p-adic measures and square roots of special values of triple product L-functions

Digital Object Identifier (DOI) 10.1007/s002080100188

Math. Ann. 320, 127–147 (2001) Mathematische Annalen

p-adic measures and square roots of special valuesof triple product L-functions

Michael Harris � · Jacques Tilouine��

Received January 13, 1998 / Revised July 26, 2000 /Published online February 5, 2001 – © Springer-Verlag 2001

Introduction

Letp be a prime number. In this note, we combine the methods of Hida with theresults of [HK1] to define ap-adic analytic function, the squares ofwhose specialvalues are related to the values of triple productL-functions at their centersof symmetry. More precisely, letf , g, andh be classical normalized cuspidalHecke eigenforms of level 1 and (even) weightsk, �, andm, respectively, withk ≥ � ≥ m; assumek ≥ � + m. Let L(s, f, g, h) be the triple productL-function [G1, G2, PSR]; its center of symmetry is the points = k+�+m−2

2 . Let〈•, •〉k be the normalized Petersson inner product for modular forms of weightk. LetQ{f, g, h} be the field generated overQ by the Fourier coefficients off ,g, andh. Using the integral representation forL(s, f, g, h) [op. cit], Kudla andone of the authors have shown that the quotient

L(k+�+m−2

2 , f, g, h)

π2k 〈f, f 〉2k · C(k, �,m)is a square inQ{f, g, h}. HereC(k, �,m) ∈ Q is a universal constant, dependingonly onk, �, andm. We constructp-adic measures which interpolate the squareroot of this quotient, as (the ordinary eigenform associated to)f varies in aHida family f . These are actuallygeneralized measures, in the sense of [H1,II]:elements of the finite normal algebra extensions of the Iwasawa algebra which

M. HarrisU.F.R. de Mathématiques, Université Paris 7, 2, place Jussieu, 75251 Paris Cedex 05, France

J. TilouineUniversite Paris-XIII-Institut Galilee, Mathematiques - Bât. B, Avenue Jean Baptiste Clément,93430 Villetaneuse, France

� Institut de Mathematiques de Jussieu-U.M.R. 7586 du CNRS. Supported in part by the Na-tional Science Foundation, through Grant DMS-9203142.

�� Membre, Institut Universitaire de France

128 M. Harris, J. Tilouine

arise in Hida’s theory of the ordinary Hecke algebra. In the simplest case, weobtain the following formula (cf. Theorem 2.2.8):(

DH(f , g, h)(k)H(k) ·K(k)

)2

= L(w+12 , fk, g, h

)π2k · 〈fk, fk〉2k · C(k, �,m).

Herefk is (the primitive form associated to) the specialization in weightk off , DH(f , g, h) is the analytic function associated to ourp-adic measure,w =k + �+m− 3, andH(k) andK(k) are normalizing factors depending onf .

The construction of this measure is a modification of Hida’s approach top-adic interpolation of Rankin products [H1,II]. Indeed, when the cusp formgis replaced by an appropriate Eisenstein series, Hida’s method defines a three-variable Rankin product: one variable for the value ofs and the other two forthep-adic variation off andh. However, there is a subtle difference betweenour p-adic construction and that of Hida. LetZ = Z×

p and letX be ap-adicmanifold on whichZ acts. Starting with measuresdE onZ andµ onX, Hidaconstructed the convolutiond(E ∗ µ) by the formula∫

Z×Xφ(z, x)d(E ∗ µ) =

∫Z×X

φ(z, z−1x)dE(z)dµ(x).

ThemeasuredE takes values in the space ofp-adic modular forms. The fact thatdE is supported onZ×

p forces∫Zφ(z)dE(z) to be of level divisible byp. Hence

thep-adic symmetric squareL-function constructed in [H3] comeswith anEulerfactor atp that gives a trivial zero ats = 1. In our present construction,we insteadform theproductof a measure and a fixed function. The calculation leading to(2.2.7) yields the correct Euler factor atp, and thus to thep-adic interpolationformula (2.2.9). The same idea was used by one of the authors, together withR. Greenberg, to construct a modified symmetric squareL-function, with thetrivial zero removed, and thus to obtain a formula for the derivative ats = 1of Hida’s symmetric squareL-function ats = 1, with arithmetically interestingimplications (see [GT], [HTU]).

The article [HK1] also considers the central critical values whenk < �+m.Thep-adic interpolation of the critical values of such triple products, as opposedto the square roots, has recently been obtained by B¨ocherer and Panchishkinwhen each of the rank 2 motivesM(f ),M(g), andM(h) associated tof , g, andh is ordinary, cf. [P].

Thenotion that thesquare rootsof central critical valuesofL-functionsshouldhavep-adic interpolations seems to have first arisen in connection with the thesisofA. Mori [M1, M2]. Mori showed that, iff is a holomorphic modular form andK is an imaginary quadratic field inwhichp splits, then the value off atHeegnerpoints associated toK, suitably normalized, is naturally an Iwasawa function.Whenf is a new form, it should be possible to useWaldspurger’s results in [W]to show that this Iwasawa functionp-adically interpolates the square roots ofthe central critical values of theL-functionsL(s, fK, χ), wherefK is the base

p-adic measures and square roots of special values 129

change off toK, as above, andχ runs through a continuous family of algebraicHecke characters forK. Other examples have been studied by Stevens [St], Hida[H5], Sofer [So] and Villegas [V].

The first author would like to take this opportunity to thank his colleaguesat the Universit´e de Paris-Sud at Orsay, where most of this article was writtenin 1993, for providing a uniquely congenial working environment. Both authorswould like to thank the referee, whose careful reading uncovered a significanterror in the previous version, and whose suggestions led to a number of improve-ments. In particular, Sect. 1.4 has been thoroughly rewritten on the basis of thereferee’s suggestions.

1. p-adic measures associated to three modular forms

(1.1)Review of p-adic modular forms.Let O be an algebra of finite-rank overZp, K = O ⊗Zp Qp. Let N be a positive integer prime top and letZN =lim←−(Z/NpαZ)×; M = M(N,O) denotes the completeO-algebra ofO-valued(p-adic)modular forms of levelNp∞ and anyweight (i.e.,p-adicmodular formsin the sense of Katz; see [H1 I or II,§1]). For any integersk, � ≥ 0, and anyringR, we letM�(Np

k, R) denote the module of modular forms of weight� forΓ1(Np

k) – we say of levelNpk, for short – whoseq-expansion at the cusp atinfinity lies inR[[q]]. Let

M�(Np∞, R) =

⋃k

M�(Npk, R).

Similarly, we defineS�(Npk, R), S�(Np∞, R), andS = S(N,O) to be thecorresponding modules of cusp forms. Ifg ∈ S, we write itsq-expansiong =∑∞

n=1 a(n, g)qn,witha(n, g) ∈ O for alln.Weuse thesamenotation formodular

forms with complex Fourier coefficients.

The following operators onM�(Np∞,O) are standard, preserve the submodules

of cusp forms, and extend to the completionM (cf. [H3,§1] and [Go] for details):(1.1.1) For any prime numberλ, the Hecke operatorsT (λ) and, forλ relativelyprime toN , T (λ, λ), whose action on theq-expansion is given by [H2, (1.13a)]; more generally, for anyλ relatively prime toN , T (λ) andT (λ, λ) can bedefined by the usual formulas.(1.1.2) For any prime numberλ relatively prime toNp, thediamond operators〈λ〉�, whose action onM�(Np

∞,O) is given by

〈λ〉� = T (λ)2 − T (λ2)λ�−1

= T (λ, λ)

λ�−1;

we let〈λ〉 = 〈λ〉0.


(1.1.3) By continuity, the mapλ �→ 〈λ〉 extends to an action of the groupZN onM(Np∞,O) given by〈z〉 = zp · T (z, z). It coincides with that induced byZNacting on thep-adic moduli problem (cf. [H3,§1] and [Go] for details).

Moreover(1.1.4) Thedifferential operatord = q d

dq, operating on theq-expansion

sends classical forms top-adic modular forms.

(1.2)Review of Hecke algebras.We retain the notation from the previous section.

(1.2.1) Definition.The Hecke algebra

h = h(Np∞,O) ⊂ EndO(S(N,O))is theO-subalgebra generated by the Hecke operatorsT (λ) for all primesλ andbyT (λ, λ) for λ relatively prime toN . Similarly,

h�(Npα,O) ⊂ EndO

(⊕k≤�

Sk(Npα,K)

)∩ O[[q]]),

the Hecke algebra of weight� and levelNpα, is theO-subalgebra generated bytheT (λ) for all primesλ and by theT (λ, λ) for λ relatively prime toN .

There are canonical surjections ofh(Np∞,O) ontoh�(Npα,O) for all �, α,and (cf. [H2, pp 243 ff.])

(1.2.2) h(Np∞,O) = lim←−h�(Npα,O).Let h�(Npα,O) be theO-algebra generated by the Hecke operators acting

onS�(Npα,O) (note the difference betweenh�(Npα,O) andh�(Npα,O)). Lete ∈ h denote Hida’s ordinary idempotent [H2, (1.17 b)], and letho = eh bethe universal ordinaryp-adic Hecke algebra of levelN . We setSo� (Np

α,O) =eS�(Npα,O), for α ≤ ∞, and letho�(Np

α,O) = eh�(Npα,O).Let γ = 1+Np ∈ ZN ; thenX = γ − 1 is a topologically nilpotent element

in ΛN = lim←− αO[(Z/NpαZ)×]. Similarly, one letsΛ = O[[Z×p ]]. LetΛ be the

Iwasawa algebraO[[X]] ⊂ ΛN . The map

λ �→ 〈λ〉with the right hand side defined as in (1.1.3), makesh and thusho into continuousΛ-modules, and one of Hida’s main theorems is that

(1.2.3) Theorem.The Hecke algebraho is free of finite rank overΛO. Moreover,let � ≥ 2, � ≡ 1 (mod p−1); let P� denote the element(1+X)− (1+Np)� ∈ΛO. Then the projection defines a canonical isomorphism

ho/P�ho∼−→ho�(Np,O).


(1.2.4) Remark.Note thatΛO/P�ΛO is canonically isomorphic toO for everyinteger� ≥ 1 and everyO.

LetMo = M

o(N,O) = e·M ⊂ M, S

o = So(N,O) = e· S ⊂ S. We define

anO-bilinear pairing

(1.2.5) 〈•, •〉 : So × ho → O; 〈g, T 〉 = a(1, g|T )where, as usual, we writeg �→ g|T for the action of the Hecke operatorT . Then(1.2.5) is a perfect pairing, with respect to which the action ofho is (tautologi-cally) symmetric. For each� ≥ 2, we obtain by restriction a perfect pairing

(1.2.6) 〈•, •〉 : So� (Np,O)× ho�(Np,O) → Odefined by the same formula as (1.2.5). The pairing (1.2.6) identifiesSo� (Np,O)as theP�-torsion submodule ofS

o.

(1.3)Congruence modules andΛ-adic forms.Henceforward we assumeO to be the ring of integers of a finite extensionK

of Qp. We letL denote the fraction field ofΛ, L′ a finite extension ofL, and letI be the integral closure ofΛ in L′. We denote byX (I ) the set of prime idealsof I of height 1, and let

Xk(I ) = {P ∈ X (I )|P ∩ΛO = Pk}.ForP ∈ X (I ) letOP = I/P ; we letk(P ) = k if P ∈ Xk(I ).

Let τ = τf : ho → I be a homomorphism ofΛO-algebras. The subscriptfrefers implicitly to a family ofp-adic modular forms, or to aΛ-adic modularform, in the sense of Wiles, cf. [H4].

(1.3.1) Definition.We sayτf , or f , is N -primitive if, for some integerk > 1(equivalently, for allk > 1) and for someP ∈ Xk(I ), the homomorphism

τf (modPk) : ho/Pkho → OP = I/P

(cf. Remark (1.2.4)) is the homomorphismassociated to amodular formofweightk of level dividingNp which is primitive at all primes dividingN .

From now on, we consider such anN -primitive form f .

(1.3.2) Proposition [H2, Cor. 3.7].If f is primitive, then the homomorphismτf ⊗ 1L′ : ho ⊗Λ L′ → L′ is split overL′.

Thus there are an idempotent 1f ∈ ho ⊗Λ L′ and an isomorphism 1f · ho ⊗Λ

L′ ∼−→L′ such that the homomorphismτf ⊗ 1L′ is given by multiplication by 1f .We write

(1.3.3) ho ⊗Λ L′ ∼−→L′ ⊕ B;


thenτf corresponds to projection on the first factor.LethoI = ho⊗Λ I , h(B) = im(hoI ) ⊂ B with respect to the second projection

in (1.3.3). It follows fromTheorem (1.2.3) that (1.3.3) induces an injectionhoI ↪→I ⊕ h(B) of I -modules. Thecongruence module(1.3.4) C = (I ⊕ h(B)) /hoI .is anI -torsion module. On the other hand, if we take the natural embedding

i : I ↪→ I ⊕ h(B); λ �→ (λ,0)

thenHf = I ∩ i−1(hoI ) ⊂ I is an ideal inI . Then i induces an isomorphism

C ∼−→I/Hf of I -modules. Thus

(1.3.5) Denominator(1f ) = {a ∈ I |a1f ∈ hoI } = Hf .

Once and for all we fix an elementH ∈ Hf and define

Tf = Tf ,H = H · 1f ∈ hoI .

Remark:By assumption,f primP is N -primitive. Letω : (Z/pZ)× → O be the Teich-

muller character. Write

ho = Πa∈Z/(p−1)Zho(ωa),

whereho(ωa) ⊂ ho is the subalgebra on which(Z/pZ)× ⊂ ZN acts via theathpowerωa of ω; let hoI (ω

a) = ho(ωa)⊗Λ I . SinceI is assumed to be an integraldomain, it follows thatτf : hoI → I factors through the natural projection onhoI (ω

a), for exactly onea = a(f ), say. For anyk ≥ 2 and anya ∈ Z/(p − 1)Zthere is an isomorphism

ho(ωa)/Pkho(ωa)∼−→hok(Np,ω

a−k,O),wherehok(Np,ω

a−k,O) ⊂ hok(Np,O) is the subalgebra on which the quotientΓ1(N)∩Γ0(p)Γ1(Np)

� (Z/pZ)× acts viaωa−k [H2, p.249]. IfP ∈ Xk(I ), f primP is thusunramified atp if and only ifa ≡ k(modp−1), unlessk = 2, in which case thep-component of the automorphic representation attached tof may be special; cf.[H1, II, p. 37].

(1.4)Arithmetic p-adic measures.LetZ be ap-adic manifold of the formZrp×(finite group). LetO beas in (1.1), and letC(Z,O)denote the spaceof continuousO-valued functions onZ. For any subringR ⊂ O letLC(Z,R) denote the spaceof locally constantR-valued functions onZ.

LetMes(Z,M) = HomO(C(Z,O),M)


be the set ofp-adic measures onZ with values inM. The measureµ ∈ Mes(Z,M) is calledordinary (respcuspidal) if it takes values inM

o(resp.inS). As

usual, we write∫Zφdµ in place ofµ(φ). We assumeZ given with a continuous

ZN -action, denoted(z, x) �→ z · x, for z ∈ ZN, x ∈ Z. We letz �→ zp be theprojection ofz ∈ ZN on itsp-adic partzp ∈ Z×

p .

(1.4.1) Definition.Let κ be an integer andξ : ZN → O× a character of finiteorder. Ap-adic measure onZ with values inM is arithmetic with character(κ, ξ) (cf. [H1, II, (5.1)] if

(a) for all φ ∈ LC(Z,O ∩ Q)∫Z

φdµ ∈ Mκ(Np∞,Q);

(b) For all φ as above(∫Z

φdµ

)|z = zκpξ(z)

∫Z

φ|zdµ

whereφ|z(x) = φ(z · x) for all z ∈ ZN .(c) Letd denote the operator of (1.1.4). There is a continuous functionν :

Z→O such thatν|z = z2p · ν

for all z ∈ ZN and such that, for anyφ as above,

dr(∫

Z

φdµ

)=∫Z

νrφdµ.

LetO−κ,ξ−1 beO, viewed asO[[ZN ]]-module via the linear extension of thecharacterz �→ z−κp ξ−1(z). Condition (b) can be rephrased:(b′) The map ∫

Z

• dµ : C(Z,O) → S0 ⊗ O−κ,ξ−1

is ΛN -linear, whereΛN acts onC(Z,O) byO-linear extension of the actionφ �→ φ|z.Comments:Let us give a useful reformulation of this definition whenZ = Z×

p .

The action ofZN onM given byz �→ 〈z〉 endowsho with a naturalΛN -algebrastructure. LetΛ = O[[Z×

p ]] andψ : ZN → Λ×, z �→ zκpξ(z)[zp]2

For anyΛN -algebraA, letA(ψ) = A⊗ΛN Λ


whereΛN → Λ is the natural extension ofψ . We viewA(ψ) as aΛN -algebra.Then, the group of ordinary cuspidal measures onZ×

p with character(κ, ξ) can

be identified with theψ-isotypicΛ-submodule of(So)ψ of S

o ⊗ Λ defined by

(So)ψ =

⋂z∈Z×

p

(So ⊗ Λ)z⊗1=1⊗ψ(z)

As an example, we can take the following(1.4.2) Example. HereZ = Z×

p andν in (c) is the tautological inclusionZ ↪→ O.The action ofZN onZ is given by

z · x = z2px z ∈ ZN, x ∈ Z(note the square!). LetR be the ring of algebraic integers inC, and letg ∈S�(N,R) for some� and someN relatively prime top. We assumeg has neben-typus characterξ . Let gp be the twist ofg by the trivial character(mod p):If g = ∑∞

n=1 a(n, g)qn, we havegp = ∑

(n,p)=1 a(n, g)qn. For any function

φ ∈ LC(Z,R) set

(1.4.2.1)∫Z

φdµg =∑

(n,p)=1

φ(n)a(n, g)qn.

Extend this by continuity toC(Z,O) for varyingO. Hida has verified (cf. [H1, I,Prop. 8.1]) thatdµg is arithmetic with character(�, ξ); its “moments" are givenby

(1.4.2.2)∫Z

xrdµg = dr(gp);

here and in what follows we writexr = ν(x)r .(1.4.3) Example. HereZ,R,N, andg are as in (1.4.2), andwe leth ∈ Sm(N,R)for somem. We let ξg and ξh denote the nebentypus characters ofg andh,respectively. The measureh · dµg is defined by

(1.4.3.1)∫Z

φ h · dµg = h ·∞∑n=1

φ(n)a(n, g)qn.

It follows from (1.4.2) thath · dµg is arithmetic with character(�+m, ξg · ξh);its “moments" are then given by

(1.4.3.2)∫Z

xr h · dµg = h · dr(gp).

(1.4.4)Contractions of arithmetic measures by Hida families.Unless otherwisespecified, we assume


(G) Z is ap-adic group containingZ×p as an open subgroup of finite index,

and with an action of(Z/NZ)×.Think for instance ofZ = Z×

p with trivial action of(Z/NZ)×, as we will assumein the next chapter.

The identification of the completed group algebraO[[Z]] with the spaceof continuousO-valued distributions onX, as for example in [H4], yields anisomorphism

O[[Z]] ∼−→HomO(C(Z,O),O).By extension of scalars, we may thus identify

(1.4.4.1){ ordinary cuspidal measures

onZ with values inM} ∼= HomO(HomO(O[[Z]],O), So).

Let HomO(HomO(O[[Z]],O), So)κ,ξ be theO-submodule of the right-handside of (1.4.4.1) corresponding to arithmetic measures with character(κ, ξ).

Let I and So

I be as in§1.3 and let(κ, ξ) be as in Definition 1.4.1. LetMes(Z, S

o)κ,ξ = (S

o)ψ be the set of arithmetic measures onZ with charac-

ter (κ, ξ). Suppose

The pairing (1.2.5) induces by extending the scalars toΛ, a pairing

〈·, ·〉 : (So ⊗ Λ)⊗Λ(ho ⊗ Λ) → Λ

henceMes(Z, S

o)κ,ξ ⊗ΛN(ψ) h

o(ψ) → ΛN(ψ)

We base change it toho(ψ) and obtain

(1.4.4.3) Mes(Z, So)κ,ξ ⊗ΛN(ψ) h

o(ψ)⊗ΛN(ψ) ho(ψ) → ho(ψ)

Wenow twist theHida familyτf : ho → I byψ .We thus obtain anho(ψ)-algebraI (ψ). We use this algebra to base change(1.4.4.3). We get

(1.4.4.4) Mes(Z, So)κ,ξ ⊗ΛN(ψ) h

o(ψ)⊗ΛN(ψ) I (ψ) → I (ψ)

(1.4.5) Definition.Let�f : (So)ψ → I (ψ)

be theΛN(ψ)-linear map given by

µ �→ 〈µ, Tf ⊗ 1〉 .We call it thecontraction against the Hida familyf .

Applying (1.4.5) to µ = h · dµg ∈ Mes(Z, So)κ,ξ = (S

o)ψ the desired

elementDH(f , g, h) ∈ I (ψ).


(1.5) Evaluation ofDH(f , g, h) at certain arithmetic points.

(1.5.1)Notation.

For any formh ∈ Sk(Np,Q), let hρ = ∑∞n=1 a(n, h)q

n, wherez �→ z

denotes complex conjugation;hρ is also an element ofSk(Np,Q). We let

(1.5.1.1) h = hρ |k(

0 1−Np 0

).

Let us denote by〈•, •〉pm,k the Petersson inner product forSk(Npm,O), nor-malized to be linear in the first variable and anti-linear in the second. The formulafor 〈•, •〉pm,k is given as usual by:

(1.5.1.2) 〈f1, f2〉pm,k =∫Γ0(pm)\H

f1(z)f 2(z)yk−2dxdy

wheneverf1 andf2 aremodular forms of weightk for Γ0(pm)with sameNeben-typus, and one of the two is a cusp form.

(1.5.2)Evaluation.In what follows, we letZ = Z×p . We denote the set of height

1 prime ideals ofI byX (I ). Any element ofI (ψ) defines a function onX (I ).Let f (or τf ) be as in 1.3, withO′ = O, and letg ∈ S�(N,R) andh ∈

Sm(N,R) be as in (1.4.3), whereR is the ring of integers in some number field,which we assume contained inO. The ordinary projectione(h · dµg) of themeasureh · dµg is naturally an ordinary cuspidal measure onZ of character(κ, ξ) for ξ = ξg · ξh andκ = �+m.

We shall compute special values at arithmetic points ofDH(f , g, h). ForP ∈ Xk(I ), for somek ≥ 2, let fP be thee-eigenform associated tof primP , withq-expansion

∑∞n=1 a(n, fP )q

n, where

a(npr, fP ) = α(f primP )r · a(n, f primP ) if (n, p) = 1,

whereα(f primP ) is thep-adic unit root of the Hecke polynomial off primP atp. Ifthe nebentypus off primP is non-trivial thenf primP = fP . In particular,fP is of levelexactlyNp.

Let

X adm = {P ∈ X (I ) | ∃ k = k(P ) ≥ 2, k ≡ 1 (mod p − 1), P ∈ Xk(I )}.

Then the setX adm is Zariski dense inX (I ). Therefore, the elementDH ∈ I (ψ) isdetermined by its values at points inX adm. For anyP ∈ X adm, letH(P ) ∈ OP

denote the reduction ofH moduloP . Let P ∈ X adm; let Tf ,P = H(P ) · 1fP ∈hok(Np,O), wherek = k(P ). Let 2r = k− �−m. Observe that by definition of


ψ : [z] �→ z�+mp ·ξgξh(z) · [z2p], the image ofP under the twisting mapI → I (ψ)is abovePk−�−m

2= Pr ∈ ΛO. Hence, by definition (1.4.5), we have

1.5.2.1

DH(f , g, h)(P ) = �fP

(e

(h ·∫Z

xrdµg

))=⟨e

(h ·∫Z

xrdµg

), Tf ,P

⟩= ⟨e(h · drgp) , Tf ,P

⟩= �fP (e(h · drgp))

by compatibility of the pairings (1.2.5) and (1.2.6).TheMaass operatorsδr� , r = 1,2, . . . , defined by Maass and Shimura, are

the differential operators on the upper half plane given by the formula

δ� = 1

2πi

(�

2iy+ d

dz

); δr� = δ�+2r−2 ◦ · · · ◦ δ�+2 ◦ δ�.

For any congruence subgroupΓ , δr� takes holomorphic cusp forms of weight� for Γ to C∞ functions on the upper half plane, rapidly decreasing at infinityand “nearly holomorphic" in Shimura’s sense [S], which transform underΓ likemodular formsofweight�+2r.We refer to such functions as nearly holomorphiccusp forms. Iffi are nearly holomorphic cusp forms of weightsmi , i = 1,2,then the productf1f2 is a nearly holomorphic cusp form of weightm1 +m2.

If G is a nearly holomorphic cusp form of weightk, then the holomorphicprojectionH(G) is the unique holomorphic cusp form of weightk that satisfies

〈G, f 〉k = 〈H(G), f 〉kfor all holomorphic cusp formsf . LetG = h · δr�g andGp = h · δr�gp. It followsfrom [H1, I,p. 185; II, Lemma 6.5, (iv)] that

e(h · drgp) = e(H(Gp)).

Thus, returning to formula (1.5.2.1), we find that

DH(f , g, h)(Q) = �f ,P ◦ e(H(Gp)).Finally, appealing to [H1, I,prop. 4.5, II, 7.6] , we find that

(1.5.2.2) DH(f , g, h)(Q) = H(P ) · α(f primP )−1 · pk−1 ·

⟨Gp, fP (pz)

⟩p2,k⟨

fP , fP⟩p2,k

.


2. Triple product L-functions

2.1. A formula for the central critical value

We retain the notation of the previous section. Letf ∈ Sk(N,R), g ∈ S�(N,R),h ∈ Sm(N,R) be three modular forms of levelN , with k ≥ � ≥ m. We writetheir standard HeckeL-functions as follows:

L(s,?) =∏

(q,N)=1

Lp(s,?)×∏q|NLq(s,?), ?= f, g, h

where, for(q,N) = 1 the local Euler factors are of the form

(2.1.1)

Lq(s, f ) = [(1− α1,qq−s)(1− α2,qq−s)]−1,

Lq(s, g) = [(1− β1,qq−s)(1− β2,qq−s)]−1,

Lq(s, h) = [(1− γ1,qq−s)(1− γ2,qq−s)]−1,

Here ourL-functions are normalized so that|αi,q | = qk−12 , |βi,q | = q

�−12 ,

|γi,q | = qm−12 , i = 1,2, for any archimedean absolute value. The triple product

L-function is the convolution of these three:

2L(s, f, g, h) =∏

(q,N)=1

∏i,i′,i′′=1,2

(1− αi,qβi′,qγi′′,qq−s)

−1

×∏q|NLq(s, f, g, h)(2.1.2)

where the factorsLq(s, f, g, h) for q|N are the local ArtinL-factors of thecorresponding Weil-Deligne group representations, defined by reference to thelocal Langlands correspondence forGL(2).

In what follows we restrict attention to the caseN = 1, i.e., we assume ourforms are all of level 1. This implies in particular that the weightsk, �,m areall even. We assume thatf , g, andh correspond to cuspidal automorphic rep-resentationsπ(f ), π(g), andπ(h), respectively, ofGL(2)Q, with trivial centralcharactersξ(f ), ξ(g), ξ(h), respectively. denote the respective central charac-ters.

The analytic continuation of the triple productL-function has been proved bythe method of Langlands-Shahidi [Sha] and by a variant of the Rankin method,due to Garrett [G1,G2] and generalized by Piatetski-Shapiro and Rallis [PSR].It is known to satisfy a functional equation of the usual type, relating the valuesat s andw + 1− s, wherew = k + �+m− 3.

We assume henceforward that

(2.1.3) k ≥ �+m.


Under hypothesis (2.1.3), a formula is obtained in [HK1] – the Main Identity 9.2– for the central critical valueL

(w+12 , f, g, h

)of the triple productL-function.

The value is expressed as an integral of theta lifts off, g, h to the orthogonalgroup attached to the split quaternion algebraM(2)Q overQ; i.e. to the splitform of O(4). The exact formula depends on several auxiliary choices. LetH

denote the algebraic group(GL(2) × GL(2))/d(Gm), whered is the diagonalembedding. ThenH is naturally isomorphic to the identity component of thegroup of orthogonal similitudes of the split quaternion algebra. We letS be thespace of Schwartz-Bruhat functions onM(2)(A). To anyφ ∈ S that satisfiesappropriate finiteness properties (K-finite for a maximal compact subgroupKof GL(2,R) × O(2,2) with respect to the Weil representation) and any auto-morphic formF onGL(2,Q)\GL(2,A), the theta correspondence associatesan automorphic formθφ(F ) onH(Q)\H(A) (see [HK2], (5.1.12) for the preciseformula, which also depends on the choice of a measure, specified in [HK1]).

Let r = k−�−m2 , which by (2.1.3) is a positive integer. Letf ι be the normal-

ized newform whose Hecke eigenvalues are the complex conjugates of those off . In fact f ι = f , sinceN = 1, but we leave the notationf ι with a view tofuture generalizations. Thenf

ιis an antiholomorphic form with the same Hecke

eigenvalues asf ; in other words,fιlifts to an elementf ! of π(f ). Similarly,

let g!(i) andh! be liftings of δi�(g), 0 ≤ i ≤ r andh, respectively, to auto-morphic forms onGL(2,Q)R×

+\GL(2,A); i.e., to elements ofπ(g) andπ(h),respectively. Here we are using the fact that the Maass operators correspond toelements ofLie(GL(2)), cf. [HK1, Lemma 12.5] and the references cited there.It follows from [HK1, Theorem 7.2] that, for appropriate choices ofφi ∈ S,i = 1,2,3, we can arrange that

(2.1.4) θφ1(f!) = f !; θφ2(g!(0)) = g!(r); θφ3(h!) = h!

We abbreviateΦ = (φ1, φ2, φ3), F = (f !, g!(r), h!). Let dµ be theGL(2,A)-invariant Haar measure onA× · GL(2,Q)\GL(2,A) with total volume 1.Then we have the following formula:

MAIN IDENTITY. ([HK1, 9.2]):

(2.1.5) Z∞(F,Φ) · ·L(w + 1

2, f, g, h

)= 2ζ(2)2 · I (f !, g!(r), h!)2,

where

(2.1.6) I (f !, g!(r), h!) =∫A×·GL(2,Q)\GL(2,A)

f ! · g!(r) · h!dµ

Hereζ(2) is the value ats = 2 of the Riemann zeta function, andZ∞(•, •)is the value ats = 0 of the normalized local zeta integrals, defined by Garrettand Piatetski-Shapiro-Rallis. The nature ofZ∞(•, •) will be discussed in thenext section; here wemerely remark that the notation of [HK1] has been slightlysimplified in the present account.


2.2. p-adic interpolation of certain central critical values

The Main Identity (2.1.6) can be rewritten

(2.2.1)

(⟨h · δr�(g), f

⟩k

〈f, f 〉k

)2

= Z∞(F,Φ)2ζ(2)2

L(w+12 , f, g, h

)(〈f, f 〉k)2

.

The left handsideof (2.2.1) hasalmost the same formas the squareof a specialvalue (1.5.2.2) of thep-adic measure constructed in 1.5. The only modificationnecessary is to replacef by the valuefP at a primeP of an ordinary Hida family,and to incorporate the twistfP �→ fP .

WriteG = h · δr�(g) andGp = h · δr�(gp), as in§1.5.2. Letf , fP , andf primP beas in Sect. 1.3. In what follows, we letf = f primP . Recall that we have fixed theauxiliary levelN to be 1. Letα1 = α1(fP ) be thep-adic unit root of the Heckepolynomial off primP atp, and letα2 = α2(fP ) denote its other root. Recall that〈•, •〉pm,k (m ≥ 0) has been defined in (1.3.11).

Proposition 2.2.2.With notations as above, the following formula is valid:⟨Gp, fP

⟩p2,k⟨

fP , fP⟩p,k

= Ep(fP , g, h)

p1− k2α1

(1− α2

α1

) (1− α2

pα1

) ·⟨G, f primP

⟩1,k⟨

f primP , f primP

⟩1,k

.

where

Ep(fP , g, h) = p−k(p2α21 − α2ap)− p2− k+�+m2 α1bpcp

+p1− k+�−m2 b2p + p1−mc2p − α2p1− k+m

2 cp − 1

Proof.The elements of this calculation are certainly well known to specialists.However, we were unable to find a complete comparison of the two sides in theliterature, so we are including all details.

We extend the Petersson inner product〈φ,ψ〉pm,k toC∞ forms of and weightk, level pm with trivial Nebentypus, one of them decreasing rapidly at cusps.Recall thatfP = f − α2 · f |[p] where:

a(p, f ) = α1 + α2, α1α2 = pk−1 andα1 is ap-adic unit, and•φ|[m] = φ(mz) = m− �

2 · φ|(m 00 1

))•

for anyφ of weight� and anym ≥ 1.


From

fP = fP |(0 −1p 0

)using the equality

(0 1

−p 0

)=(0 −11 0

)(p 00 1

), we find

fP (pz) = p− k2 ·(f |(p2 00 1

)− α2 · p− k

2f |(p 00 1

)).

SetA(pm) =(pm 00 1

). We will repeatedly use the following

Lemma 2.2.3.The following formulas are valid: 1) Given anyC∞ formsφ andψ of weightw, level1, withφ eigen forTpm of eigenvalueλpm :

(i) 〈φ,ψ |w A(pm)〉pm,w = pm(1−w2 ) · λpm 〈φ,ψ〉1,w .

(ii) 〈φ,ψ〉pm,w = [Γ0(pm) : SL(2,Z)] · 〈φ,ψ〉1,w .(iii) 〈φ|w A(p), ψ |w A(p)〉p,w = (p + 1) 〈φ,ψ〉1,w .

2) Similarly, ifψ has levelp andφ level1 and is eigen forTp, one has:

(iv) 〈φ,ψ |w A(p)〉p2,w = p1−w2 · λp · 〈φ,ψ〉1,w − < φ|wA(p), ψ >p,w .

Proof (of Lemma 2.2.3).We proceed as in [PR,4.2 or H1, II, Lemma 5.3] byobserving

SL(2,Z) A(pm)Γ0(pm) = SL(2,Z) A(pm)

and, ifφ,ψ ∈ Sk(SL(2,Z)) andγ ∈ GL(2,Q) has positive determinant,⟨φ,ψ |w[SL(2,Z)γ Γ0(pm)]

⟩pm,w

= ⟨φ|w[Γ0(pm)γ ιSL(2,Z)], ψ

⟩1,w .

Here γ �→ γ ι is the main involution

(a b

c d

)ι=(d −b

−c a). Moreover, one

checks that if

Γ0(pm)

(pm 00 1

)SL2(Z) =

∐i

Γ0(pm)αi

then

SL2(Z)

(pm 00 1

)SL2(Z) =

∐i

SL2(Z)αi

and sinceφ has level 1, one finds

pm(w2 −1)

⟨φ,ψ |k A(pm)

⟩pm,w

=< φ|Tpm, ψ >1,w= λp 〈φ,ψ〉1,k ,which proves (i).


Next, assertion (ii) is obvious, and (iii) is similar, whenΓ0(pm) is replacedbyA(pm)Γ0(pm)A(pm)−1, which has the same index inSL(2,Z).

For assertion (iv), one observes the equality of sets

Γ0(p)A(p) = Γ0(p)A(p)Γ0(p2)

then one uses the adjunction formula for

[Γ0(p)A(p)Γ0(p2)]together with the fact thatΓ0(p2)A(p)ιΓ0(p) andUp = Γ0(p)A(p)

ιΓ0(p) admita same set of representatives.

Step 1: Computation of⟨Gp, fP

⟩p2,k

We have⟨Gp, fP

⟩p2,k

= p−k2 · 〈f |A(p2)− α2 · p− k

2f |A(p), h · δr�(g|ιp)〉p2,kwhere, ifg = ∑

n≥1 bnqn, one hasg|ιp = ∑

(n,p)=1 bnqn andr = k − �−m.

Let us observe that sinceg is a Hecke eigenform,

g|ιp = g|(1− Tp[p] + p�−1[p2])Therefore,

(2.2.4) h·δr�(g|ιp) = h·δr�g−bpp− �2h·δr�

(g|(p 00 1

))+p−1h·δr�

(g|(p2 00 1

))Recall by the way that

δr� (g|�α) = (δr�g

) |�+2rα.

Now, by substituting (2.2.4) in the inner product

〈f |A(p2)− α2 · p− k2f |A(p), h · δr�(g|ιp)〉p2,k

one obtains a sum of six terms that we compute separately. LetG = h · δr�g.T1 = 〈f |A(p2),G〉p2,k•

SinceG has level 1, we can apply Lemma 2.2.3 (i); one finds

T1 = p2−k · 〈f,G|Tp2〉1,k = p2−k(a2p − pk−1) · 〈f,G〉1,k

T2 = −α2p− k2 〈f |A(p),G〉p2,k•

Observe

〈f |A(p),G〉p2,k = p · 〈f |(p 00 1

),G〉p,k


Then, by the same reasoning as above, one has

T2 = −α2p− k2p · p1− k

2 · 〈f,G|Tp〉1,kso,

T2 = −α2p2−kap · 〈f,G〉1,k

T3 = −bp · p− �2 · 〈f |A(p2), h · δr�g|A(p)〉p2,k•

One rewritesT3 as

−bp · p1− �+m2 ·

⟨(f |(p 00 1

)δr�gy

k−m)

|Γ0(p)(p 00 1

)Γ0(p

2), h

⟩p2,m

By Lemma 2.2.3 (iv), one gets

pm2 −1〈φ|A(p), h〉p2,m = 〈φ, h|Up〉p,m

whereh|Up = h|Tp − pm2 −1 · h|A(p). Thus, one has

T3 = −p1− �+m2 bpcp · 〈f |A(p) · δr�gyk−m, h〉p,m

+p− �2bp〈f |A(p) · δr�gyk−m, h|A(p)〉p,m

and finally,

T3 = −p2− k+�+m2 apbpcp · 〈f,G〉1,k + p1− k+�−m

2 b2p · 〈f,G〉1,k

T4 = α2bpp− �+k

2 · 〈f |A(p), h · δr�g|A(p)〉p2,k•We rewrite it as

α2bpp− �+k

2 · 〈(f δr�gyk−m) |A(p), h〉p2,m.That is,

T4 = α2bpcpp2− k+�+m

2 · 〈f,G〉1,k

T5 = p−1 · 〈f |A(p2), h · δr�g|A(p2)〉p2,k.•By the same calculation, we get

T5 = p1−m(c2p − pm−1) · 〈f,G〉1,k

T6 = −α2p−1− k2 · 〈f |A(p), h · δr�g|A(p2)〉p2,k•


which is equal to

−α2p−1− k2p1−

m2 〈(f δr�gyk−m) |A(p), h〉p2,m.

Hence by adjunction

T6 = −α2p−1− k2

[p1−

m2 cp〈f δr�gyk−m, h〉p,m − 〈f δr�gyk−m, h|A(p2))〉p,m

]Note that〈f δr�gyk−m, h〉p,m = p · 〈f δr�gyk−m, h〉1,m and

〈f δr�g|A(p) · yk−m, h|A(p)〉p,m = 〈f, (δr�g · h) |A(p)〉p,k = p1−k2ap · 〈f,G〉1,k

Therefore,

T6 = −α2p1− k+m2 cp · 〈f,G〉1,k + α2app−k · 〈f,G〉1,k

The sum of the termsTi (i = 1, . . . ,6) is the product of〈f,G〉1,k byp2−k(a2p−pk−1)−α2p2−kap−p2− k+�+m

2 apbpcp+p1− k+�−m2 b2p+α2bpcpp2−

k+�+m2 +

+p1−m(c2p − pm−1)− α2p1− k+m2 cp + α2app−k

That is,

(2.2.5)

⟨Gp, fP

⟩p2,k

〈G, f 〉1,k= Ep(fP , g, h)

Step 2: Computation of⟨fP , fP

⟩p,k⟨

fP , fP⟩p,k

= 〈f, f |kA(p))〉p,k − α2 · p k2 〈f |kA(p), f |kA(p)〉p,k

−α2 · p k2 〈f, f 〉p,k + α22 · pk 〈f |kA(p), f 〉p,k .

It follows from Lemma 2.2.3 that⟨fP , fP

⟩p,k

〈f, f 〉1,k= p− k

2 (p · ap − 2(p + 1)α2 + p1−kα22 · ap).

Sincef is of level 1,ap = ap andα1 · α2 = pk−1. Thus,

(2.2.6)

⟨fP , fP

⟩p,k

〈f, f 〉1,k= p1−

k2α1

(1− α2

α1

)(1− α2

pα1

).

Proposition 2.2.2 now follows immediately by combining (2.2.5) and (2.2.6).


Recall we put

Ep(fP , g, h) = p−k(p2α21 − α2ap)− p2− k+�+m2 α1bpcp

+p1− k+�−m2 b2p + p1−mc2p − α2p1− k+m

2 cp − 1

LetS(P ) =(1− α2

α1

) (1− α2

pα1

). It follows from Lemma2.2.2 that the right-

hand side of (1.5.2.2) equals

(2.2.7) H(P ) · α−21 p

k−2Ep(fP , g, h)S(P )

· 〈G, f 〉1,k〈f, f 〉1,k

.

Let

K(P ) = α−21 p

k−2Ep(fP , g, h)S(P )

Combining (1.5.2.2) with (2.2.1), we then obtain our main result.

Theorem 2.2.8.Let f be ap-adic family of ordinary cusp forms, in the senseof 1.3, unramified outsidep. Let g and h be cusp forms of weights� andm,respectively, of level1. LetH be an annihilator of the congruence module at-tached tof , and letDH(f , g, h) be the generalizedp-adic measure constructedin Sect. 1.4. For any integerk ≥ 2, k ≡ 1 (mod p − 1) and forP ∈ Xk(I ),the value ofDH(f , g, h) at P is related to the central critical values = w+1

2 of

L(s, f primP , g, h) by the following formula:

(2.2.9)

(DH(f , g, h)(P )H(P ) ·K(P )

)2

= Z∞(F,Φ)2ζ(2)2

L(w+12 , f

prim

P , g, h)

(⟨f primP , f primP

⟩k

)2 .

2.3. Refinement of the main formula

In order to compare the results described in themain formula to accepted conjec-tures onp-adicL-functions, or to formulate reasonable conjectures regarding thesquare roots ofp-adicL-functions along “anti-cyclotomic" variables, it wouldbe necessary to determine thep-adic behavior of the archimedean zeta integralZ∞(F,Φ) as the weightk varies. The local nature of the calculations in [HK1]makes it clear thatZ∞(F,Φ) depends only on the weightsk, �,m. Our choicesof archimedean data are dictated by thep-adic construction, and a full calcu-lation of the archimedean integral would require summingr = k−�−m

2 separateterms for givenk. Ikeda has recently computed archimedean triple product zetaintegrals under very general hypotheses [I]. In the casek ≥ �+m his inputs arenot quite the same are ours, but his techniques may be applicable to determineZ∞(F,Φ) explicitly.We note thatZ∞(F,Φ) has been determined up to rational


multiples in [HK1]. Bearing in mind the slightly different normalization used in[HK1], we find that

Z∞(F,Φ)π4−2k

∈ Q×.

Sinceζ(2) = π2

6 , we recover the statement of the introduction.We have restricted attention to formsf , g, andh of level 1. Allowing rami-

fication at primes different fromp will modify the final formula. We may treatbad finite placesv as we have treated the infinite place, choosing local Schwartz-Bruhat functionsφi,v. Then nothing will change on the right-hand side of theMain Identity (2.1.6), but the left-hand side will include additional zeta integralsZv(F,Φ), reflecting these choices. Just as in the archimedean case, these localzeta integrals will depend only on the local components of the automorphic rep-resentations associated tof , g, andh. The qualitative variation of these ramifiedcomponents in a Hida family has been determined by Hida [H3, pp. 129-133](the variation in a Hida family attached to Hecke characters of an imaginaryquadratic field can be seen quite explicitly). In any case, for fixed conductorN ,the number of distinct possible bad non-archimedean components is finite, so thepossible denominators created by the local integralsZv(F,Φ) remain bounded.

Our restriction to level 1 is more serious at the primep. Requiring thatfP beunramified atp for all P amounts to restricting attention to a single branch of theHida family f , namely to thosek congruent toa(f ) (modp− 1). In general, onewants to allow the conductor offP to be divisible byp but not byp2. Removingthe restriction thatfP be unramified atp only makes sense if we also allowg andh to have conductor divisible byp. In that case, the Main Identity will involvea zeta integralZp(F,Φ), which might introduce additionalp-adic zeroes orpoles. Explicit determination of such a local integral is also extremely difficult.The case of three special representations is considered in the article [GK] ofGross and Kudla; its explicit calculation is one of the most intricate in the theoryof L-functions.

Anyone who successfully undertakes these calculations should find it easy,using the methods of this paper, to construct ap-adic measure in three variables(allowingf , g, h and the nebentypus characters to vary, always subject toξ(f ) ·ξ(g) · ξ(h) = 1), whose moments interpolate the square roots of normalizedcentral critical values of triple productL-functions.

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