1. introduction l a g - brandeis...

THE RALLIS INNER PRODUCT

FORMULA AND p-ADIC L-FUNCTIONS

To Steve Rallis, the oracle

Michael Harris1

UFR de Mathematique, Universite Paris 7, 2 Pl. Jussieu 75251 Paris cedex 05, FRANCE

Jian-Shu Li2

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

Christopher M. Skinner3

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 casesthat congruences satisfied by normalized special values of L-functions of geometricautomorphic forms on unitary groups give rise to non-trivial elements of certainGalois cohomology groups. More precisely, letting p be an odd prime number,the Galois cohomology groups, or generalized Selmer groups, can be viewed asmodules over appropriate Iwasawa algebras, and, possibly up to p-power torsion, theassociated p-adic L-functions divide the characteristic power series, in the sense ofIwasawa theory, of the corresponding Selmer groups. Divisibility in both directionsis a version of the main conjecture of Iwasawa theory.

Divisibility results of this kind were first obtained by Ribet for class groups ofcyclotomic fields. Ribet observed that divisibility of normalized values of DirichletL-functions by p gave rise to congruences modulo p between holomorphic Eisen-stein series and cusp forms on GL(2). Here two holomorphic modular forms fand g are considered congruent if their Fourier expansions at the cusp at infinityhave algebraic integer coefficients an(f) ≡ an(g) (mod p) for all n ≥ 0. Ribet’stechnique was vastly generalized by Mazur and Wiles, who obtained the first proofof Iwasawa’s main conjecture for cyclotomic fields, and Wiles extended these ar-guments to abelian extensions of totally real fields. More recently, this techniquehas been applied successfully by Urban to congruences of Siegel modular forms,

1Institut de Mathematiques de Jussieu, U.M.R. 7586 du CNRS. Membre, Institut Universitairede France.

2Partially supported by NNSFC Grant No. 19928103, RGC-CERG grants HKUST6126/00P,

HKUST6115/02P, and the Cheung Kong Scholars Programme3Partially supported by a grant from the National Science Foundation and fellowships from

the Sloane Foundation and the David and Lucile Packard Foundation.

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1

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-functionsof degree greater than 1 – specifically, the symmetric square p-adic L-functions forelliptic modular forms. Admitting some expected consequences of the stable traceformula for certain unitary groups, Skinner and Urban have used similar techniquesto prove the main conjecture for the standard p-adic L-function associated to anelliptic modular form for a large class of such forms. The principle underlying theseconstructions is the following: the normalized value of the L-function in questionappears as the constant term of a certain holomorphic Eisenstein series. In veryfavorable cases, vanishing of the constant term modulo pr implies congruence of theEisenstein series with a cusp form modulo pr. This in turn implies isomorphism ofthe semisimplified reductions modulo pr of the Galois representations associated tothe cusp form and the Eisenstein series, and following Ribet this isomorphism givesrise to the non-trivial Galois cohomology class, as will be explained in §8. Veryfavorable cases are unfortunately very rare, however, and the greatest challenge inthis approach, as well as in the one described below, is to make the argument workin relatively unfavorable cases.

Our approach is based on a different principle. We work with the standardL-function L(s, π) of an automorphic representation π of the unitary group of ahermitian vector space V ′ of dimension n over a CM field K. These L-functionscan and will be viewed as the L-functions of the corresponding automorphic repre-sentation BC(π) of GL(n,K) obtained by stable base change. Suppose there is ann+ 1-dimensional positive-definite hermitian vector space V over K such that thetheta lift Θ(π) of π to U(V ) is non-trivial. Then the Rallis inner product formulaidentifies 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 thesquare 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 explicitcorrection factors. By an endoscopic automorphic representation of U(V ) (moreprecisely, an endoscopic representation of type (n, 1)), we will understand an au-tomorphic representation of the form Θ(π); stable automorphic representations ofU(V ) will be automorphic representations not of this form. A principle originat-ing 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 formsin other representations π′ of U(V ′) (resp. U(V )). Thus p-divisibility of the ratio<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 ). Apply-ing Ribet’s construction to these congruences one then obtains Galois cohomologyclasses as before.

Much work is required in order to sort out what we mean by a very favorablecase, not to mention making the technique work more generally. We actually needto study congruences in the Iwasawa algebra, which can be viewed as a powerseries ring over a p-adic integer ring in up to n-variables. This means we need toconstruct the p-adic L-functions for continuous families – Hida families – of p-adicautomorphic forms on U(V ′) and U(V ). We also need to obtain a version of theRallis inner product formula adapted to these Hida families. These results arenearly complete, at least for the one- and two-dimensional families that suffice for

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

our purposes, though some questions remain concerning the behavior of bad localzeta integrals. The most serious problem has to do with the p-adic properties of thetheta 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 itselfdivisible by p. This problem can be resolved when n = 2. It is likely that we canobtain 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 agood integral structure on the cohomology of the Shimura variety attached to theunitary group U(V ′). This is well understood when V ′ is positive-definite, or whenn = 2, but otherwise is completely mysterious. We will restrict our attention to thecase of positive-definite V ′, with some remarks about the case n = 2. When K is animaginary quadratic field, which we assume henceforward, the case n = 2 includesthe 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 onunitary groups was first proposed by Richard Taylor in a message to one of theauthors in 1993. The connection to the theta correspondence, and specifically tothe Rallis inner product formula, was first observed in 2000 or 2001. Throughout theproject 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 adviceand encouragement this work would have been simply impossible.

It gives us great pleasure to dedicate this article to Steve Rallis. Steve’s gen-erosity 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 Howeduality correspondence and incorporates his results obtained with Piatetski-Shapiroon the standard L-functions of classical groups, on the one hand, and the resultsof his collaboration with Kudla on the extended Siegel-Weil formula, on the otherhand, in its version for unitary groups due to Ichino. The applications of theseresults to arithmetic are just beginning to be explored. Yet all this work is at leastten years old. Since then Steve has, if anything, been even busier. The possibilityis 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 ev-erything goes over with little change to general totally real fields. Thus K is animaginary quadratic field, given with a fixed embedding as a subfield of C, V ′ andV are hermitian spaces of dimension n and n + 1, respectively, over K. We takeV to be positive-definite. In the applications V ′ will usually be positive-definite aswell, but in the present report we assume the signature of the hermitian form onV ′ to be (a, b), with a+ b = n. We let H = U(V ), G = U(V ′), viewed as algebraicgroups over Q, and let K∞ ' U(a) × U(b) be a maximal compact subgroup ofG(R). Let εK be the quadratic character attached to the extension K/Q and c thenon-trivial automorphism of K (complex conjugation). We fix a prime p, assumedto split in K.

For the moment we keep V and H in reserve and restrict our attention to au-tomorphic 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 aone-dimensional representation (τ,W ) of G(R) = U(a, b):

(2.1) Hab(Lie(G)(R), K∞;π∞ ⊗W ) 6= 0

(since π∞ is in the discrete series, all the cohomology is concentrated in the middledimension ab). There are more general coefficient systems but we will ignore these.

Via twisted (unitary) stable base change, π can be identified – unconditionally forn ≤ 3 [Rog], conditionally4 for general n [HL] – with an automorphic representationBC(π) = πK of GL(n,K), unramified at p, which is cohomological at the uniquearchimedean place and which satisfies

(polarization) BC(π)∨∼−→ BC(π)c.

Arithmetically, this corresponds – again, under some conditions for n > 3 – to acontinuous representation σ : Gal(Q/K) → GL(A) on an n-dimensional Qp-vectorspace A which is unramified outside a finite set and is crystalline at p. This canbe viewed either as the Galois representation attached to BC(π) or to π; there isno difference. But the former has an easier interpretation: if BC(π) = ⊗vBC(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 |Γv∼−→ L(BC(π)v)

up to semisimplification, where v is any finite prime of K not dividing p, Γv is adecomposition group at v and L is the local Langlands correspondence, appropri-ately normalized, taking irreducible representations of GL(n,Kv) to n-dimensionalrepresentations of Γv.5

The Galois representation σ is realized, up to an explicit abelian twist, on thecohomology of some Shimura variety, not necessarily attached to the group G.6

Suppose for the moment that the coefficient representation τ is trivial. Then theweight of the p-adic representation σ, determined by Frobenius eigenvalues at un-ramified primes, is n − 1, the Hodge-Tate weights are 0, 1, . . . , n − 1, and therepresentation is regular in the following sense: each Hodge-Tate weight has mul-tiplicity one (over the coefficient field). This regularity hypothesis, which remainsvalid for general τ (the list of Hodge-Tate weights depends on τ) is the principalrestriction imposed by the use of Shimura varieties.

We continue to assume τ trivial. The polarization condition has one degree offreedom, namely twisting by a Hecke character χ of K which is anticyclotomic in

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 theoremapplies 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.


the sense that χ factors through the map a 7→ a/ac. To χ we can associate a weightk (basically χ(z) = (z/zc)−k) and a p-adic incarnation σp(χ), so that the p-adicrepresentation σ ⊗ σp(χ) is pure of weight w = n − 1 + k. The L-function witharithmetic normalization:

(2.3)

L(s− n− 1 + k

2, BC(π), χ) (GL(n) version)

= L(s− n− 1 + k

2, π, χ) (U(n) version)

= L(s, σ ⊗ σp(χ)) (Galois version )

then has critical special values, in the sense of Deligne, for most k, the nature of theperiods depending on k. In the first line of (2.3) we are working with the principalL-function of GL(n), whereas in the second line we are working with the LanglandsL-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 therepresentation generated by an elliptic modular newform f of weight 2. Then ifk = 0 (so w = 1) L(s, σ ⊗ σp(χ)) has a critical value at the center s = 1 of Birch-Swinnerton-Dyer type whose associated Deligne period is roughly the Peterssonnorm < f, f >. If k = 3, so w = 4, L(s, σ ⊗ σp(χ)) has a critical value at thenear-central point s = 3 whose Deligne period depends only on χ – i.e., on CMperiods – 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 withk = s = n + 1; in particular, the critical value L(n + 1, σ ⊗ σp(χ)) is an algebraicmultiple of a fixed CM period independent of π.

Fixing k eliminates the archimedean degree of freedom but χ can still vary ina one-dimensional p-adic (anticyclotomic) family. Henceforward we incorporate kinto the representation τ – i.e., we drop the assumption that τ be trivial. Wetake L-functions in the unitary normalization, so that Re(s) = 1

2 is the axis ofsymmetry, and s = 1 is the possible pole for L-functions of tempered automorphicrepresentations of GL(n). We write L(s, π, χ) = L(s,BC(π), χ) as above. In whatfollows, we let χ = ⊗vχv vary among Hecke characters satisfying:

(2.4) χ |A×Q = εn+1K ; |χ∞| = 1.

If χ0 and χ are two characters satisfying (2.4), then χ · χ−10 is anticyclotomic in

the sense defined above, so that a family of χ satisfying (2.4) can be viewed as theproduct of a fixed base point χ0 by the family of all anticyclotomic Hecke characters.We write χpf = ⊗v 6=p,∞χv, and view χ as a family of characters of finite order with

values in Q× ⊂ C×p , where Cp is the p-adic completion of Qp. Consider the familyXK of p-adically continuous characters satisfying (2.4):

χ : A×K → C×p

with fixed χpf . This can be viewed as the set of Cp-valued points of a rigid-analyticspace over Qp. 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. Wecan 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, isdefined by means of the doubling method of Piatetski-Shapiro and Rallis, studiedin some detail, though insufficient for our purposes, by Shimura in two recentbooks ([GPR], [Sh1], [Sh2]; see also [Gar]). Let −V ′ be the space V ′ with itshermitian form multiplied by −1, let 2V ′ = V ′ ⊕ (−V ′), and define two totallyisotropic subspaces of 2V ′ by V ′,d = (v, v) | v ∈ V ′, V ′d = (v,−v) | v ∈ V ′. LetP ⊂ U(2V ′) be the stablizer of V ′,d; then the stabilizer M ⊂ P of V ′d is a Levicomponent, canonically isomorphic to GL(V ′,d) ' GL(V ′). Let m : GL(V ′) → Mdenote the canonical isomorphism. We write P = M · UP ; the exponential mapidentifies UP with the group Hern of n × n-hermitian matrices over K, viewed asan 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 ofU(2V ′)(A):

I(χ, s) = Ind(χ| · |sK)∼−→ ⊗v Iv(χv| · |sv),

the induction being normalized; the local factors Iv, as v runs over places of Q, arelikewise defined by normalized induction. At archimedean places we assume oursections to be K∞,2-finite. For a section f(h;χ, s) ∈ I(χ, s) we form the Eisensteinseries

Ef (h;χ, s) =∑

γ∈P (Q)\U(2V ′)(Q)

f(γh;χ, s)

This series is absolutely convergent for Re(s) > n−12 , and it can be continued to a

meromorphic function on the entire plane.Let (π,Hπ) be a cuspidal automorphic representation of G, (π∨, Hπ∨) its contra-

gredient, which we assume given with compatible isomorphisms of G(A)-modules

π∼−→ ⊗v πv, 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 (•, •)πvdenote the canonical bilinear pairing πv ⊗ π∨v → C.

Let f(h;χ, s) be a section, as above, ϕ ∈ Hπ, ϕ′ ∈ Hπ, 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)

Ef ((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 [Hi02]. 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 Sf ofQ such that, for any non-archimedeanv /∈ Sf , the representations πv, the characters χv, and the fields Kw, for wdividing v, are all unramified;

(b) The section f admits a factorization f = ⊗vfv, and f∞ is an eigenvectorfor a one-dimensional representation of K∞,2;

(c) The functions ϕ, ϕ′ admit factorizations ϕ = ϕSf ⊗v/∈Sf ϕv, ϕ′ = ϕSf ⊗v/∈Sfϕ′v

(d) For v /∈ Sf non-archimedean, the local vectors fv, ϕv, and ϕ′v, are the nor-malized spherical vectors in their respective representations, with (ϕv, ϕ

′v)πv =

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 = ∞ ∪ Sf . Define

dSn(s, χ) =n−1∏

r=0

LS(2s+ n− r, εn−1+rK ) =

∏

v/∈Sdn,v(s, χ),

Q0V ′(ϕ⊗ ϕ′) =

∫

G(Q)\G(A)

ϕ(g)ϕ′(g)dg;

(2.7) ZS(s, ϕ, ϕ′, f, χ) =∏

v∈S

∫

G(Qv)

fv((gv, 1);χ, s)(πv(gv)ϕ, ϕ′)dgv;

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

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

dSn(s, χ)Z(s, ϕ, ϕ′, f, χ) = Q0V ′(ϕ⊗ ϕ′)ZS(s, ϕ, ϕ′, f, χ)LS(s+

1

2, π, χ).

Here and throughout the superscript S means that we omit the Euler factors at theplaces 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 Ef a function Ef (z, uf , s),meromorphic in s, on X(n, n) × U(2V ′)(Af ) × C, with values in C (because f∞belongs to a one-dimensional K∞,2-type), which satisfies a familiar automorphyrelation with respect to the left action of U(2V ′)(Q). In what follows we call Ef

the classical Eisenstein series associated to Ef .


When V ′ is positive definite the zeta integral (2.5) is just a finite sum of valuesof the classical Eisenstein series Ef . When n = 1, the basic identity reduces tothe expression of the Hecke L-function on the right hand side as a finite sum ofspecial values of the normalized elliptic modular Eisenstein series on the left. Thisexpression is familiar to number theorists for its role in the proof of Damarell’s the-orem and its generalizations by Shimura, the starting point for Katz’ constructionof 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 fv, ϕv, andϕ′v at primes v ∈ Sf , the basic identity yields the following result:

Theorem (2.9). The expression

π−n(n+3)

2 Q0V ′(ϕ⊗ ϕ′)ZS(

1

2, ϕ, ϕ′, f, χ)LS(1, π, χ)

is an algebraic number.

In order for this to hold, the zeta integral has to be identified with a natural cupproduct pairing in coherent cohomology, the Eisenstein series has to be Q-rationalin the sense of having algebraic Fourier coefficients, and the functions φ, φ′ haveto be identified with Q-rational modular forms. These conditions are discussed insections 3 and 4 below, (cf. condition (3.3)(ii), for example). The power of π in thedenominator is an algebraic multiple of the factor dSn( 1

2, χ). We prefer not to be

precise about choices of measures so the power of π is somewhat spurious. Laterwe will implicitly assume that measures have been chosen so that the archimedeanfactor Z∞( 1

2 , ϕ, ϕ′, f, χ) is an algebraic number (even a p-unit); any discrepancy is

of course incorporated into the definition of Q0V ′ . The non-archimedean bad local

factors in ZS( 12 , ϕ, ϕ

′, f, χ) can easily be arranged to be algebraic.

3. p-adic Eisenstein measures.

Henceforward we assume πp is unramified. We also assume Sf = SK∐Sπ,χ∪p,

where SK is the set of primes ramified in K and Sπ,χ is the set of primes at whicheither π or χ is ramified. We make the hypothesis

Hypothesis (3.0). Every prime in Sπ,χ splits in K.

In the cases we consider the classical Eisenstein series Ef (z, uf , s) is a holomor-phic modular form at the point s = 1

2 , corresponding to the value L(1, π, χ) in thebasic identity. When ϕ and ϕ′ correspond to (anti)-holomorphic modular forms onthe Shimura variety associated to G, the basic identity, combined with rationalityproperties of the Fourier coefficients of holomorphic Eisenstein series, thus providesan expression of the special values L(1, π, χ), as χ varies with χ∞ fixed, as algebraicmultiples of a fixed period factor:

(3.1) L(1, π, χ) = Lalg(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 sen-tences, the first of which is highly misleading:

(i) As χ varies over a p-adic family X, for instance the XK defined above, oneexpects the values Lalg(1, π, χ) to vary p-adically analytically, and thus toextend to an analytic function on X with values in Cp.

(ii) The properties of this analytic function depend strongly on the p-adic ab-solute values of the Satake parameters of the unramified representation πp.

When one actually tries to construct p-adic L-functions, one immediately realizesthat the values Lalg(1, π, χ) need to be modified in order for something like (i) to betrue. The modification involves multiplication by p-adic and archimedean correctionfactors, whose conjectural form is proposed by Coates in [Co]. We have chosen tokeep the real components of π and χ fixed, so the archimedean correction factorscan be ignored. In the ultimate theory π∞ and χ∞ will vary and the archimedeancorrection terms – mainly products of factorials – will have to be addressed.

On the other hand, we allow χp to be arbitrarily ramified, so p-adic correctionfactors are essential. This amounts to including the prime p in the set Sπ,χ of badprimes, and making specific choices of local data that allow explicit calculation ofthe zeta integral

(3.2) Zp(s, ϕp, ϕ′p, fp, χp) =

∫

G(Qp)

fp((gp, 1);χ, s)(πp(gp)ϕp, ϕ′p)dgp.

at the point s = 1. The choices of vectors ϕp, ϕ′p are more or less imposed by

Hida theory. Let P⊂GL(n) be a standard parabolic subgroup, of parabolic rank`, say. In the most general setting, what Hida theory provides is a p-adic manifold

XP of dimension ` and a countable Zariski dense subset XalgP ⊂ XP (Cp) of points

corresponding to triples (πx, ϕx, χx), with πx an automorphic representation of Gof fixed infinity type and fixed central character, ϕx ∈ πx a factorizable vector, andχx a Hecke character of K satisfying (2.4). The pairs (φx, χx) extend to an analyticfamily (ϕ, χ) of some sort of p-adic modular forms on G(A). The problem is to findfp(x) = fp(φx, χx) such that the corresponding Eisenstein series also extend to ap-adic analytic map from XP to p-adic modular forms on U(2V ′).

One of the rules of the game is that the fv, for v 6= p, are constant as functionson XP . Specifically,

(i) When v 6= S hypothesis (2.6)(d) remains true;(ii) When v ∈ Sπ,χ is prime to p, fv is chosen to make all zeta integrals constant

and algebraic;

(3.3)

(iii) The section f∞ is chosen to make the classical Eisenstein series Ef (z, uf , s)holomorphic.8

8More generally, when π∞ is allowed to vary, one will want Ef (z, uf , s) to have holomorphic

restriction to the Shimura variety associated to G×G. We actually work with unitary similitudegroups in order to define Shimura varieties, but in fact the theory of the standard L-functions

only depends on the unitary groups.


(iv) When v ∈ SK, we fix fv.(v) Finally, we will assume below that f = ⊗fv is a Siegel-Weil section for some

(n+ 1)-dimensional hermitian space V .

Only fp 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. ForU(2V ′) the definition is relatively simple. We restrict our attention to Eisensteinseries. The classical Eisenstein series Ef (z, uf ) = Ef (z, uf ,

12 ) – we only work

with Eisenstein series holomorphic at s = 12, corresponding to the value of the

L-function at s = 1 – is completely determined by its Fourier expansion. Let

E∗f = π−n(n+3)

2 dSn( 12 , χ)Ef . Then the Fourier expansion of E∗f is

(3.4) E∗f (z, uf ) =∑

β∈Hern(Q)

cf,β(uf )qβ(z), z ∈ X(n, n) , uf ∈M(AS)×∏

v∈SfKv

Here Kv ∈ GL(V ′v) is a special maximal compact subgroup (by strong approxi-mation one does not need all values of the argument uf ∈ U(2V ′)(Af ) to defineE∗f , the values in M(AS) × ∏v∈Sf Kv suffice). The symbol qβ = qβ(z) denotes

the function z 7→ e2πiαTr(β·z), where α ∈ Q×, assumed to be a p-unit, is a freeparameter corresponding to the choice of an additive character. Holomorphy andthe Koecher principle imply that E∗f,β = 0 unless β is positive semidefinite. Ourchoices of fp provide a stronger vanishing condition:

(3.5) cf,β 6= 0⇒ det β 6= 0;

as well as a factorization condition9

(3.6) cf,β(uf ) = c∞,β ·∏

v 6=∞cfv,β,v(uv).

Here uv is the v-component of uf 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 ⊂ Hn(Q) denote the lattice of hermitian matriceswith entries in OK, Λ∗ the dual lattice

Λ∗ = B ∈ Hern(Q) | tr(BΛ)⊂OK.

For any v let Tv denote the characteristic function of Λ∗v.

Proposition (3.7). (a) For A ∈ GL(V ′v), v /∈ S.

cfv,β,v(m(A)) = Tv(αtAβA), v /∈ S.

(b) Let c0f,β = c∞(β) · ∏v∈Sπ,χ,v 6=p cfv,β,v. Under hypotheses (3.3) one can

find lattices Lv ∈ Hern(Qv) and a constant γ0 ∈ Q such that, for all m(A) ∈9Compare [Sh1, Proposition 19.2]. Shimura’s hypothesis c 6= g, which has to be read to-

gether 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.


∏v∈Sπ,χ∪∞,v 6=pKv, c0f,β(m(A)) = γ0 · det(β) · χ det(A) for β ∈ ∏v∈Sπ,χ L

∗v and

equals zero otherwise.

Here L∗v is the dual lattice.We only really care about the p-adic valuation of cf,β(uf ), and Proposition (3.7)

tells us that the contributions of primes outside p ∪ SK to this valuation arebounded. Moreover, the constant γ0 is a product of explicit local constants. Thearchimedean contribution is a p-unit if p > n + 1 and if det β is a p-unit, whichwill be true for our choice of fp. The other local contributions are indices of opencompact subgroups in G(Qv), whose p-adic valuations can be determined easily. Atthe time of writing we do not have good choices of fv for v ∈ SK. However, we canagain guarantee at least that the corresponding factor cKf,β(uf ) =

∏v∈SK cfv,β,v(uf )

has p-adic absolute value bounded above and below, and we will ignore these factorsfor the time being.

It remains to consider the factor cfp,β,p, which depends on explicit choices oflocal factors fp. Consider a partition n = n1 + · · ·+ n`, corresponding to a stan-dard parabolic subgroup P ⊂ GL(n). Let Mn(Qp) denote the space of n × n-matrices with values in Qp, and let KP ⊂ GL(n,Zp) ⊂ Mn(Qp) be the standardparahoric subgroup corresponding to the parabolic P . Let ν = (ν1, . . . , ν`) be an

`-tuple of characters of Z×p of finite order, with values in Q× ⊂ C×. The groupKP consists of matrices Z = (Zij)1≤i,j≤`, written according to the chosen parti-tion, with Zii ∈ GL(ni,Zp) for all i, with above-diagonal entries p-integral, and

below-diagonal entries divisible by p. We define a function φν : KP → Q× byφν(Z) =

∏1≤i≤` νi(detZii). We extend φν by zero to a function, still denoted φν ,

on Mn(Qp).Let w1, w2 denote the two primes of K dividing p. The local component χp of the

Hecke character maps K×p = K×w1×K×w2

∼−→ Q×p ×Q×p to Q×. By restriction to thefirst factor (more precisely, by fixing w1 rather than w2) χp becomes a character ofZ×p of finite order. The group

M(Qp) = GL(V ′p)∼−→ GL(V ′w1

)×GL(V ′w2)∼−→ GL(n,Qp)×GL(n,Qp),

and elements of M(Zp) will be denoted m(A,B) for A,B ∈ GL(n,Zp). There is achoice of basis implicit in the constructions below.

Proposition (3.8). For every ` + 1-tuple (ν1, . . . , ν`, χp) of characters of Z×p of

finite order, there exists a section fp(ν, χp; s) ∈ Ip(χp, s) such that, setting s = 12 ,

cfp(ν,χp),β,p(m(A,B)) = χp(det(BA−1))φν(tAβB−1).

for any A,B ∈ GL(n,Zp), provided β is p-integral with determinant a p-unit, andvanishes otherwise.

The construction of fp was inspired by Katz’ construction [Katz] when n = 1.It involves tensoring φν with its Fourier transform and viewing the result as aSchwartz-Bruhat function Φν for M(n, 2(n+1)), hence a datum for the local Siegel-Weil formula for the theta lift from U(n+1)(Qp) = GL(n+1,Qp) to U(2V ′)(Qp) =GL(2n,Qp). The details are quite complicated, but we would like to point out that,


although the partial Fourier transform does not preserve p-integrality, the result-ing local coefficient in (3.8) is p-integral. We write f(ν, χ) = ⊗v 6=pfv ⊗ fp(ν, χp),E∗fp,χ,ν = E∗f(ν,χ).

One can define a p-adic modular form on U(2V ′) to be a formal expressionof the form (3.4), where now the qβ are symbols satisfying the obvious relations

qβqβ′

= qβ+β′ , and the coefficients cf,β(uf ) are Cp-valued functions on M(Af) ∩(M(AS)×M(Zp)×

∏v∈Sf ,v 6=pKv) with p-adic absolute values uniformly bounded

in β as well as uf . In order to qualify as a p-adic modular form, the formal series hasto arise as the p-adic limit of Fourier expansions of genuine holomorphic modularforms 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 itis this interpretation that allows us to restrict p-adic modular forms to the Shimuravarieties attached to G×G, and hence to give a p-adic interpretation of the doublingmethod.10

We stress that, in the setting of Shimura varieties of higher dimension, the foun-dations 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 general-izing 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 L-functions will appear in a forthcoming article of Hida.

Let M(2V ′) denote the Cp-Banach space of p-adic modular forms on U(2V ′).Let Z` = (Z×p )`+1 and let XP denote the set of C×p -valued characters on Z`. Theset XP has the structure of the set of Cp-valued points of a rigid-analytic variety

over Qp whose subset XalgP of characters of finite order is Zariski-dense. A measuredµ on Z` with values in M(2V ′) is a finitely additive continuous map from theBanach space of continuous Cp-valued functions F on Z` to M(2V ′):

C(Z`,Cp) 3 φ 7→∫

Z`φdµ ∈ M(2V ′).

Such a measure is determined uniquely by its restriction to XP⊂C(Z`,Cp), and

even by its restriction to the Zariski-dense subset XalgP .

Theorem (3.9). Fix fp = ⊗v 6=pfv as above. There is a (unique) p-adic measure

dµEis(fp) on Z` with values in M(2V ′) such that, for any (ν, χ) ∈ XalgP ,

∫

Z`(ν, χ)dµEis(f

p) = E∗fp,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 productof local coefficients; Proposition 3.7 shows that for fixed f p, the product of localcoefficients prime to p is bounded uniformly in β and uf ; and Proposition 3.8

asserts that the local coefficient at p varies p-adically continuously on XalgP . Let

10Note that p-adic integrality of the Fourier coefficients is not in general preserved by the actionof the full group U(2V ′)(Qp), 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.


F = (φ, ξ) : Z`+1 → Cp be a continuous function where φ depends on the first `variables and ξ on the last (so F (z1, ..., z`, x) = φ(z1, ..., z`)ξ(x)). Define a functionφF : IP → Cp by composing φ with the natural map

Ip → Z`p; Z 7→ (detZ11, . . . , detZ`,`).

Let T(Lv)(β,A) : Hern(Q)×GL(V ′Apf) → Cp be the function that takes the value 1

if αtAvβAv ∈ L∗v for all v 6∈ S, β ∈ L∗v for all p 6= v ∈ Sπ,χ, and β is integral at pand det(β) ∈ Z×p , and otherwise takes the value 0. Then

∫

Z`F dµEis(f

p) =∑

β

cβ(u)qβ

where cβ(u) is the function of u ∈ GL(V ′Af) = M(Af), up = (Ap, Bp) ∈ M(Zp)

and uv ∈ Kv for all p 6= v ∈ Sf , given by(3.10)cβ(u) = γ0 det(β)ξ(detupSπ,χ)χp(det(BpA

−1p ))φF (tApβB

−1p )× T(Lv)(β, u)cβ,K(u),

where upSπ,χ =∏p6=v∈Sπ,χ uv and cβ,K(u) =

∏v∈SK cβ,v(uv).

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

The rigid-analytic variety XP introduced above breaks up as the disjoint union of(p− 1)`+1 connected components, indexed by the distinct characters of the torsionsubgroup ((Z/pZ)×)`+1 ⊂ Z`. We fix such a character ν0 = (ν0

1 , . . . , ν0` , χ

0) andthe associated connected component X0

P ⊂ XP . For our purposes, a Hida family oftype P on U(V ′) will correspond to

(1) A rigid-analytic variety Y and a finite flat morphism wt : Y → X0P ;

(2) An analytic family (ϕy, χy) over Y of pairs where ϕy is a p-adic modular formon U(V ′) and χy : Z×p → C×p is a continuous character whose restriction to

the torsion subgroup (Z/pZ)× is χ0;

(3) When wt(y) = (ν, χ) ∈ XalgP ∩X0P , χy = χ and ϕy is a holomorphic automor-

phic form on U(V ′), belonging to an irreducible automorphic representationπy, of archimedean type π∞ (as in §2) and of type ν at p (see below);

(4) The family ϕy is P -nearly ordinary in Hida’s sense (see below).

Of coures, if (ϕy, χy) is any Hida family then one obtains another by multiplyingby any holomorphic rigid-analytic function f on Y: (f(y)ϕy, χy). We call a Hidafamily primitive if it can not be obtained from another by multiplication by anon-constant f .

We will not define p-adic modular forms on a unitary group of general signatureexcept to say that (i) they are uniform limits of classical holomorphic modularforms of variable level and/or weight which are Q-rational in a sense compatiblewith 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) ∈ XalgP as classicalpoints and the associated ϕy as classical modular forms. The character ν in (3) isassumed to restrict to ν0 = (ν0

1 , . . . , ν0` ) 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) withπp in the principal series, and if it is an eigenvector for P (Zp) ⊂ IP ⊂ U(V ′)(Qp) =GL(n,Qp) with eigenvalue Z 7→ ν(Z) = ν1(det(Z11)) · · ·ν`(det(Z`,`)). It is alsoinvariant under a certain open compact subgroup Kν ⊂ IP containing the kernelof ν in P (Zp) – in particular, Kν contains the Zp-points RuP (Zp) of the unipotentradical of P . In many cases ν extends to a character of Kν and (Kν , ν) is asemisimple type in the sense of Bushnell-Kutzko, but this is (unfortunately!) onlythe case when the conductors of the νi are monotonic (increasing or decreasing; so` ≤ 2 is OK).

Finally, the P -ordinarity condition is a condition on the inducing character of

the principal series πp. Suppose α = α1 ⊗ · · · ⊗ αn : B → Q× is a character ofthe standard Borel subgroup of GL(n,Qp), each αi a character of Q×p , and πp isthe principal series representation unitarily induced from α. In practice πp will beessentially tempered, so the principal series is necessarily irreducible. Condition (3)implies that the αi can be ordered in such a way that their restrictions αi to Z×psatisfy

αj = νj , n1 + · · ·+ ni−1 < j ≤ n1 + · · ·+ ni.

Fix an embedding of Q in Cp, i.e. a p-adic valuation |·|p on Q, and let µj = |αj(p)|p.Condition (4) above is that

(4.1)

n1+n2+···+ni∑

j=1

µj =

n1+n2+···+ni∑

j=1

j − 1 + k , i = 1, . . . , `

where k is a fixed constant (this is the same k we fixed to be n+1 in §2). A conditionof this sort is the starting point of Hida’s deformation theory and corresponds toa condition on the p-adic Galois representation associated to π, but we will haveno use for its specific form here. We note that (4.1) together with condition (3)imposes a partial order on the characters αj : one can permute the αj ’s within theni-th block (n1 + · · ·+ni−1 < j ≤ n1 + · · ·+ni) for any i, but no permutations areallowed between blocks.

We henceforward assume that the partition n = n1 + · · ·+ n` associated to P isa refinement of the partition n = a+ b associated to the signature of V ′; i.e. thata = n1 + · · ·+nm, b = nm+1 + · · ·+n` for some index m. Let Yalg = (wt)−1(X0,alg),so that every y ∈ Yalg defines an irreducible automorphic representation πy ofinfinity type π∞. One wants to guarantee that the map from Yalg to automorphicrepresentations is injective, and in particular we want a way to single out vectorsϕy ∈ πy, up to scalar multiples. At places outside S we assume ϕy is an unramifiedvector. At the infinite place ϕy is determined by holomorphicity. Every v ∈ S(π, χ)splits in K, hence πy,v is a representation ofGL(n,Qv), and we assume the conditionof being a semi-simple type vector determines the v-component of ϕy uniquely. Thisis possible as long as πv is fully induced from a supercuspidal representation of someLevi component, provided we remember to interpret ϕy as a vector valued modularform 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 ∈ SK unless πv is spherical forsome special maximal compact subgroup – and even then there are two conjugacyclasses 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 ϕy is fixed up to scalarmultiples.

Finally, and most importantly, Hida has shown that conditions (3) and (4) de-termine the p-component of ϕy up to scalars, and we can identify it explicitly asa (Kν , ν)-type when the latter is a type. The conditions guarantee that ϕy is afactorizable vector for any y ∈ Yalg. We determine vectors ϕ′y ∈ π∨y by the dualconditions and the requirement that the Petersson norms be the same. We wouldlike to apply the basic identify (2.8). However, since we have defined Hida familiesso that ϕy, y ∈ Yalg, is holomorphic (at least when ab 6= 0), we apply the basicidentity not to this pair but to the corresponding pair ϕy, ϕy′ , where the formeris antiholomorphic and the latter is defined again by the dual conditions and bothare normalized so that Q0

V (ϕy ⊗ ϕ′y) = Q0V (ϕy ⊗ ϕ′y).11 When ab = 0 holomorphy

(resp. antiholomorphy) is replaced by the condition of taking values in a certainfinite-dimensional representation of U(n) (resp. the dual representation).

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

Proposition (4.2). The factor ZpS(1, ϕy, ϕ′y, f, χ) is an algebraic number whose

p-adic absolute value is uniformly bounded above and below as a function of y andχ.

(The superscript ′p′ denotes the ommision of the relevant factors at p.) One ac-tually knows more. If we avoid small primes the archimedean contribution is ap-adic unit, and for the choices made (without explanation) in (3.3)(ii), the factorscorresponding to primes in S(π, χ) are just group indices again. Any variation isagain due to SK.

It remains to determine the local zeta integral at p. We let Pa,b be the standardparabolic subgroup of GL(n) corresponding to the partition n = a + b. We write

πp = IndGL(n,Qp)

Pa,b(Qp) πa ⊗ πb (normalized induction) where πa is a principal series

representation of GL(a,Qp) and πb a principal series representation of GL(b,Qp).In what follows, ` = 1 is only allowed if U(V ′)(R) is definite; i.e. if Pa,b = GL(n).

Proposition 4.3. Write

Za(1, π) = εp(0, πa ⊗ χ, ψ−1p )

Lp(1, π∨a ⊗ χ−1)

Lp(0, πa ⊗ χ)

Zb(1, π) = εp(0, π∨b ⊗ χ−1, ψ−1

p )Lp(1, πb ⊗ χ)

Lp(1, π∨b ⊗ χ−1)

Suppose ` = 2. Then for any y ∈ Yalg

(4.3.1) Zp(1

2, ϕy, ϕ

′y, fp, χp) = A(a, b)pN(πp) · Za(1, π) · Zb(1, π),

11This has the effect of replacing L(s, π, χ) by L(s, π, χ) in the Basic Identity. To avoidchanging all the formulas we make the simplifying hypothesis that πf is isomorphic to its complex

conjugate.


where A(a, b) ∈ Q is an explicit constant depending only on a and b and N(πp) ∈ Zdepends only on the conductors of the characters νi and νiν

−1j . The formula (4.3.1)

remains true for any `, for a Zariski dense set of points y ∈ Yalg.

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 powerN(πp) of p corresponds exactly to his normalization. For purposes of comparison,the standard L-factor at p is just

LU(V ′)p (s, π, χ) = LGL(n)

p (s, π, χ)LGL(n)p (s, π∨, χ−1).

The p-adic normalization process, sometimes called p-stabilization, selects half theEuler 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 ofGauss sums). Here the presence of the denominator corresponds precisely to thepartial Fourier transform used to define the section fp, and appears in the courseof applying the Godement-Jacquet functional equation for the standard L-functionof GL(n).

Finally, Hida theory provides a way to pair the Eisenstein measure of the pre-ceding with the Hida family ϕy (or rather ϕy ⊗ϕ′y) in such a way that at points in

Yalg the result is the global zeta integral involving ϕy and ϕ′y. Let

Lp(1, πy, χy) = LS(1, π, χ)ZpS(1/2, ϕy, ϕ′y, f, χy).

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

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

Lp(1, •) = ϕ ∗ dµEis : Y → Cpsuch that

Lp(1, y) = A(a, b)qN(πy,p) · Za(1, πy) · Zb(1, πy)Lp(1, πy, χy)/Ωyfor all y ∈ Yalg.

Here Ωy is a period essentially equal to the quotient of the Petersson norm of ϕyby a congruence number measuring congruences between ϕy and other forms on G.

The notation Lp(1, y) indicates the expectation that the p-adic L-function ex-tends to an analytic function in the first argument (s = 1) but we have not con-sidered values other than s = 1. We also write Lp(1, π, χ) = Lp(1, y) if π = πy,χ = χy.

(4.5) Hecke algebras in Hida’s theory. Rigid analytic geometry is not reallynecessary for Hida’s theory but it provides a useful geometric picture of p-adicvariation. Let ΛP and HY be the rings of functions on X0 and Y, respectively. Hidaactually works with integral versions of these rings, in which case the former is anIwasawa algebra in rank(P ) variables and the latter, finite and torsion-free overthe former, is a p-adic arithmetic Hecke algebra. In our rigid-analytic setting, therings ΛP and HY are obtained from Hida’s rings upon tensoring with Qp. We haveconstructed Lp(1, y) to be an element in HY ⊗ Cp (and even in HY).

The irreducible components of Y correspond to Hida families, whereas the con-nected components of Y correspond to collections of Hida families admitting con-gruences modulo maximal ideals in ΛP . We will make more precise the notion ofsuch 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 exposi-tion, we will henceforth assume n to be even. Let χ be an (adelic) Hecke characterof K satisfying (2.4), so

χ |A×Q = εK.

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). TheArthur conjectures predict the existence of an automorphic representation Πχ(π×ξ)of H, with the property that BC(Πχ(π × ξ)) is induced from the representation(BC(π)⊗ χ det)× BC(ξ) of GL(n)×GL(1):

(5.1) L(s,BC(Πχ(π × ξ))) = L(s,BC(π)⊗ χ det)L(s,BC(ξ)).

Here recall that BC(ξ) is the character z 7→ ξ(z/zc), where c denotes complexconjugation. The Πχ(π ⊗ ξ) are the endoscopic representations of the title (moreproperly attached to the elliptic endoscopic group U(n)×U(1)). Note that Πχ(π×ξ)comes along with a conjectural multiplicity (proved for n = 2) which may be zero,but in general is not. Also Πχ(π × ξ) is actually a packet, but in the exampleswe consider there will be only one representation. Indeed, H(R) is compact, soL-packets at ∞ are singletons.12

The theta correspondence constructs these representations “by hand”. Thetensor product V ′ ⊗ V has a non-degenerate hermitian form <,>V ′⊗V , and thechoice of a non-zero element ג ∈ K with trK/Q(ג) = 0 transforms <,>V ′⊗V into

a skew-hermitian form 1−ג <,>V ′⊗V . Thus V = RK/Q(V ′ ⊗ V ) can be viewed asa symplectic vector space over Q of dimension 4n(n + 1), and there is a naturalmap G × H = U(V ′) × U(V ) → Sp(V). For our purposes, the metaplectic coverMp(V)(A) is the extension of Sp(V)(A) by the circle. The character χ introducedabove 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 rightsign at∞, one can then define the metaplectic representation ωψ,χ of G(A)×H(A)and, for any automorphic representation π of G, the theta lift Θψ,χ(π) as an auto-morphic representation of H.

Assume π is in the domain of the theta correspondence; i.e., that Θψ,χ(π) is notidentically zero. As explained below, this can be guaranteed by a global condition(which will be automatically satisfied) and by local conditions (essentially at primesin SK). The local components of Θψ,χ(π) at places outside S can be determinedin terms of those of π and χ, by a procedure introduced by Rallis in [Rallis HDC].We have

(5.2) ΘSψ,χ(π) = ΠS

χ−1(π × 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 πv is spherical and χv is unramified; therefore

packets are singletons at such places as well. Here as previously, we don’t really know what to sayabout places v ramified in K, but the explicit construction of endoscopic representations described

in this section yields well-defined representations of H(Qv).


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

(5.3) LS(s,Θψ,χ(π)) = LS(s, π ⊗ χ−1)ζSK(s) = LS(s, π ⊗ χ)ζK(s).

We may thus define Πχ−1(π × 1) to be Θψ,χ(π).Not every Πχ−1(π×ξ) occurs as such a theta-lift; Θψ,χ(π) = 0 unless π∞ belongs

to the domain of the local theta correspondence for U(a, b) × U(n + 1, 0). Forexample, suppose n = 2 and V ′ is positive-definite, so a = 2, b = 0, and π is of theform π0 ⊗ β where π0

∞ 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, π0) = L(s, f)L(s, f, εK)

(i.e., π0 = 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 tothe Hecke character ? (a binary theta function) then

(L(s,Θψ,χ(π)) =)L(s, π ⊗ χ)ζK(s) = L(s, f, θ(BC(β) · χ))ζK(s)

where the first term on the right-hand side is the Rankin-Selberg product. ThenΘψ,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’stheorem implies that

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

Moreover, the Petersson norm in the denominator can be expressed in terms ofperiods 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 thenotation θψ,χ,Φ0

(ϕ) to designate the theta lift of ϕ to an element of Θψ,χ(π), anautomorphic form on the definite unitary group U(V ). As indicated, the notationdepends on a Schwartz-Bruhat function Φ0, which in turn depends on a choice ofpolarization of the symplectic vector space V = RK/Q(V ′ ⊗ V ). The constructionbecomes more canonical when we consider a pair of forms ϕ ∈ π, ϕ′ ∈ π∨; theirtheta lift

θψ,χ,Φ(ϕ⊗ ϕ′) ∈ Θψ,χ(π)⊗Θψ,χ(π)∨

is defined in terms of a choice of maximal isotropic subspace X of the skew-hermitianspace 2(V ′⊗V ), and for X one can choose the diagonal subspace (V ′⊗V )d, whichwe identify with V ′ ⊗ V .

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

(6.1)

U(2V ′) U(V )× U(V )x

x

U(V ′)× U(V ′) U(V )


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

(6.2) QV (θψ,χ,Φ(ϕ⊗ ϕ′)) def=

∫

U(V )(Q)\U(V )(A)

θψ,χ,Φ(ϕ⊗ ϕ′)(h)dh

=

∫

U(V ′)×U(V ′)(Q)\U(V ′)×U(V ′)(A)

θψ,χ,Φ(1)(g1, g2) · ϕ(g1)⊗ ϕ′(g2)dg1dg2

Here ϕy and ϕ′y are as in the discussion preceding Proposition 4.2. For appropriatechoices of a Schwartz-Bruhat function Φ, the theta lift θψ,χ,Φ(ϕ⊗ϕ′) can be factoredas a product of forms on the two copies of U(V ′), and the first integral can be writtenas a genuine inner product. The θψ,χ,Φ(1) appearing in the second integral denotesthe 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, expres-sion (6.2) becomes the Rallis inner product formula when the function θψ,χ,Φ(1)can be identified with a Siegel Eisenstein series on U(2V ′). This identification isprovided 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 Eisen-stein series on symplectic groups. For appropriate choices of Φ we obtain precisely13

the value at s = 12 of the Eisenstein series attached to the section f = ⊗vfv defined

in §3; this is the meaning of condition (3.3)(v). The precise formula is then

(6.3)

dSn(1

2, χy)QV (θψ,χy,Φ(ϕy ⊗ ϕ′y))

= dSn(1

2, χy)

∫Ef ((g1, g2);χ,

1

2)ϕy(g1)⊗ ϕ′y(g2)dg1dg2

= Q0V ′(ϕy ⊗ ϕ′y)Lp(1, y)Ωy,

where the integral is over U(V ′)× U(V ′)(Q)\U(V ′)× U(V ′)(A) and Ωy is the pe-riod appearing in Theorem 4.4.. The appearance of Lp(1, y) suggests that the Rallisinner product formula has a variant in terms of p-adic L-functions. This is indeedthe 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 Q-rational structure compatible with the theory of canon-ical models. We can do the same on H×H, but since H(R) is compact, the locallysymmetric variety attached to H and an open compact subgroup K⊂H(Af) is thefinite set KSh(H) = H(Q)\H(A)/H(R)×K. The theta lift Θψ,χ(π) to H of anautomorphic representation π of G of our fixed infinity type π∞ is an automorphicrepresentation Π with Π∞ fixed and, with our choice of τ , necessarily of dimension1. 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 Q-rational structure, andeven an integral structure over the ring Z of algebraic integers in Q. (Of course L

13Up to multiplication by 12

, which we incorporate into the constants.


is constant on connected components of KSh(H), which are just points!) We letSh(H) be the profinite set lim←−K KSh(H).14

It is an important fact that the theta lift θψ,χ,Φ(ϕ⊗ ϕ′) almost never preserves

arithmeticity. If ϕ ⊗ ϕ′ is a holomorphic modular form on G × G which is Q-rational and belongs to an irreducible automorphic representation π ⊗ π∨, thenone can define without much difficulty an almost certainly transcendental factor

qπ ∈ C×, well-defined up to Q×, such that

(6.4.1) q−1π θψ,χ,Φ(ϕ⊗ ϕ′) ∈ H0(Sh(H),L)(Q)

Unfortunately, unless either (i) n = 2 or (ii) ab = 0 we know no sensible way todefine a variant of qπ that is well-defined up to p-integers. More precisely, as longas π is unramified at p (which will not generally be the case!) there are naturalchoices of qπ so that q−1

π θψ,χ,Φ(ϕ ⊗ ϕ′) ∈ H0(Sh(H),L)(Z(p)), but no choice isknown that does not compromise valuable information about congruences betweenholomorphic forms on G×G, except in cases (i) or (ii). We will say no more aboutthis 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 dµEis introduced in Theorem 3.9, depends on the p-adic family of sections fp = fp(ν, χ) described after Proposition 3.8, which inturn depends on a certain collection of Schwartz-Bruhat functions Φν on V(Qp) 'M(n, 2(n+ 1))(Qp) depending on characters ν of finite order. We have not checkedenough details to remove the question mark in the following Proposition(?), but itis undoubtedly true. Note that the period Ωy appears squared in the denominator.

Proposition(?) (6.4.2). Let (φy, χy) be a Hida family over Y, as in §4. LetΦν ∈ S(M(n, 2(n+ 1))(Qp)) and Φp ∈ S(V(Ap

f )) be the Schwartz-Bruhat functions

used to define the Eisenstein measure dµEis. Then there is a Hida family F θ(y) =F θϕy⊗ϕ′y,χy of p-adic forms in H0(Sh(H)×Sh(H),L⊗L∨)(Cp), parametrized by Y,

such that

F θ(y) = (dSn(1/2, χy)/Ω2y)θψ,χy,Φwt(y)

(ϕy ⊗ ϕ′y)

for all y ∈ Yalg.

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

Proposition (6.4.3). There is a rigid meromorphic function gθ on Y such thatF (y) = gθ(y)F θ(y) is a primitive Hida family.

Much of our work at present is concerned with finding conditions under which gθcan 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

QalgV ′ (φy ⊗ φ′y) = Ω−1y Q0

V ′(φy ⊗ φ′y)

whenever y ∈ Yalg.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 QV and QalgV ′ extend to natural pairings onp-adic modular forms on U(V )× U(V ) and U(V ′)× U(V ′), respectively, such that

QV (F (y)) = gθ(y)QV (F θ(y))

= gθ(y)QalgV ′ (ϕy ⊗ ϕ′y)Lp(1, y).

Remark 6.5 We note in passing that there is no natural way to separate ϕy fromϕ′y in the above discussion, mainly because the doubling method cannot be appliedto pairs ϕ1 ⊗ ϕ2, with ϕ1 ∈ π1, ϕ2 ∈ π2, where ϕ2 is congruent to a form in π∨1but is not actually in π∨1 . This is the source of serious problems whose resolutionbrings 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. Fixa level subgroup K =

∏qKq with Kp determined by the level of the character χ,

Kq hyperspecial maximal at all inert primes, and some other local conditions ifnecessary. The space MK(L,C) = H0(KSh(H),L) of K-fixed vectors in the spaceof all automorphic forms on H of type Π∞ contains a natural O = OK,p-latticeL(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 thefinite set KSh(H) giving each point mass 1, is a unimodular hermitian form (moreprecisely, we work with the perfect pairing between L(L) and L(L−1)).

Now we write

(7.1) H0(KSh(H),L) = H0(KSh(H),L)end ⊕H0(KSh(H),L)st

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

L(L) ⊃ L(L)end ⊕ L(L)st

where L(L)? = H0(KSh(G),L)? ∩ L(L). This inclusion is an equality if and onlyif there are no pairs fend ∈ L(L)end, fst ∈ L(L)st of p-primitive forms such thatp−1[fend − fst] ∈ L(L), i.e. if and only if there are no congruences between endo-scopic and stable forms of the form

fend ≡ fst (mod p).

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

We now allow the level subgroup Kp to vary, always assuming Kp ⊃ RuP (Zp).Applying Hida’s constructions, we obtain modules of P -nearly ordinary formsLP−ord(L) ⊃ LP−ord(L)end ⊕ LP−ord(L)st of finite type over ΛP , the ring of func-tions on X0. The torsion ΛP -module

C = LP−ord(L)/(LP−ord(L)st ⊕ LP−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 HY (a Hecke algebra; see (4.5)) acts faithfully on LP−ord(L) and henceacts on C. (To make all this true we have to consider Hida families on H consistingsimply of p-adic forms, not pairs of p-adic forms and characters.) Let

I = AnnHY(C).

Let (φy, χy) over Y0 be an irreducible Hida family of modular forms on G.Let R = HY0

. In general this is finite and integral over ΛP , but for the sake ofsimplicity we will assume that R = ΛP . Recall that we can view Lp(1, •) as anelement of R (see (4.5)). For the appropriate choice of the prime-to-p levels Kq,if we admit Proposition 6.4.2 then the endoscopic Hida family F of Proposition6.4.3 is associated to an irreducible component of Y, which identifies R as a ΛP -quotient of HY. In fact, this identifies Y0 as a component of Y. The considerationsintroduced in the first paragraph of this section apply to the unimodular pairing

(7.2) LP−ord(L)P ⊗ LP−ord(L)P → ΛP,P,

where P ⊂ ΛP is any height one prime. In particular, the discriminant of therestriction of the pairing (7.2) to LP−ord(L)end,P ⊗ LP−ord(L)end,P divides IRP.Concretely, we obtain the following theorem (admitting Proposition 6.4.2).

Theorem (7.3). IR ⊆ (gθ(•)Lp(1, •)).

Assume we can eliminate the function gθ from the above statement. Then thetheorem asserts that the p-adic L-function divides the ideal measuring congruencesbetween endoscopic and stable forms (really Hida families). In the next sectionwe will briefly explain how such congruences give rise to non-trivial classes in Ga-lois cohomology, as predicted by the appropriate generalization of Iwasawa’s mainconjecture.

In order to eliminate gθ it suffices to show that the normalized theta lifts F θϕy⊗ϕ′y ,χyare already p-primitive (integral, but not divisible by any fractional power of p).The methods available for proving the arithmeticity of F θϕy⊗ϕ′y ,χy – i.e., for deter-

mining the transcendental factors qπy introduced in (6.4) – can be applied to provethe desired primitivity, provided one knows that sufficiently many special values ofauxiliary L-functions are not divisible by p. It appears that one can derive whatone needs from results of Vatsal in the case n = 2.

(7.4) Hecke algebras on H and congruences. Let P ⊂ R = ΛP be anyprime ideal containing IR. It follows that localization at P of the quotient of HYcorresponding to all the families of stable forms (i.e., the quotient acting faithfullyon LP−ord(L)st) is non-trivial. This can be interpreted as meaning that there isa component of Y corresponding to a family of stable forms that intersects theendoscopic component Y0 “at the prime P.”

(7.5) Theorem (7.3) as a statement about forms on G. As written, Theorem7.3 asserts that the p-adic L-function of a Hida family on G = U(V ′) controlscongruences between endoscopic and stable forms on H = U(V ). The applications


in the next section concern the Galois representations attached to the Hida family onG. Thus neither the input (the Hida family on G) nor the output (the constructionof Galois cohomology classes) makes mention of the group H, and one would liketo be able to eliminate H and the hermitian space V from the statement. In otherwords:

Question (7.5.1). Does the typical classical point y = (πy, χy) in the Hida familyadmits a non-zero lifting to H = U(V ), for some positive-definite hermitian spaceV of dimension n+ 1?

As Waldspurger was the first to observe, and as Rallis explained in great general-ity, there are both global and local obstructions. The Rallis inner product formulaaccounts for both obstructions: if QV (θψ,χ,Φ(ϕ ⊗ ϕ′)) 6= 0 for some choice of datathen the theta lift is non-trivial. It suffices to restrict our attention to Hida familiessuch that πy is tempered for all (classical) y, and for which no πy is itself a theta liftfrom some U(W ) with dimW < n. Under our running assumption that πy admitsa base change to GL(n)K, the L-function L(s, π, χ) does not vanish at s = 1, hencethere is no global obstruction. The local obstructions are given by the productof the local terms in the Rallis inner product formula. There is a genuine localobstruction at the archimedean prime, since we have to take V positive-definite,but the local component πy,∞ has been chosen to eliminate this obstruction. It isshown in [HKS] that there is a non-archimedean local obstruction at a prime p ifand only p is inert or ramified in K and the local theta lift of πp (determined by χp,as well as the additive character ψp and (ג to U(V )(Qp) is trivial. It is also shownin [loc. cit.] that for at least one choice of local space Vp of dimension n+ 1 up toisometry – there are two isometry classes – the theta lift of πp to Vp is non-trivial.Thus Question (7.5.1) comes down to whether the set of local Vp’s to which πp liftscan be realized as the local components of a given global positive-definite hermitianspace over K. In principle there is a parity obstruction; however, at least when nis even, “most” πp lift to either isometry class, which means that “most” π (withthe given π∞) lift to some global U(V ). Finally, it is not difficult to see that, byvarying the choice of χ (away from p) one can always arrange that the theta lift toa 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 specificpair (π, χ). At present we do not know the answer to Question (7.5.1), which hassomething to do with the nature of the values of “incoherent” Eisenstein series atthe point s = 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 Hidafamily (φy, χy) over Y0 is an analytic family of n-dimensional Galois representationsσ : Gal(K/K) → GLn(R) such that σ(y) = σ(πy ⊗ χy), the latter being the n-dimensional representation corresponding to BC(π) ⊗ χ as in section 2. At thispoint, we make a number of assumptions on p, S, and σ, the most important ofwhich is that the mod p reduction of some (and therefore all) σ(πy ⊗ χy) in ourfamily is absolutely irreducible.

Let P ⊂ R = ΛP be any height one prime containing I. Let σP be σ viewed asa representation into GLn(RP) and let σP be the reduction of σP modulo P. This


is a representation into GLn(k(P)), where k(P) = RP/P is the residue field. Ourassumptions imply that σ′ is irreducible.

From Theorem (7.3) it follows that there is some irreducible component Y′ cor-responding to a Hida family of stable forms that intersects the endoscopic familyY0 at P (see (7.4)) (recall that for these families we have dropped the extra char-acter). Again, for simplicity we shall assume that R′ = HY′ (a quotient of HY) isalso isomorphic to ΛP . Then P determines a prime of R′ and we assume that thereis an n + 1-dimensional representation σ′ : Gal(K/K) → GLn+1(R′) such that its

specializations at points in Y′alg are the n+ 1-dimensional representations (conjec-turally) associated to the representations BC(π′) as in section 2. The existence ofσ′ 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, whenn+ 1 ≤ 3 the existence of the representations σ′(y) has been shown by Blasius andRogawski [BR], and they are expected to exist in general.

Define σ′P and σ′P as we did σP and σP. The properties of these representationsconnecting Hecke operators and eigenvalues of Frobenius elements then implies that

(σ′P)ss = σP ⊕ 1,

where the “ss” denotes semi-simplification. But we actually get more. The rep-resentations associated to stable forms are expected to have “large” image. Whenthis is so, after possibly replacing σ′P with a conjugate over the field of fractions ofRP, we have that

σ′P =

(σP ∗

1

).

with the “∗” in the upper-right corner non-trivial. This “∗” determines a non-trivialclass in H1(K, σ−1

P ). With more work this class can be shown to have nice ramifi-cation properties at the primes above p, and its existence can be reinterpreted assaying that localization at P of the Bloch-Kato/Greenberg Selmer group associatedto an integral version of σ has length ≥ 1 as a RP-module.

An argument originally due to Wiles, as generalized by Urban, shows that byworking with all stable components congruent to the endoscopic component Y0

modulo P it is possible to prove that the length of the Selmer group localized at Pis at least ordP(IRP).If we assume the function gθ of Proposition 6.4.3 to be a scalar, then the congruenceideal I is divisible by the p-adic L-function Lp(1, •), and therefore the length of theSelmer group localized at any height one prime P of ΛP is at least ordP(Lp(1, •)).Provided we can justify all the assumptions we have made up to this point !

REFERENCES

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[Co] J. Coates, Motivic p-adic L-functions, in J. Coates and M. J. Taylor, eds., L-functions and Arithmetic London Mathematical Society Lecture Note Series, 153Cambridge: Cambridge University Press (1991), 141-172.

[Gar] P. B. Garrett, Integral representations of Eisenstein series and L-functions, inNumber theory, trace formulas and discrete groups (Oslo, 1987), Boston: AcademicPress (1989), 241-264.

[GPR] S. Gelbart, I. Piatetski-Shapiro, and S. Rallis, Explicit constructions ofautomorphic L-functions, Lecture Notes in Math., 1254 (1987).

[H97] M. Harris, L-functions and periods of polarized regular motives, J. reineangew. Math., 483 (1997), 75-161.

[HKS] M. Harris, S. Kudla, and W. J. Sweet, Theta dichotomy for unitary groups,J. Am. Math. Soc., 9 (1996) 941-1004.

[HL] M. Harris and J.-P. Labesse, Conditional base change for unitary groups,manuscript (2003).

[HT] M. Harris and R. Taylor, The Geometry and Cohomology of Some SimpleShimura Varieties, Annals of Math. Studies, 151, Princeton: Princeton UniversityPress (2001).

[Hi81] H. Hida, Congruence of cusp forms and special values of their zeta functions,Invent. Math., 63 (1981) 225-261.

[Hi00] H. Hida, Irreducibility of generalized Igusa towers, manuscript (2000); avail-

able at http://www.math.ucla.edu/hida.

[Hi02] H. Hida, Control theorems of coherent sheaves on Shimura varieties of PELtype, J. Inst. Math. Jussieu, 1 (2002) 1-76.

[Hi IHP] H. Hida, p-adic automorphic forms on reductive groups, Notes of a course

at the IHP in 2000, Asterisque (to appear); available at http://www.math.ucla.edu/hida.

[Ich] A. Ichino, A regularized Siegel-Weil formula for unitary groups (manuscript,2001-2003).

[Katz] N. Katz, p-adic L-functions for CM fields, Invent. Math., 49 (1978) 199-297.

[Kudla] S. Kudla, Splitting metaplectic covers of dual reductive pairs, Israel J.Math., 87 (1994) 361-401.

[L92] J.-S. Li, Non-vanishing theorems for the cohomology of certain arithmeticquotients, J. reine angew. Math., 428 (1992), 177-217.

[N] A. Nair, Intersection cohomology, Shimura varieties, and motives, Manuscript(2003).

[Rallis HDC] S. Rallis, On the Howe duality conjecture, Compositio Math., 51(1984) 333-399.

[Rog] J. Rogawski, Automorphic Representations of Unitary Groups in Three Vari-ables, Annals of Math. Studies, 123, Princeton: Princeton University Press (1990).

[Sh1] G. Shimura, Euler products and Eisenstein series, CBMS Regional Confer-ence Series in Mathematics, 93, Providence, R.I.: American Mathematical Society(1997).


[Sh2] G. Shimura, Arithmeticity in the theory of automorphic forms, Mathemati-cal Series and Monographs, 82, Providence, R.I.: American Mathematical Society(2000).

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