne, for most k, the nature of the periods depending on k. In the first line of (2.3) we are working with the principal L-function of GL(n), whereas in the second line we are working with the Langlands L-function attached to π and the standard representation of the L-group of U(n). Suppose n = 2, for instance, in which case BC(π) is the base change to K of the representation generated by an elliptic modular newform f of weight 2. Then if k = 0 (so w = 1) L(s, σ ⊗ σ_{p}(χ)) has a critical value at the center s = 1 of Birch-Swinnerton-Dyer type whose associated Deligne period is roughly the Petersson norm < f, f >. If k = 3, so w = 4, L(s, σ ⊗ σ_{p}(χ)) has a critical value at the near-central point s = 3 whose Deligne period depends only on χ – i.e., on CM periods – up to algebraic factors. For general n, if V′ is supposed positive-definite, but k is fixed and τ is trivial, we only obtain the analogue of the latter case with k = s = n + 1; in particular, the critical value L(n + 1, σ ⊗ σ_{p}(χ)) is an algebraic multiple of a fixed CM period independent of π.

Fixing k eliminates the archimedean degree of freedom but χ can still vary in a one-dimensional p-adic (anticyclotomic) family. Henceforward we incorporate k into the representation τ – i.e., we drop the assumption that τ be trivial. We take L-functions in the unitary normalization, so that Re(s) = \frac{1}{2} is the axis of symmetry, and s = 1 is the possible pole for L-functions of tempered automorphic representations of GL(n). We write L(s, π, χ) = L(s, BC(π), χ) as above. In what follows, we let χ = ⊗_{v}χ_{v} vary among Hecke characters satisfying:

(2.4)  χ |_{A_{ℚ}^{×}} = ε_{K}^{n+1}; |χ_{∞}| = 1.

If χ_{0} and χ are two characters satisfying (2.4), then χ · χ_{0}^{−1} is anticyclotomic in the sense defined above, so that a family of χ satisfying (2.4) can be viewed as the product of a fixed base point χ_{0} by the family of all anticyclotomic Hecke characters. We write χ_{f}^{p} = ⊗_{v≠p,∞}χ_{v}, and view χ as a family of characters of finite order with values in ℚ̄^{×} ⊂ ℂ_{p}^{×}, where ℂ_{p} is the p-adic completion of ℚ̄_{p}. Consider the family X_{K} of p-adically continuous characters satisfying (2.4):

χ : A_{K}^{×} → ℂ_{p}^{×}

with fixed χ_{f}^{p}. This can be viewed as the set of ℂ_{p}-valued points of a rigid-analytic space over ℚ_{p}. Our results concern the p-adic L-function – which we need to construct for n > 2 – corresponding to the values L(1, π, χ) as a function of χ. We can also let π vary in an appropriate Hida family.^{7}

The p-adic L-function, like the standard L-function on which it is based, is defined by means of the doubling method of Piatetski-Shapiro and Rallis, studied in some detail, though insufficient for our purposes, by Shimura in two recent books ([GPR], [Sh1], [Sh2]; see also [Gar]). Let −V′ be the space V′ with its hermitian form multiplied by −1, let 2V′ = V′ ⊕ (−V′), and define two totally isotropic subspaces of 2V′ by V^{′,d} = {(v, v) | v ∈ V′}, V′_{d} = {(v, −v) | v ∈ V′}. Let P ⊂ U(2V′) be the stablizer of V^{′,d}; then the stabilizer M ⊂ P of V′_{d} is a Levi component, canonically isomorphic to GL(V^{′,d}) ≃ GL(V′). Let m : GL(V′) → M denote the canonical isomorphism. We write P = M · U_{P}; the exponential map identifies U_{P} with the group Her_{n} of n × n-hermitian matrices over K, viewed as an algebraic vector group over ℚ.

The inclusion V′ ⊕ (−V′) ⊂ 2V′ defines a natural embedding G × G ↪ U(2V′). The group U(2V′) is quasi-split, and U(2V′)(ℝ) can be identified with U(n, n). Choose a maximal compact subgroup K_{∞,2} ⊂ U(2V′)(ℝ) so that K_{∞,2} ∩ (G × G)(ℝ) = K_{∞,G} × K_{∞,G}.

Let χ be any unitary Hecke character of K, viewed as a character of M(A) ≅ GL(V′)(A) via composition with det(•). Consider the induced representation of U(2V′)(A):

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 ℚ, are likewise defined by normalized induction. At archimedean places we assume our sections to be K_{∞,2}-finite. For a section f(h; χ, s) ∈ I(χ, s) we form the Eisenstein series

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

This series is absolutely convergent for Re(s) > \frac{n−1}{2}, and it can be continued to a meromorphic function on the entire plane.

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

π ⟶^{∼} ⊗_{v} π_{v}, π^{∨} ⟶^{∼} ⊗_{v} π_{v}^{∨},

the tensor products taken over places v of ℚ, with π_{∞} an admissible (𝔤_{∞}, K_{∞,G})- module of cohomological type, with lowest K_{∞}-type τ_{∞}. 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:

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

7Especially in §3 and §4, we will make use, without explanation, of notions from Hida’s theory of (nearly) ordinary families of modular forms. Most of the results we use are contained in [Hi 02]. There is no simple introduction to Hida theory in higher dimensions, but the reader can refer to [Hi IHP] for background.

with dg = dg′ the Tamagawa measure on G(A).

The theory of this function was worked out (for trivial χ) by Li [L92] and more generally in [HKS,§6]. We make the following hypotheses:

Hypotheses (2.6)

(a) There is a finite set of finite places S_{f} of ℚ such that, for any non-archimedean 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}, and f_{∞} is an eigenvector for a one-dimensional representation of K_{∞,2};

(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}

(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) The vector φ_{∞} (resp. φ′_{∞}) is a non-zero highest (resp. lowest) weight vector in τ_{∞} (resp. in τ_{∞}^{∨}), such that (φ_{∞}, φ′_{∞})_{π_{∞}} = 1.

We let S = {∞} ∪ S_{f}. Define

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

Q_{V′}^{0}(φ ⊗ φ′) = ∫_{G(ℚ)\G(A)} φ(g)φ′(g)dg;

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

The integrals (2.5, 2.7) converge absolutely in a right halfplane and admit meromorphic continuations to all s, and we have the following identity of meromorphic functions on ℂ:

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

d_{n}^{S}(s, χ)Z(s, φ, φ′, f, χ) = Q_{V′}^{0}(φ ⊗ φ′)Z_{S}(s, φ, φ′, f, χ)L^{S}(s + \frac{1}{2}, π, χ).

Here and throughout the superscript S means that we omit the Euler factors at the places in S.

Let X(n, n) be the tube domain associated to the unitary group U(n, n); there is a homogenous isomorphism U(2V′)/K_{∞,2} = U(n, n)/U(n) × U(n) ⟶^{∼} X(n, n). A well-known procedure associates to the Eisenstein series E_{f} a function E_{f}(z, u_{f}, s), meromorphic in s, on X(n, n) × U(2V′)(A_{f}) × ℂ, with values in ℂ (because f_{∞} belongs to a one-dimensional K_{∞,2}-type), which satisfies a familiar automorphy relation with respect to the left action of U(2V′)(ℚ). In what follows we call E_{f} the classical Eisenstein series associated to E_{f}.

When V′ is positive definite the zeta integral (2.5) is just a finite sum of values of the classical Eisenstein series E_{f}. When n = 1, the basic identity reduces to the expression of the Hecke L-function on the right hand side as a finite sum of special values of the normalized elliptic modular Eisenstein series on the left. This expression is familiar to number theorists for its role in the proof of Damarell’s theorem and its generalizations by Shimura, the starting point for Katz’ construction of p-adic L-functions of Hecke characters of CM fields. The generalization of Katz’ construction is described in the following section.

In general, with the appropriate normalization and choices of data f_{v}, φ_{v}, and φ′_{v} at primes v ∈ S_{f}, the basic identity yields the following result:

Theorem (2.9). The expression

π^{− \frac{n(n+3)}{2}} Q_{V′}^{0}(φ ⊗ φ′)Z_{S}(\frac{1}{2}, φ, φ′, f, χ)L^{S}(1, π, χ)

is an algebraic number.

In order for this to hold, the zeta integral has to be identified with a natural cup product pairing in coherent cohomology, the Eisenstein series has to be \overline{ℚ}-rational in the sense of having algebraic Fourier coefficients, and the functions ϕ, ϕ′ have to be identified with \overline{ℚ}-rational modular forms. These conditions are discussed in sections 3 and 4 below, (cf. condition (3.3)(ii), for example). The power of π in the denominator is an algebraic multiple of the factor d_{n}^{S}(\frac{1}{2}, χ). We prefer not to be precise about choices of measures so the power of π is somewhat spurious. Later we will implicitly assume that measures have been chosen so that the archimedean factor Z_{∞}(\frac{1}{2}, φ, φ′, f, χ) is an algebraic number (even a p-unit); any discrepancy is of course incorporated into the definition of Q_{V′}^{0}. The non-archimedean bad local factors in Z_{S}(\frac{1}{2}, φ, φ′, f, χ) can easily be arranged to be algebraic.

3. p-adic Eisenstein measures.

Henceforward we assume π_{p} is unramified. We also assume S_{f} = S_{K} ∐ S_{π,χ}∪{p}, where S_{K} is the set of primes ramified in K and S_{π,χ} is the set of primes at which either π or χ is ramified. We make the hypothesis

Hypothesis (3.0). Every prime in S_{π,χ} splits in K.

In the cases we consider the classical Eisenstein series E_{f}(z, u_{f}, s) is a holomorphic modular form at the point s = \frac{1}{2}, corresponding to the value L(1, π, χ) in the basic identity. When φ and φ′ correspond to (anti)-holomorphic modular forms on the Shimura variety associated to G, the basic identity, combined with rationality properties of the Fourier coefficients of holomorphic Eisenstein series, thus provides an expression of the special values L(1, π, χ), as χ varies with χ_{∞} fixed, as algebraic multiples of a fixed period factor:

(3.1)  L(1, π, χ) = L^{alg}(1, π, χ) · p(π, χ_{∞}),

where p(π, χ_{∞}) is essentially the Petersson norm < φ, φ′ >, up to elementary factors [Sh1,H97].

The philosophy of p-adic L-functions is summarized by the following two sentences, the first of which is highly misleading:

(i) As χ varies over a p-adic family X, for instance the X_{K} defined above, one expects the values L^{alg}(1, π, χ) to vary p-adically analytically, and thus to extend to an analytic function on X with values in ℂ_{p}.

(ii) The properties of this analytic function depend strongly on the p-adic ab-

solute values of the Satake parameters of the unramified representation π_{p}. When one actually tries to construct p-adic L-functions, one immediately realizes that the values L^{alg}(1, π, χ) need to be modified in order for something like (i) to be true. The modification involves multiplication by p-adic and archimedean correction factors, whose conjectural form is proposed by Coates in [Co]. We have chosen to keep the real components of π and χ fixed, so the archimedean correction factors can be ignored. In the ultimate theory π_{∞} and χ_{∞} will vary and the archimedean correction terms – mainly products of factorials – will have to be addressed.

On the other hand, we allow χ_{p} to be arbitrarily ramified, so p-adic correction factors are essential. This amounts to including the prime p in the set S_{π,χ} of bad primes, and making specific choices of local data that allow explicit calculation of the zeta integral

(3.2)  Z_{p}(s, φ_{p}, φ′_{p}, f_{p}, χ_{p}) = ∫_{G(ℚ_{p})} f_{p}((g_{p}, 1); χ, s)(π_{p}(g_{p})φ_{p}, φ′_{p})dg_{p}.

at the point s = 1. The choices of vectors φ_{p}, φ′_{p} are more or less imposed by Hida theory. Let P ⊂GL(n) be a standard parabolic subgroup, of parabolic rank ℓ, say. In the most general setting, what Hida theory provides is a p-adic manifold X_{P} of dimension ℓ and a countable Zariski dense subset X_{P}^{alg} ⊂ X_{P} (ℂ_{p}) of points corresponding to triples (π_{x}, φ_{x}, χ_{x}), with π_{x} an automorphic representation of G of fixed infinity type and fixed central character, φ_{x} ∈ π_{x} a factorizable vector, and χ_{x} a Hecke character of K satisfying (2.4). The pairs (ϕ_{x}, χ_{x}) extend to an analytic family (φ, χ) of some sort of p-adic modular forms on G(A). The problem is to find f_{p}(x) = f_{p}(ϕ_{x}, χ_{x}) such that the corresponding Eisenstein series also extend to a p-adic analytic map from X_{P} to p-adic modular forms on U(2V′).

One of the rules of the game is that the f_{v}, for v ≠ p, are constant as functions on X_{P}. Specifically,

(i) When v ≠ S hypothesis (2.6)(d) remains true;

(ii) When v ∈ S_{π,χ} is prime to p, f_{v} is chosen to make all zeta integrals constant and algebraic;

(3.3)

(iii) The section f_{∞} is chosen to make the classical Eisenstein series E_{f}(z, u_{f}, s) holomorphic.^{8} (iv) When v ∈ S_{K}, we fix f_{v}.

8More generally, when π_{∞} is allowed to vary, one will want E_{f} (z, u_{f} , s) to have holomorphic restriction to the Shimura variety associated to G × G. We actually work with unitary similitude groups in order to define Shimura varieties, but in fact the theory of the standard L-functions only depends on the unitary groups.

(v) Finally, we will assume below that f = ⊗f_{v} is a Siegel-Weil section for some

(n + 1)-dimensional hermitian space V. Only f_{p} needs to vary.

At this point we need to provide at least a rough definition of a p-adic modular form and of a p-adic family of modular forms, on U(2V′) as well as on G. For U(2V′) the definition is relatively simple. We restrict our attention to Eisenstein series. The classical Eisenstein series E_{f}(z, u_{f}) = E_{f}(z, u_{f}, \frac{1}{2}) – we only work with Eisenstein series holomorphic at s = \frac{1}{2}, corresponding to the value of the L-function at s = 1 – is completely determined by its Fourier expansion. Let E_{f}^{∗} = π^{− \frac{n(n+3)}{2}} d_{n}^{S}(\frac{1}{2}, χ)E_{f}. Then the Fourier expansion of E_{f}^{∗} is

(3.4) E_{f}^{∗}(z, u_{f}) = ∑_{β∈Her_{n}(ℚ)} c_{f,β}(u_{f})q^{β}(z), z ∈ X(n, n) , u_{f} ∈ M(A^{S}) × ∏_{v∈S_{f}} K_{v}

Here K_{v} ∈ GL(V′_{v}) is a special maximal compact subgroup (by strong approximation one does not need all values of the argument u_{f} ∈ U(2V′)(A_{f}) to define E_{f}^{∗}, the values in M(A^{S}) × ∏_{v∈S_{f}} K_{v} suffice). The symbol q^{β} = q^{β}(z) denotes the function z ↦ e^{2πiαT r(β·z)}, where α ∈ ℚ^{×}, assumed to be a p-unit, is a free parameter corresponding to the choice of an additive character. Holomorphy and the Koecher principle imply that E_{f,β}^{∗} = 0 unless β is positive semidefinite. Our choices of f_{p} provide a stronger vanishing condition:

(3.5)  c_{f,β} ≠ 0 ⇒ det β ≠ 0;

as well as a factorization condition^{9}

(3.6)  c_{f,β}(u_{f}) = c_{∞,β} · ∏_{v≠∞} c_{f_{v},β,v}(u_{v}).

Here u_{v} is the v-component of u_{f} and the term c_{∞,β} is a function of β alone.

Since we are working with E^{∗} rather than E, the normalized coefficient in (3.6) is a finite Euler product. Let Λ ⊂ H_{n}(ℚ) denote the lattice of hermitian matrices with entries in O_{K}, Λ^{∗} the dual lattice

Λ^{∗} = {B ∈ Her_{n}(ℚ) | tr(BΛ)⊂O_{K}}.

For any v let T_{v} denote the characteristic function of Λ_{v}^{∗}.

Proposition (3.7). (a) For A ∈ GL(V′_{v}), v ∉ S.

c_{f_{v},β,v}(m(A)) = T_{v}(α^{t}ĀβA), v ∉ S.

(b) Let c_{f,β}^{0} = c_{∞}(β) · ∏_{v∈S_{π,χ},v≠p} c_{f_{v},β,v}. Under hypotheses (3.3) one can find lattices L_{v} ∈ Her_{n}(ℚ_{v}) and a constant γ_{0} ∈ \overline{ℚ} such that, for all m(A) ∈ ∏_{v∈S_{π,χ}∪∞,v≠p} K_{v}, c_{f,β}^{0}(m(A)) = γ_{0} · det(β) · χ ∘ det(A) for β ∈ ∏_{v∈S_{π,χ}} L_{v}^{∗} and equals zero otherwise.

9Compare [Sh1, Proposition 19.2]. Shimura’s hypothesis 𝔠 ≠ 𝔤, which has to be read together with the definition (18.4.8, 18.4.9) and various other definitions in [Sh1], guarantees the factorization condition for the Eisenstein series he considers.

Here L_{v}^{∗} is the dual lattice.

We only really care about the p-adic valuation of c_{f,β}(u_{f}), and Proposition (3.7) tells us that the contributions of primes outside {p} ∪ S_{K} to this valuation are bounded. Moreover, the constant γ_{0} is a product of explicit local constants. The archimedean contribution is a p-unit if p > n + 1 and if det β is a p-unit, which will be true for our choice of f_{p}. The other local contributions are indices of open compact subgroups in G(ℚ_{v}), whose p-adic valuations can be determined easily. At the time of writing we do not have good choices of f_{v} for v ∈ S_{K}. However, we can again guarantee at least that the corresponding factor c_{f,β}^{K}(u_{f}) = ∏_{v∈S_{K}} c_{f_{v},β,v}(u_{f}) has p-adic absolute value bounded above and below, and we will ignore these factors for the time being.

It remains to consider the factor c_{f_{p},β,p}, which depends on explicit choices of local factors f_{p}. Consider a partition n = n_{1} + ⋯ + n_{ℓ}, corresponding to a standard parabolic subgroup P ⊂ GL(n). Let M_{n}(ℚ_{p}) denote the space of n × nmatrices with values in ℚ_{p}, and let K_{P} ⊂ GL(n, ℤ_{p}) ⊂ M_{n}(ℚ_{p}) be the standard parahoric subgroup corresponding to the parabolic P. Let ν = (ν_{1}, …, ν_{ℓ}) be an ℓ-tuple of characters of ℤ_{p}^{×} of finite order, with values in \overline{ℚ}^{×} ⊂ ℂ^{×}. The group K_{P} consists of matrices Z = (Z_{ij})_{1≤i,j≤ℓ}, written according to the chosen partition, with Z_{ii} ∈ GL(n_{i}, ℤ_{p}) for all i, with above-diagonal entries p-integral, and below-diagonal entries divisible by p. We define a function ϕ_{ν} : K_{P} → \overline{ℚ}^{×} by ϕ_{ν}(Z) = ∏_{1≤i≤ℓ} ν_{i}(det Z_{ii}). We extend ϕ_{ν} by zero to a function, still denoted ϕ_{ν}, on M_{n}(ℚ_{p}).

Let w_{1}, w_{2} denote the two primes of K dividing p. The local component χ_{p} of the Hecke character maps K_{p}^{×} = K_{w_{1}}^{×} × K_{w_{2}}^{×} ⟶^{∼} ℚ_{p}^{×} × ℚ_{p}^{×} to \overline{ℚ}^{×}. By restriction to the first factor (more precisely, by fixing w_{1} rather than w_{2}) χ_{p} becomes a character of ℤ_{p}^{×} of finite order. The group

M(ℚ_{p}) = GL(V′_{p}) ⟶^{∼} GL(V′_{w_{1}}) × GL(V′_{w_{2}}) ⟶^{∼} GL(n, ℚ_{p}) × GL(n, ℚ_{p}),

and elements of M(ℤ_{p}) will be denoted m(A, B) for A, B ∈ GL(n, ℤ_{p}). There is a choice of basis implicit in the constructions below.

Proposition (3.8). For every ℓ + 1-tuple (ν_{1}, …, ν_{ℓ}, χ_{p}) of characters of ℤ_{p}^{×} of finite order, there exists a section f_{p}(ν, χ_{p}; s) ∈ I_{p}(χ_{p}, s) such that, setting s = \frac{1}{2},

c_{f_{p}(ν,χ_{p}),β,p}(m(A, B)) = χ_{p}(det(BA^{−1}))ϕ_{ν}(^{t}AβB^{−1}).

for any A, B ∈ GL(n, ℤ_{p}), provided β is p-integral with determinant a p-unit, and vanishes otherwise.

The construction of f_{p} was inspired by Katz’ construction [Katz] when n = 1. It involves tensoring ϕ_{ν} with its Fourier transform and viewing the result as a Schwartz-Bruhat function Φ_{ν} for M(n, 2(n+1)), hence a datum for the local Siegel-Weil formula for the theta lift from U(n+1)(ℚ_{p}) = GL(n+1, ℚ_{p}) to U(2V′)(ℚ_{p}) = GL(2n, ℚ_{p}). The details are quite complicated, but we would like to point out that, although the partial Fourier transform does not preserve p-integrality, the resulting local coefficient in (3.8) is p-integral. We write f(ν, χ) = ⊗_{v≠p}f_{v} ⊗ f_{p}(ν, χ_{p}), E_{f^{p},χ,ν}^{∗} = E_{f(ν,χ)}^{∗}.

One can define a p-adic modular form on U(2V′) to be a formal expression of the form (3.4), where now the q^{β} are symbols satisfying the obvious relations q^{β}q^{β′} = q^{β+β′}, and the coefficients c_{f,β}(u_{f}) are ℂ_{p}-valued functions on M(A_{f}) ∩ (M(A^{S}) × M(ℤ_{p}) × ∏_{v∈S_{f} ,v≠p} K_{v}) with p-adic absolute values uniformly bounded in β as well as u_{f}. In order to qualify as a p-adic modular form, the formal series has to arise as the p-adic limit of Fourier expansions of genuine holomorphic modular forms on U(2V′), whose weights and levels are allowed to vary. This condition, first introduced by Serre, was given a rigid-analytic interpretation by Katz, and it is this interpretation that allows us to restrict p-adic modular forms to the Shimura varieties attached to G×G, and hence to give a p-adic interpretation of the doubling method.^{10}

We stress that, in the setting of Shimura varieties of higher dimension, the foundations of the theory of p-adic modular forms are entirely due to Hida (especially [Hi00, Hi02, Hi IHP]). Moreover, the possibility of applying these results to generalizing Katz’ construction of p-adic L-functions is explicitly stated in Hida’s articles, and – at least when V′ is positive-definite – the construction of these p-adic Lfunctions will appear in a forthcoming article of Hida.

Let M(2V′) denote the ℂ_{p}-Banach space of p-adic modular forms on U(2V′). Let Z^{ℓ} = (ℤ_{p}^{×})^{ℓ+1} and let 𝕏_{P} denote the set of ℂ_{p}^{×}-valued characters on Z^{ℓ}. The set 𝕏_{P} has the structure of the set of ℂ_{p}-valued points of a rigid-analytic variety over ℚ_{p} whose subset 𝕏_{P}^{alg} of characters of finite order is Zariski-dense. A measure dμ on Z^{ℓ} with values in M(2V′) is a finitely additive continuous map from the Banach space of continuous ℂ_{p}-valued functions F on Z^{ℓ} to M(2V′):

C(Z^{ℓ}, ℂ_{p}) ∋ ϕ ↦ ∫_{Z^{ℓ}} ϕdμ ∈ M(2V′).

Such a measure is determined uniquely by its restriction to 𝕏_{P} ⊂C(Z^{ℓ}, ℂ_{p}), and even by its restriction to the Zariski-dense subset 𝕏_{P}^{alg}.

Theorem (3.9). Fix f^{p} = ⊗_{v≠p}f_{v} as above. There is a (unique) p-adic measure dμ_{Eis}(f^{p}) on Z^{ℓ} with values in M(2V′) such that, for any (ν, χ) ∈ 𝕏_{P}^{alg},

∫_{Z^{ℓ}}(ν, χ)dμ_{Eis}(f^{p}) = E_{f^{p},χ,ν}^{∗}.

This is an immediate consequence of the above results on Fourier coefficients. Indeed, (3.4) and (3.5) show that the Fourier coefficients factorize as a product of local coefficients; Proposition 3.7 shows that for fixed f^{p}, the product of local coefficients prime to p is bounded uniformly in β and u_{f}; and Proposition 3.8 asserts that the local coefficient at p varies p-adically continuously on 𝕏_{P}^{alg}. Let F = (ϕ, ξ) : Z^{ℓ+1} → ℂ_{p} be a continuous function where ϕ depends on the first ℓ variables and ξ on the last (so F (z_{1}, ..., z_{ℓ}, x) = ϕ(z_{1}, ..., z_{ℓ})ξ(x)). Define a function ϕ_{F} : I_{P} → ℂ_{p} by composing ϕ with the natural map

10Note that p-adic integrality of the Fourier coefficients is not in general preserved by the action of the full group U(2V′)(ℚ_{p}), but only by the subgroup that stabilizes the “cusp at infinity.” In the theories of Katz and Hida this is interpreted in terms of the geometry of the Igusa varieties.

I_{p} → ℤ_{p}^{ℓ}; Z ↦ (det Z_{11}, …, det Z_{ℓ,ℓ}).

Let T_{(L_{v})}(β, A) : Her_{n}(ℚ) × GL(V′_{A_{f}^{p}}) → ℂ_{p} be the function that takes the value 1

if α^{t}Ā_{v}βA_{v} ∈ L_{v}^{∗} for all v ∉ S, β ∈ L_{v}^{∗} for all p ≠ v ∈ S_{π,χ}, and β is integral at p and det(β) ∈ ℤ_{p}^{×}, and otherwise takes the value 0. Then

∫_{Z^{ℓ}} F dμ_{Eis}(f^{p}) = ∑_{β} c_{β}(u)q^{β}

where c_{β}(u) is the function of u ∈ GL(V′_{A_{f}}) = M(A_{f}), u_{p} = (A_{p}, B_{p}) ∈ M(ℤ_{p}) and u_{v} ∈ K_{v} for all p ≠ v ∈ S_{f}, given by (3.10) c_{β}(u) = γ_{0} det(β)ξ(det u_{S_{π,χ}}^{p})χ_{p}(det(B_{p}A_{p}^{−1}))ϕ_{F} (^{t}A_{p}βB_{p}^{−1}) × T_{(L_{v})}(β, u)c_{β,K}(u),

where u_{S_{π,χ}}^{p} = ∏_{p≠v∈S_{π,χ}} u_{v} and c_{β,K}(u) = ∏_{v∈S_{K}} c_{β,v}(u_{v}).

4. p-adic L-functions for unitary groups.

The rigid-analytic variety 𝕏_{P} introduced above breaks up as the disjoint union of (p − 1)^{ℓ+1} connected components, indexed by the distinct characters of the torsion subgroup ((ℤ/pℤ)^{×})^{ℓ+1} ⊂ Z^{ℓ}. We fix such a character ν^{0} = (ν_{1}^{0}, …, ν_{ℓ}^{0}, χ^{0}) and the associated connected component 𝕏_{P}^{0} ⊂ 𝕏_{P}. For our purposes, a Hida family of type P on U(V′) will correspond to

(1) A rigid-analytic variety 𝕐 and a finite flat morphism wt : 𝕐 → 𝕏_{P}^{0} ;

(2) An analytic family (φ_{y}, χ_{y}) over 𝕐 of pairs where φ_{y} is a p-adic modular form on U(V′) and χ_{y} : ℤ_{p}^{×} → ℂ_{p}^{×} is a continuous character whose restriction to the torsion subgroup (ℤ/pℤ)^{×} is χ^{0};

(3) When wt(y) = (ν, χ) ∈ 𝕏_{P}^{alg} ∩𝕏_{P}^{0} , χ_{y} = χ and φ_{y} is a holomorphic automorphic form on U(V′), belonging to an irreducible automorphic representation π_{y}, of archimedean type π_{∞} (as in §2) and of type ν at p (see below);

(4) The family φ_{y} is P -nearly ordinary in Hida’s sense (see below). Of coures, if (φ_{y}, χ_{y}) is any Hida family then one obtains another by multiplying by any holomorphic rigid-analytic function f on 𝕐: (f(y)φ_{y}, χ_{y}). We call a Hida family primitive if it can not be obtained from another by multiplication by a non-constant f.

We will not define p-adic modular forms on a unitary group of general signature except to say that (i) they are uniform limits of classical holomorphic modular forms of variable level and/or weight which are \overline{ℚ}-rational in a sense compatible with Shimura’s theory of canonical models, and (ii) they can be defined without recourse to q-expansions. We refer to the points y such that wt(y) ∈ 𝕏_{P}^{alg} as classical points and the associated φ_{y} as classical modular forms. The character ν in (3) is assumed to restrict to ν^{0} = (ν_{1}^{0}, …, ν_{ℓ}^{0}) on the torsion subgroup. A form is of type ν at p if it belongs to an irreducible automorphic representation π of U(V′)(A) with π_{p} in the principal series, and if it is an eigenvector for P (ℤ_{p}) ⊂ I_{P} ⊂ U(V′)(ℚ_{p}) = GL(n, ℚ_{p}) with eigenvalue Z ↦ ν(Z) = ν_{1}(det(Z_{11})) ⋯ ν_{ℓ}(det(Z_{ℓ,ℓ})). It is also invariant under a certain open compact subgroup K_{ν} ⊂ I_{P} containing the kernel of ν in P (ℤ_{p}) – in particular, K_{ν} contains the ℤ_{p}-points R_{u}P (ℤ_{p}) of the unipotent radical of P. In many cases ν extends to a character of K_{ν} and (K_{ν}, ν) is a semisimple type in the sense of Bushnell-Kutzko, but this is (unfortunately!) only the case when the conductors of the ν_{i} are monotonic (increasing or decreasing; so ℓ ≤ 2 is OK).

Finally, the P -ordinarity condition is a condition on the inducing character of the principal series π_{p}. Suppose α = α_{1} ⊗ ⋯ ⊗ α_{n} : B → \overline{ℚ}^{×} is a character of the standard Borel subgroup of GL(n, ℚ_{p}), each α_{i} a character of ℚ_{p}^{×}, and π_{p} is the principal series representation unitarily induced from α. In practice π_{p} will be essentially tempered, so the principal series is necessarily irreducible. Condition (3) implies that the α_{i} can be ordered in such a way that their restrictions α̃_{i} to ℤ_{p}^{×} satisfy

α̃_{j} = ν_{j}, n_{1} + ⋯ + n_{i−1} < j ≤ n_{1} + ⋯ + n_{i}.

Fix an embedding of \overline{ℚ} in ℂ_{p}, i.e. a p-adic valuation |·|_{p} on \overline{ℚ}, and let μ_{j} = |α_{j}(p)|_{p}. Condition (4) above is that

(4.1)  ∑_{j=1}^{n_{1}+n_{2}+···+n_{i}} μ_{j} = ∑_{j=1}^{n_{1}+n_{2}+···+n_{i}} j − 1 + k , i = 1, …, ℓ

where k is a fixed constant (this is the same k we fixed to be n+1 in §2). A condition of this sort is the starting point of Hida’s deformation theory and corresponds to a condition on the p-adic Galois representation associated to π, but we will have no use for its specific form here. We note that (4.1) together with condition (3) imposes a partial order on the characters α_{j}: one can permute the α_{j}’s within the n_{i}-th block (n_{1} + ⋯ + n_{i−1} < j ≤ n_{1} + ⋯ + n_{i}) for any i, but no permutations are allowed between blocks.

We henceforward assume that the partition n = n_{1} + ⋯ + n_{ℓ} associated to P is a refinement of the partition n = a + b associated to the signature of V′; i.e. that a = n_{1} +⋯+n_{m}, b = n_{m+1} +⋯+n_{ℓ} for some index m. Let 𝕐^{alg} = (wt)^{−1}(𝕏^{0,alg}), so that every y ∈ 𝕐^{alg} defines an irreducible automorphic representation π_{y} of infinity type π_{∞}. One wants to guarantee that the map from 𝕐^{alg} to automorphic representations is injective, and in particular we want a way to single out vectors φ_{y} ∈ π_{y}, up to scalar multiples. At places outside S we assume φ_{y} is an unramified vector. At the infinite place φ_{y} is determined by holomorphicity. Every v ∈ S(π, χ) splits in K, hence π_{y,v} is a representation of GL(n, ℚ_{v}), and we assume the condition of being a semi-simple type vector determines the v-component of φ_{y} uniquely. This is possible as long as π_{v} is fully induced from a supercuspidal representation of some Levi component, provided we remember to interpret φ_{y} as a vector valued modular form with values in the space dual to the (tensor product over v ∈ S(π, χ) of) the type spaces. We have no good choice for v ∈ S_{K} unless π_{v} is spherical for some special maximal compact subgroup – and even then there are two conjugacy classes of special maximal compacts if n is odd, so we have to choose one – so in this exposition we simply ignore this issue and pretend φ_{y} is fixed up to scalar multiples.

Finally, and most importantly, Hida has shown that conditions (3) and (4) determine the p-component of φ_{y} up to scalars, and we can identify it explicitly as a (K_{ν}, ν)-type when the latter is a type. The conditions guarantee that φ_{y} is a factorizable vector for any y ∈ 𝕐^{alg}. We determine vectors φ′_{y} ∈ π_{y}^{∨} by the dual conditions and the requirement that the Petersson norms be the same. We would like to apply the basic identify (2.8). However, since we have defined Hida families so that φ_{y}, y ∈ 𝕐^{alg}, is holomorphic (at least when ab ≠ 0), we apply the basic identity not to this pair but to the corresponding pair φ̄_{y}, φ̄_{y′}, where the former is antiholomorphic and the latter is defined again by the dual conditions and both are normalized so that Q_{V}^{0} (φ̄_{y} ⊗ φ̄′_{y}) = Q_{V}^{0} (φ_{y} ⊗ φ′_{y}).^{11} When ab = 0 holomorphy (resp. antiholomorphy) is replaced by the condition of taking values in a certain finite-dimensional representation of U(n) (resp. the dual representation).

Corresponding to Proposition 3.7, we have that, with an appropriate choice of measures,

Proposition (4.2). The factor Z_{S}^{p}(1, φ̄_{y}, φ̄′_{y}, f, χ) is an algebraic number whose p-adic absolute value is uniformly bounded above and below as a function of y and χ.

(The superscript ′p′ denotes the ommision of the relevant factors at p.) One actually knows more. If we avoid small primes the archimedean contribution is a p-adic unit, and for the choices made (without explanation) in (3.3)(ii), the factors corresponding to primes in S(π, χ) are just group indices again. Any variation is again due to S_{K}.

It remains to determine the local zeta integral at p. We let P_{a,b} be the standard parabolic subgroup of GL(n) corresponding to the partition n = a + b. We write π_{p} = Ind_{P_{a,b}(ℚ_{p})}^{GL(n,ℚ_{p})}π_{a} ⊗ π_{b} (normalized induction) where π_{a} is a principal series representation of GL(a, ℚ_{p}) and π_{b} a principal series representation of GL(b, ℚ_{p}). In what follows, ℓ = 1 is only allowed if U(V′)(ℝ) is definite; i.e. if P_{a,b} = GL(n).

Proposition 4.3. Write

Z_{a}(1, π) = ε_{p}(0, π_{a} ⊗ χ, ψ_{p}^{−1})\frac{L_{p}(1, π_{a}^{∨} ⊗ χ^{−1})}{L_{p}(0, π_{a} ⊗ χ)}

Z_{b}(1, π) = ε_{p}(0, π_{b}^{∨} ⊗ χ^{−1}, ψ_{p}^{−1})\frac{L_{p}(1, π_{b} ⊗ χ)}{L_{p}(1, π_{b}^{∨} ⊗ χ^{−1})}

Suppose ℓ = 2. Then for any y ∈ 𝕐^{alg}

(4.3.1)  Z_{p}(\frac{1}{2}, φ̄_{y}, φ̄′_{y}, f_{p}, χ_{p}) = A(a, b)p^{N(π_{p})} · Z_{a}(1, π) · Z_{b}(1, π),

conjugate.

11This has the effect of replacing L(s, π, χ) by L(s, π̄, χ) in the Basic Identity. To avoid changing all the formulas we make the simplifying hypothesis that π_{f} is isomorphic to its complex

where A(a, b) ∈ ℚ is an explicit constant depending only on a and b and N(π_{p}) ∈ ℤ depends only on the conductors of the characters ν_{i} and ν_{i}ν_{j}^{−1}. The formula (4.3.1) remains true for any ℓ, for a Zariski dense set of points y ∈ 𝕐^{alg}.

The shape of the local factor (4.3.1) is roughly as predicted by Coates [Co], although we have not checked that the constants A(a, b) and (especially) the power N(π_{p}) of p corresponds exactly to his normalization. For purposes of comparison, the standard L-factor at p is just

L_{p}^{U(V′)}(s, π, χ) = L_{p}^{GL(n)}(s, π, χ)L_{p}^{GL(n)}(s, π^{∨}, χ^{−1}).

The p-adic normalization process, sometimes called p-stabilization, selects half the Euler factors, places the others (with s replaced by 1 − s) in the denominator, and for good measure multiplies the whole by a product of ε factors (products of Gauss sums). Here the presence of the denominator corresponds precisely to the partial Fourier transform used to define the section f_{p}, and appears in the course of applying the Godement-Jacquet functional equation for the standard L-function of GL(n).

Finally, Hida theory provides a way to pair the Eisenstein measure of the preceding with the Hida family φ_{y} (or rather φ_{y} ⊗ φ′_{y}) in such a way that at points in 𝕐^{alg} the result is the global zeta integral involving φ̄_{y} and φ̄′_{y}. Let

L^{p}(1, π_{y}, χ_{y}) = L^{S}(1, π, χ)Z_{S}^{p}(1/2, φ̄_{y}, φ̄′_{y}, f, χ_{y}).

The final result (see the remarks following Theorem 2.9) is:

Theorem 4.4. There is an analytic function (generalized measure)

L_{p}(1, •) = φ ∗ dμ_{Eis} : 𝕐 → ℂ_{p}

such that

L_{p}(1, y) = A(a, b)q^{N(π_{y,p})} · Z_{a}(1, π_{y}) · Z_{b}(1, π_{y})L^{p}(1, π_{y}, χ_{y})/Ω_{y}

for all y ∈ 𝕐^{alg}.

Here Ω_{y} is a period essentially equal to the quotient of the Petersson norm of φ_{y} by a congruence number measuring congruences between φ_{y} and other forms on G.

The notation L_{p}(1, y) indicates the expectation that the p-adic L-function extends to an analytic function in the first argument (s = 1) but we have not considered values other than s = 1. We also write L_{p}(1, π, χ) = L_{p}(1, y) if π = π_{y}, χ = χ_{y}.

(4.5) Hecke algebras in Hida’s theory. Rigid analytic geometry is not really necessary for Hida’s theory but it provides a useful geometric picture of p-adic variation. Let Λ_{P} and H_{𝕐} be the rings of functions on 𝕏^{0} and 𝕐, respectively. Hida actually works with integral versions of these rings, in which case the former is an Iwasawa algebra in rank(P) variables and the latter, finite and torsion-free over the former, is a p-adic arithmetic Hecke algebra. In our rigid-analytic setting, the rings Λ_{P} and H_{𝕐} are obtained from Hida’s rings upon tensoring with ℚ_{p}. We have constructed L_{p}(1, y) to be an element in H_{𝕐} ⊗ ℂ_{p} (and even in H_{𝕐}).

The irreducible components of 𝕐 correspond to Hida families, whereas the connected components of 𝕐 correspond to collections of Hida families admitting congruences modulo maximal ideals in Λ_{P}. We will make more precise the notion of such congruences when we return to these considerations in §7.

5. Endoscopic forms and the Rallis inner product formula

Formulas are different depending on the parity of n, and to shorten the exposition, we will henceforth assume n to be even. Let χ be an (adelic) Hecke character of K satisfying (2.4), so

χ |_{A_{ℚ}^{×}} = ε_{K}.

Recall that we have fixed a definite hermitian space V at the beginning of the text, with unitary group H. Let π×ξ be an automorphic representation of G×U(1). The Arthur conjectures predict the existence of an automorphic representation Π_{χ}(π×ξ) of H, with the property that BC(Π_{χ}(π × ξ)) is induced from the representation (BC(π) ⊗ χ ∘ det) × BC(ξ) of GL(n) × GL(1):

(5.1)  L(s, BC(Π_{χ}(π × ξ))) = L(s, BC(π) ⊗ χ ∘ det)L(s, BC(ξ)).

Here recall that BC(ξ) is the character z ↦ ξ(z/z^{c}), where c denotes complex conjugation. The Π_{χ}(π ⊗ ξ) are the endoscopic representations of the title (more properly attached to the elliptic endoscopic group U(n)×U(1)). Note that Π_{χ}(π×ξ) comes along with a conjectural multiplicity (proved for n = 2) which may be zero, but in general is not. Also Π_{χ}(π × ξ) is actually a packet, but in the examples we consider there will be only one representation. Indeed, H(ℝ) is compact, so L-packets at ∞ are singletons.^{12}

The theta correspondence constructs these representations “by hand”. The tensor product V′ ⊗ V has a non-degenerate hermitian form <, >_{V′⊗V}, and the choice of a non-zero element ℷ ∈ K with tr_{K/ℚ}(ℷ) = 0 transforms <, >_{V′⊗V} into a skew-hermitian form ℷ^{−1} <, >_{V′⊗V}. Thus V = R_{K/ℚ}(V′ ⊗ V) can be viewed as a symplectic vector space over ℚ of dimension 4n(n + 1), and there is a natural map G × H = U(V′) × U(V) → Sp(V). For our purposes, the metaplectic cover Mp(V)(A) is the extension of Sp(V)(A) by the circle. The character χ introduced above can be used to define a splitting G(A) × H(A) → Mp(V)(A) (cf. [Kudla]). Given χ as above and an additive adelic character ψ, which has to have the right sign at ∞, one can then define the metaplectic representation ω_{ψ,χ} of G(A)×H(A) and, for any automorphic representation π of G, the theta lift Θ_{ψ,χ}(π) as an automorphic representation of H.

Assume π is in the domain of the theta correspondence; i.e., that Θ_{ψ,χ}(π) is not identically zero. As explained below, this can be guaranteed by a global condition (which will be automatically satisfied) and by local conditions (essentially at primes in S_{K}). The local components of Θ_{ψ,χ}(π) at places outside S can be determined in terms of those of π and χ, by a procedure introduced by Rallis in [Rallis HDC]. We have

(5.2)  Θ_{ψ,χ}^{S}(π) = Π_{χ^{−1}}^{S}(π̌ × 1)

12Local L-packets in general arise at places that do not split in K. We retain the hypothesis (3.0) that if v is an unramified inert place then π_{v} is spherical and χ_{v} is unramified; therefore packets are singletons at such places as well. Here as previously, we don’t really know what to say about places v ramified in K, but the explicit construction of endoscopic representations described in this section yields well-defined representations of H(ℚ_{v}).

in the sense that the left-hand side of (5.2) satisfies (5.1) at unramified places. That is,

(5.3)  L^{S}(s, Θ_{ψ,χ}(π)) = L^{S}(s, π̌ ⊗ χ^{−1})ζ_{K}^{S}(s) = L^{S}(s, π ⊗ χ)ζ_{K}(s).

We may thus define Π_{χ^{−1}}(π̌ × 1) to be Θ_{ψ,χ}(π).

Not every Π_{χ^{−1}}(π×ξ) occurs as such a theta-lift; Θ_{ψ,χ}(π) = 0 unless π_{∞} belongs to the domain of the local theta correspondence for U(a, b) × U(n + 1, 0). For example, suppose n = 2 and V′ is positive-definite, so a = 2, b = 0, and π is of the form π^{0} ⊗ β where π_{∞}^{0} is the trivial representation and β is a character of U(1). Then there exists a classical holomorphic new form f of weight 2 such that

L(s, π^{0}) = L(s, f)L(s, f, ε_{K})

(i.e., π^{0} = π(f)_{K} where π(f) is the automorphic representation attached to f). More generally, if θ(?) is the holomorphic modular form on GL(2, ℚ) attached to the Hecke character ? (a binary theta function) then

(L(s, Θ_{ψ,χ}(π)) =)L(s, π ⊗ χ)ζ_{K}(s) = L(s, f, θ(BC(β) · χ))ζ_{K}(s)

where the first term on the right-hand side is the Rankin-Selberg product. Then Θ_{ψ,χ}(π) = 0 unless θ(BC(β)·χ) is a holomorphic modular form of weight at least 4. In that case, s = 1 is a critical value of the Rankin-Selberg product and Shimura’s theorem implies that

L(1, f, θ(BC(β) · χ))/(2πi)^{∗} < θ(BC(β) · χ), θ(BC(β) · χ) > ∈ \overline{ℚ}.

Moreover, the Petersson norm in the denominator can be expressed in terms of periods of elliptic curves with CM by K.

6. The Rallis inner product formula

Let φ ∈ π be as in §4 and let χ be a splitting character as above. We use the notation θ_{ψ,χ,Φ_{0}}(φ) to designate the theta lift of φ to an element of Θ_{ψ,χ}(π), an automorphic form on the definite unitary group U(V). As indicated, the notation depends on a Schwartz-Bruhat function Φ_{0}, which in turn depends on a choice of polarization of the symplectic vector space V = R_{K/ℚ}(V′ ⊗ V). The construction becomes more canonical when we consider a pair of forms φ ∈ π, φ′ ∈ π^{∨}; their theta lift

θ_{ψ,χ,Φ}(φ ⊗ φ′) ∈ Θ_{ψ,χ}(π) ⊗ Θ_{ψ,χ}(π)^{∨}

is defined in terms of a choice of maximal isotropic subspace X of the skew-hermitian space 2(V′ ⊗ V), and for X one can choose the diagonal subspace (V′ ⊗ V)^{d}, which we identify with V′ ⊗ V.

This same polarization defines a theta lift from functions on U(V) to functions on U(2V′); i.e., there is a seesaw diagram

(6.1)  \begin{matrix}U(2V′) & U(V) × U(V) \\ ↑ & ↑ \\ U(V′) × U(V′) & U(V) \end{matrix}

Seesaw duality yields the following expression for the Petersson inner product of theta lifts to U(V):

(6.2) Q_{V} (θ_{ψ,χ,Φ}(φ ⊗ φ′)) =^{def} ∫_{U(V)(ℚ)\U(V)(A)} θ_{ψ,χ,Φ}(φ ⊗ φ′)(h)dh

= ∫_{U(V′)×U(V′)(ℚ)\U(V′)×U(V′)(A)} θ_{ψ,χ,Φ}(1)(g_{1}, g_{2}) · φ̄(g_{1}) ⊗ φ̄′(g_{2})dg_{1}dg_{2}

Here φ̄_{y} and φ̄′_{y} are as in the discussion preceding Proposition 4.2. For appropriate choices of a Schwartz-Bruhat function Φ, the theta lift θ_{ψ,χ,Φ}(φ⊗φ′) can be factored as a product of forms on the two copies of U(V′), and the first integral can be written as a genuine inner product. The θ_{ψ,χ,Φ}(1) appearing in the second integral denotes the theta lift of the constant function 1 to U(2V′).

At this stage it is more natural to work with a fixed choice of Φ. Indeed, expression (6.2) becomes the Rallis inner product formula when the function θ_{ψ,χ,Φ}(1) can be identified with a Siegel Eisenstein series on U(2V′). This identification is provided by the extended Siegel-Weil formula, proved in the present case by Ichino [Ich], following the methods introduced by Kudla and Rallis in their work on Eisenstein series on symplectic groups. For appropriate choices of Φ we obtain precisely^{13} the value at s = \frac{1}{2} of the Eisenstein series attached to the section f = ⊗_{v}f_{v} defined in §3; this is the meaning of condition (3.3)(v). The precise formula is then

(6.3)  \begin{matrix}d_{n}^{S}(\frac{1}{2}, χ_{y}) & Q_{V} (θ_{ψ,χ_{y},Φ}(φ_{y} ⊗ φ′_{y})) \\ = d_{n}^{S}(\frac{1}{2}, χ_{y}) ∫ E_{f}((g_{1}, g_{2}); χ, \frac{1}{2})φ̄_{y}(g_{1}) ⊗ φ̄′_{y}(g_{2})dg_{1}dg_{2} \\ = Q_{V′}^{0}(φ_{y} ⊗ φ′_{y})L_{p}(1, y)Ω_{y}, \end{matrix}

where the integral is over U(V′) × U(V′)(ℚ)\U(V′) × U(V′)(A) and Ω_{y} is the period appearing in Theorem 4.4.. The appearance of L_{p}(1, y) suggests that the Rallis inner product formula has a variant in terms of p-adic L-functions. This is indeed the case, but the precise formulation requires some foundational work.

(6.4) p-adic families of endoscopic forms.

On G × G we work with holomorphic modular forms, which, as we have already intimated, have a natural \overline{ℚ}-rational structure compatible with the theory of canonical models. We can do the same on H × H, but since H(ℝ) is compact, the locally symmetric variety attached to H and an open compact subgroup K⊂H(A_{f}) is the finite set _{K}Sh(H) = H(ℚ)\H(A)/H(ℝ) × K. The theta lift Θ_{ψ,χ}(π) to H of an automorphic representation π of G of our fixed infinity type π_{∞} is an automorphic representation Π with Π_{∞} fixed and, with our choice of τ, necessarily of dimension 1. We may thus view forms in Π as sections of a local system L on the finite set KSh(H), where L depends only on τ and has a natural \overline{ℚ}-rational structure, and even an integral structure over the ring ℤ̄ of algebraic integers in \overline{ℚ}. (Of course L is constant on connected components of _{K}Sh(H), which are just points!) We let Sh(H) be the profinite set \underleftarrow{lim}_{K} _{K}Sh(H).^{14}

13Up to multiplication by \frac{1}{2} , which we incorporate into the constants.

It is an important fact that the theta lift θ_{ψ,χ,Φ}(φ ⊗ φ′) almost never preserves arithmeticity. If φ ⊗ φ′ is a holomorphic modular form on G × G which is \overline{ℚ}- rational and belongs to an irreducible automorphic representation π ⊗ π^{∨}, then one can define without much difficulty an almost certainly transcendental factor q_{π} ∈ ℂ^{×}, well-defined up to \overline{ℚ}^{×}, such that

(6.4.1)  q_{π}^{−1}θ_{ψ,χ,Φ}(φ ⊗ φ′) ∈ H^{0}(Sh(H), L)(\overline{ℚ})

Unfortunately, unless either (i) n = 2 or (ii) ab = 0 we know no sensible way to define a variant of q_{π} that is well-defined up to p-integers. More precisely, as long as π is unramified at p (which will not generally be the case!) there are natural choices of q_{π} so that q_{π}^{−1}θ_{ψ,χ,Φ}(φ ⊗ φ′) ∈ H^{0}(Sh(H), L)(ℤ̄_{(p)}), but no choice is known that does not compromise valuable information about congruences between holomorphic forms on G × G, except in cases (i) or (ii). We will say no more about this issue, which for practical purposes amounts to restricting our attention to cases (i) and (ii), even though much of what we say below is true more generally.

The Eisenstein measure dμ_{Eis} introduced in Theorem 3.9, depends on the padic family of sections f_{p} = f_{p}(ν, χ) described after Proposition 3.8, which in turn depends on a certain collection of Schwartz-Bruhat functions Φ_{ν} on V(ℚ_{p}) ≃ M(n, 2(n + 1))(ℚ_{p}) depending on characters ν of finite order. We have not checked enough details to remove the question mark in the following Proposition(?), but it is undoubtedly true. Note that the period Ω_{y} appears squared in the denominator.

Proposition(?) (6.4.2). Let (ϕ_{y}, χ_{y}) be a Hida family over 𝕐, as in §4. Let Φ_{ν} ∈ S(M(n, 2(n + 1))(ℚ_{p})) and Φ^{p} ∈ S(V(A_{f}^{p})) be the Schwartz-Bruhat functions used to define the Eisenstein measure dμ_{Eis}. Then there is a Hida family F^{θ}(y) = F_{φ_{y}⊗φ′_{y},χ_{y}}^{θ} of p-adic forms in H^{0}(Sh(H)×Sh(H), L⊗L^{∨})(ℂ_{p}), parametrized by 𝕐, such that

F^{θ}(y) = (d_{n}^{S}(1/2, χ_{y})/Ω_{y}^{2})θ_{ψ,χ_{y},Φ_{wt(y)}}(φ_{y} ⊗ φ′_{y})

for all y ∈ 𝕐^{alg}.

Once one admits the above Proposition(?) the following assertion is not difficult:

Proposition (6.4.3). There is a rigid meromorphic function g_{θ} on 𝕐 such that F (y) = g_{θ}(y)F^{θ}(y) is a primitive Hida family.

Much of our work at present is concerned with finding conditions under which g_{θ} can be taken to equal 1. The reason for this is explained in the following section. Meanwhile, we can now state a p-adic version of the Rallis inner product formula. Define

Q_{V′}^{alg}(ϕ_{y} ⊗ ϕ′_{y}) = Ω_{y}^{−1}Q_{V′}^{0}(ϕ_{y} ⊗ ϕ′_{y})

whenever y ∈ 𝕐^{alg}.

14This is one place where we need to take care to work with similitude groups rather than unitary groups, but there is no real difficulty and we ignore the issue in this exposition.

Proposition (6.4.4). The pairings Q_{V} and Q_{V′}^{alg} extend to natural pairings on p-adic modular forms on U(V) × U(V) and U(V′) × U(V′), respectively, such that

\begin{aligned}Q_{V} (F (y)) & = g_{θ}(y)Q_{V} (F^{θ}(y)) \\ & = g_{θ}(y)Q_{V′}^{alg}(φ_{y} ⊗ φ′_{y})L_{p}(1, y). \end{aligned}

Remark 6.5 We note in passing that there is no natural way to separate φ_{y} from φ′_{y} in the above discussion, mainly because the doubling method cannot be applied to pairs φ_{1} ⊗ φ_{2}, with φ_{1} ∈ π_{1}, φ_{2} ∈ π_{2}, where φ_{2} is congruent to a form in π_{1}^{∨} but is not actually in π_{1}^{∨}. This is the source of serious problems whose resolution brings into play the deep structure theory of arithmetic Hecke algebras.

7 Inner products and congruences

To construct congruences, we use a variant of an idea introduced by Hida. Fix a level subgroup K = ∏_{q} K_{q} with K_{p} determined by the level of the character χ, K_{q} hyperspecial maximal at all inert primes, and some other local conditions if necessary. The space M_{K}(L, ℂ) = H^{0}(_{K}Sh(H), L) of K-fixed vectors in the space of all automorphic forms on H of type Π_{∞} contains a natural O = O_{K,𝔭}-lattice L(L) (if Π_{∞} were trivial it would have a ℤ-lattice, namely the ℤ-valued functions). The Petersson inner product on this lattice, with respect to the measure on the finite set _{K}Sh(H) giving each point mass 1, is a unimodular hermitian form (more precisely, we work with the perfect pairing between L(L) and L(L^{−1})).

Now we write

(7.1)  H^{0}(_{K}Sh(H), L) = H^{0}(_{K}Sh(H), L)_{end} ⊕ H^{0}(_{K}Sh(H), L)_{st}

ignoring endoscopic forms of other types (this can be arranged by adding local conditions). There is correspondingly an inclusion

L(L) ⊃ L(L)_{end} ⊕ L(L)_{st}

where L(L)_{?} = H^{0}(_{K}Sh(G), L)_{?} ∩ L(L). This inclusion is an equality if and only if there are no pairs f_{end} ∈ L(L)_{end}, f_{st} ∈ L(L)_{st} of p-primitive forms such that p^{−1}[f_{end} − f_{st}] ∈ L(L), i.e. if and only if there are no congruences between endoscopic and stable forms of the form

f_{end} ≡ f_{st} (mod p).

Hida’s observation, made in the setting of classical modular forms, is that this is true if and only if the restriction to L(L)_{end} of the hermitian pairing on L(L) is again unimodular [Hi81].

We now allow the level subgroup K_{p} to vary, always assuming K_{p} ⊃ R_{u}P (ℤ_{p}). Applying Hida’s constructions, we obtain modules of P -nearly ordinary forms L_{P −ord}(L) ⊃ L_{P −ord}(L)_{end} ⊕ L_{P −ord}(L)_{st} of finite type over Λ_{P}, the ring of functions on 𝕏_{0}. The torsion Λ_{P} -module

C = L_{P −ord}(L)/(L_{P −ord}(L)_{st} ⊕ L_{P −ord}(L)_{end})

measures congruences between stable and endoscopic forms.

Hida’s constructions also give a Hida family of modular forms on H over some 𝕐 such that H_{𝕐} (a Hecke algebra; see (4.5)) acts faithfully on L_{P −ord}(L) and hence acts on C. (To make all this true we have to consider Hida families on H consisting simply of p-adic forms, not pairs of p-adic forms and characters.) Let

I = Ann_{H_{𝕐}}(C).

Let (ϕ_{y}, χ_{y}) over 𝕐_{0} be an irreducible Hida family of modular forms on G. Let R = H_{𝕐_{0}}. In general this is finite and integral over Λ_{P}, but for the sake of simplicity we will assume that R = Λ_{P}. Recall that we can view L_{p}(1, •) as an element of R (see (4.5)). For the appropriate choice of the prime-to-p levels K_{q}, if we admit Proposition 6.4.2 then the endoscopic Hida family F of Proposition 6.4.3 is associated to an irreducible component of 𝕐, which identifies R as a Λ_{P} - quotient of H_{𝕐}. In fact, this identifies 𝕐_{0} as a component of 𝕐. The considerations introduced in the first paragraph of this section apply to the unimodular pairing

(7.2)  L_{P −ord}(L)_{𝔓} ⊗ L_{P −ord}(L)_{𝔓} → Λ_{P,𝔓},

where 𝔓 ⊂ Λ_{P} is any height one prime. In particular, the discriminant of the restriction of the pairing (7.2) to L_{P −ord}(L)_{end,𝔓} ⊗ L_{P −ord}(L)_{end,𝔓} divides IR_{𝔓}. Concretely, we obtain the following theorem (admitting Proposition 6.4.2).

Theorem (7.3). IR ⊆ (g_{θ}(•)L_{p}(1, •)).

Assume we can eliminate the function g_{θ} from the above statement. Then the theorem asserts that the p-adic L-function divides the ideal measuring congruences between endoscopic and stable forms (really Hida families). In the next section we will briefly explain how such congruences give rise to non-trivial classes in Galois cohomology, as predicted by the appropriate generalization of Iwasawa’s main conjecture.

In order to eliminate g_{θ} it suffices to show that the normalized theta lifts F_{φ_{y}⊗φ′_{y},χ_{y}}^{θ} are already p-primitive (integral, but not divisible by any fractional power of p). The methods available for proving the arithmeticity of F_{φ_{y}⊗φ′_{y},χ_{y}}^{θ} – i.e., for determining the transcendental factors q_{π_{y}} introduced in (6.4) – can be applied to prove the desired primitivity, provided one knows that sufficiently many special values of auxiliary L-functions are not divisible by p. It appears that one can derive what one needs from results of Vatsal in the case n = 2.

(7.4) Hecke algebras on H and congruences. Let 𝔓 ⊂ R = Λ_{P} be any prime ideal containing IR. It follows that localization at P of the quotient of H_{𝕐} corresponding to all the families of stable forms (i.e., the quotient acting faithfully on L_{P −ord}(L)_{st}) is non-trivial. This can be interpreted as meaning that there is a component of 𝕐 corresponding to a family of stable forms that intersects the endoscopic component 𝕐_{0} “at the prime 𝔓.”

(7.5) Theorem (7.3) as a statement about forms on G. As written, Theorem 7.3 asserts that the p-adic L-function of a Hida family on G = U(V′) controls congruences between endoscopic and stable forms on H = U(V). The applications in the next section concern the Galois representations attached to the Hida family on G. Thus neither the input (the Hida family on G) nor the output (the construction of Galois cohomology classes) makes mention of the group H, and one would like to be able to eliminate H and the hermitian space V from the statement. In other words:

Question (7.5.1). Does the typical classical point y = (π_{y}, χ_{y}) in the Hida family admits a non-zero lifting to H = U(V), for some positive-definite hermitian space V of dimension n + 1?

As Waldspurger was the first to observe, and as Rallis explained in great generality, there are both global and local obstructions. The Rallis inner product formula accounts for both obstructions: if Q_{V} (θ_{ψ,χ,Φ}(φ ⊗ φ′)) ≠ 0 for some choice of data then the theta lift is non-trivial. It suffices to restrict our attention to Hida families such that π_{y} is tempered for all (classical) y, and for which no π_{y} is itself a theta lift from some U(W) with dim W < n. Under our running assumption that π_{y} admits a base change to GL(n)_{K}, the L-function L(s, π, χ) does not vanish at s = 1, hence there is no global obstruction. The local obstructions are given by the product of the local terms in the Rallis inner product formula. There is a genuine local obstruction at the archimedean prime, since we have to take V positive-definite, but the local component π_{y,∞} has been chosen to eliminate this obstruction. It is shown in [HKS] that there is a non-archimedean local obstruction at a prime p if and only p is inert or ramified in K and the local theta lift of π_{p} (determined by χ_{p}, as well as the additive character ψ_{p} and ℷ) to U(V)(ℚ_{p}) is trivial. It is also shown in [loc. cit.] that for at least one choice of local space V_{p} of dimension n + 1 up to isometry – there are two isometry classes – the theta lift of π_{p} to V_{p} is non-trivial. Thus Question (7.5.1) comes down to whether the set of local V_{p}’s to which π_{p} lifts can be realized as the local components of a given global positive-definite hermitian space over K. In principle there is a parity obstruction; however, at least when n is even, “most” π_{p} lift to either isometry class, which means that “most” π (with the given π_{∞}) lift to some global U(V). Finally, it is not difficult to see that, by varying the choice of χ (away from p) one can always arrange that the theta lift to a given global positive-definite V of π is non-trivial.

All this is of little consolation, however, if one’s interests are served by a specific pair (π, χ). At present we do not know the answer to Question (7.5.1), which has something to do with the nature of the values of “incoherent” Eisenstein series at the point s = \frac{1}{2}, which (in principle) are not explained by the Siegel-Weil formula.

8. Applications to Selmer groups

We keep to the notation of section 7. We assume that associated to the Hida family (ϕ_{y}, χ_{y}) over 𝕐_{0} is an analytic family of n-dimensional Galois representations σ : Gal(K̄/K) → GL_{n}(R) such that σ(y) = σ(π_{y} ⊗ χ_{y}), the latter being the ndimensional representation corresponding to BC(π) ⊗ χ as in section 2. At this point, we make a number of assumptions on p, S, and σ, the most important of which is that the mod p reduction of some (and therefore all) σ(π_{y} ⊗ χ_{y}) in our family is absolutely irreducible.

Let 𝔓 ⊂ R = Λ_{P} be any height one prime containing I. Let σ_{𝔓} be σ viewed as a representation into GL_{n}(R_{𝔓}) and let σ̄_{𝔓} be the reduction of σ_{𝔓} modulo 𝔓. This is a representation into GL_{n}(k(𝔓)), where k(𝔓) = R_{𝔓}/𝔓 is the residue field. Our assumptions imply that σ̄′ is irreducible.

From Theorem (7.3) it follows that there is some irreducible component 𝕐′ corresponding to a Hida family of stable forms that intersects the endoscopic family 𝕐_{0} at 𝔓 (see (7.4)) (recall that for these families we have dropped the extra character). Again, for simplicity we shall assume that R′ = H_{𝕐′} (a quotient of H_{𝕐}) is also isomorphic to Λ_{P}. Then 𝔓 determines a prime of R′ and we assume that there is an n + 1-dimensional representation σ′ : Gal(K̄/K) → GL_{n+1}(R′) such that its specializations at points in 𝕐′^{alg} are the n + 1-dimensional representations (conjecturally) associated to the representations BC(π′) as in section 2. The existence of σ′ is already more difficult to arrange than that of σ, since the methods of [HL] only apply to representations that are locally supercuspidal or Steinberg at some finite place split in K, and this will in general not be the case for the points in 𝕐′^{alg}, precisely because they are congruent to endoscopic representations. However, when n + 1 ≤ 3 the existence of the representations σ′(y) has been shown by Blasius and Rogawski [BR], and they are expected to exist in general.

Define σ′_{𝔓} and σ̄′_{𝔓} as we did σ_{𝔓} and σ̄_{𝔓}. The properties of these representations connecting Hecke operators and eigenvalues of Frobenius elements then implies that

(σ̄′_{𝔓})^{ss} = σ̄_{𝔓} ⊕ 1,

where the “ss” denotes semi-simplification. But we actually get more. The representations associated to stable forms are expected to have “large” image. When this is so, after possibly replacing σ′_{𝔓} with a conjugate over the field of fractions of R_{𝔓}, we have that

σ̄′_{𝔓} = \binom{σ̄_{𝔓} ∗}{1} .

with the “∗” in the upper-right corner non-trivial. This “∗” determines a non-trivial class in H^{1}(K, σ̄_{𝔓}^{−1}). With more work this class can be shown to have nice ramification properties at the primes above p, and its existence can be reinterpreted as saying that localization at 𝔓 of the Bloch-Kato/Greenberg Selmer group associated to an integral version of σ has length ≥ 1 as a R_{𝔓}-module.

An argument originally due to Wiles, as generalized by Urban, shows that by working with all stable components congruent to the endoscopic component 𝕐_{0} modulo 𝔓 it is possible to prove that the length of the Selmer group localized at 𝔓 is at least ord_{𝔓}(IR_{𝔓}). If we assume the function g_{θ} of Proposition 6.4.3 to be a scalar, then the congruence ideal I is divisible by the p-adic L-function L_{p}(1, •), and therefore the length of the Selmer group localized at any height one prime 𝔓 of Λ_{P} is at least ord_{𝔓}(L_{p}(1, •)). Provided we can justify all the assumptions we have made up to this point !

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