A p-adic Waldspurger formula - Yale Universityyl2269/logarithm.pdfA p-ADIC WALDSPURGER FORMULA 745 By a theorem of Saito and Tunnell (see [24], [28]), either both spaces have dimen-sion

A p-ADIC WALDSPURGER FORMULA

YIFENG LIU, SHOUWU ZHANG, and WEI ZHANG

AbstractIn this article, we study p-adic torus periods for certain p-adic-valued functions onShimura curves of classical origin. We prove a p-adic Waldspurger formula for theseperiods as a generalization of recent work of Bertolini, Darmon, and Prasanna. Inpursuing such a formula, we construct a new anti-cyclotomic p-adic L-function ofRankin–Selberg type. At a character of positive weight, the p-adic L-function inter-polates the central critical value of the complex Rankin–Selberg L-function. Its valueat a finite-order character, which is outside the range of interpolation, essentiallycomputes the corresponding p-adic torus period.

Contents1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7432. Arithmetic of quaternionic Shimura curves . . . . . . . . . . . . . . . . 7533. Statements of main theorems . . . . . . . . . . . . . . . . . . . . . . . 7844. Proofs of main theorems . . . . . . . . . . . . . . . . . . . . . . . . . 797Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 816A. Compatibility of logarithm and Coleman integral . . . . . . . . . . . . . 816B. Serre–Tate local moduli for O-divisible groups (following N. Katz) . . . . 817References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 831

1. IntroductionThe aim of this article is to generalize a recent formula of Bertolini, Darmon, andPrasanna in [1] which relates the p-adic logarithm of Heegner points in Abelianvarieties parameterized by the modular curve X0.N / and certain p-adic L-valuesat a point outside its range of interpolation, for a prime p split in the imaginaryquadratic field. The paper [1] works in the same setting as the Gross–Zagier formula(see [12]) under the Heegner hypothesis. Prior to the Bertolini–Darmon–Prasanna for-mula, Rubin [23] obtained a similar formula for elliptic curves with complex multipli-

DUKE MATHEMATICAL JOURNALVol. 167, No. 4, © 2018 DOI 10.1215/00127094-2017-0045Received 19 October 2014. Revision received 31 August 2017.First published online 9 February 2018.2010 Mathematics Subject Classification. Primary 11G40; Secondary 11F67, 11G18, 11J95.

743

https://doi.org/10.1215/00127094-2017-0045

744 LIU, ZHANG, and ZHANG

cation, and after the Bertolini–Darmon–Prasanna formula, Brooks [4] also obtained asimilar formula allowing the modular curve to be a rational Shimura curve.

Our formula is for the general case concerning Heegner points on Abelian vari-eties parameterized by Shimura curves over a totally real number field F , for a primep of F split in a CM fieldE . Even in the case in which F DQ, our result is new sincewe remove all ramification restrictions from [1] and [4]. Moreover, we will place ourformula in the setting of the Waldspurger formula (see [29], [30]) which compares theglobal torus periods of automorphic forms with products of global central L-valuesand local torus periods. More precisely, we will define the relevant p-adicL-function,introduce the notion of p-adic Maass functions and their torus periods, and comparethem with products of p-adic L-values and local torus periods. For practical applica-tions of our formula, one may need a formula for local torus periods of Gross–Prasadtest vectors. Fortunately, this formula was worked out recently by Cai, Shu, and Tian[6].

To construct the p-adic L-function and prove our p-adic Waldspurger formula,we study the congruence relation for both global (torus) periods and local (torus)periods appearing in the complex Waldspurger formula. A key ingredient of our con-struction is the existence of action of the Lubin–Tate formal group on Shimura curvesat the infinite level; this allows us to use p-adic Fourier analysis from [26].

In the rest of this section, we will sketch our construction and the proof for theformula in the case of elliptic curves over Q. To be consistent with the notation in themain body of the article, we fix (1) an elliptic curve A over Q, (2) an indefinite quater-nion algebra B over Q, and (3) an imaginary quadratic field E embedded into B .

As usual, put A D R � bQ as the ring of adèles of Q, and put AE WD A˝Q E .By the modularity theorem, the elliptic curve A determines an irreducible cuspidalautomorphic representation … of GL2.A/. We assume that this representation hasa nontrivial Jacquet–Langlands correspondence �C to B�, uniquely realized on asubspace of AC.B

�/—the space of automorphic forms on B�n.B ˝Q A/�.

1.1. Complex Waldspurger formulaFirst let us review the (complex) Waldspurger formula (see [29], [30]) for the cusp-idal automorphic representation �C of .B ˝Q A/�. Let � W E�A�nA�E ! C� be anautomorphic character. Then we can form the (torus) period integrals

PC.�;�˙1/ WD

ZE�A�nA�

E

f .t/�˙1.t/ dt; � 2 �C: (1.1)

Here we adopt the Haar measure such that the total volume of E�A�nA�E is 2. Weconsider these integrals as elements in the linear dual of representation spaces asfollows:

PC.�; �/ 2HomA�E.�C˝ �;C/; PC.�; �

�1/ 2HomA�E.�C˝ �

�1;C/:

A p-ADIC WALDSPURGER FORMULA 745

By a theorem of Saito and Tunnell (see [24], [28]), either both spaces have dimen-sion 1 or they have dimension 0. Suppose that we are in the first case. Although wedo not know how to construct a canonical basis in either space, we do know how toconstruct a canonical one in their tensor product. Namely, we have the element

˛DYv�1

˛v 2HomA�E.�C˝ �;C/˝C HomA�

E.�C˝ �

�1;C/

defined via the integration of local matrix coefficients

˛v.�1; �2I�/ WDL.1; �v/L.1;…v;Ad/

�v.2/L.1=2;…v; �v/

ZQ�v nE

�v

��C.t/�1; �2

�v�v.t/ dt; (1.2)

where �DQv �v is the quadratic character corresponding to the quadratic field exten-

sion E=Q, and .�; �/DQv.�; �/v is the bilinear Petersson inner product pairing on �C

defined by the Haar measure on .B ˝Q A/� such that the total volume of B�A�n.B˝Q A/� is 2. It was proved by Waldspurger [29, Section 3] that ˛ is in fact a finiteproduct for every pair of test vectors .�1; �2/.

Thus, there is a unique constantƒ.�C; �/ 2C, depending only on �C and �, suchthat

PC.�; �/ �PC.�; ��1/Dƒ.�C; �/ � ˛.�; �I�/:

The Waldspurger formula gives an expression for ƒ.�C; �/ in terms of the Rankin–Selberg central value ƒ.1=2;…;�/.

THEOREM 1.1.1 (Waldspurger)We have

ƒ.�C; �/DƒQ.2/

2ƒ.1; �/ƒ.1;…;Ad/ƒ.1=2;…;�/:

In other words, for every pair of vectors �1; �2 2 �C, we have

PC.�1; �/PC.�2; ��1/D

ƒQ.2/ƒ.1=2;…;�/

2ƒ.1; �/ƒ.1;…;Ad/� ˛.�1; �2I�/:

Remark 1.1.2In the above theorem,ƒ stands for complete globalL-functions, that is, those as prod-ucts of local L-functions over all places. However, in the main body of the article, weuse global L-functions that are products of local L-functions over non-Archimedeanplaces, which will be denoted by L (except for �F .s/ with F a number field). Forexample, if �1.z/D .z= Nz/˙k with k � 1, then we have


ƒ.�C; �/DkŠ.k � 1/Š

.2�/2k�1��Q.2/L.1=2;…;�/

2L.1; �/L.1;…;Ad/:

It is a simple computation using the formulas in, for instance, [21, Lemma 2.3].

Remark 1.1.3Note that, unlike our unified choice of the Tamagawa measure on A�nA�E , in [29]and [30] the Haar measure in (1.1) has volume 1 on E�A�nA�E , and the productHaar measure in (1.2) has volume 2ƒ.1; �/ on E�A�nA�E . Therefore, the constantƒ.�C; �/ in their formulas differs from ours by 4ƒ.1; �/.

1.2. p-Adic Maass functionsFrom now on, we fix a prime p and equip B with an isomorphism B˝R'Mat2.R/.For each (sufficiently small) open compact subgroup U of .B ˝Q

bQ/�, the doublequotient

B�n.C nR/� .B ˝QbQ/�=U (1.3)

is the set of complex points of a Shimura curve XU defined over Q. The curve XU issmooth over Q, and it is proper if and only if B is division. We put X D lim

�UXU as

a scheme over Q with a right action of .B ˝QbQ/� under which XU DX=U .

We say that a function � W X.Cp/! Cp is a p-adic Maass function on X if itis the pullback of some locally analytic function XU .Cp/! Cp on XU . Denote byACp .B

�/ the Cp-vector space of all p-adic Maass functions on X . It is a represen-

tation of .B ˝QbQ/�.

Denote by �Cp the subspace of ACp .B�/ spanned by functions of the form

f � log! W X.Cp/f�!A.Cp/

log!��!Cp;

where f W X ! A is a nonconstant map, ! is a differential form on A˝Q Cp , andlog! is the p-adic logarithm map (see, e.g., [3]). The subspace �Cp �ACp .B

�/ is a

subrepresentation of .B ˝QbQ/�. Thus, on one hand we have a complex realization

�C, and on the other hand we have a p-adic realization �Cp . They are related as

follows. For every isomorphism � W Cp��!C, we have a canonical isomorphism

�Cp ˝Cp ;� C��! �

.2/C ; (1.4)

where �.2/C � �C is the subspace of weight 2 forms. It sends f � log! to f ��!, whichis well defined. The latter is a differential form on X ˝Q C and, hence, induces anelement in AC.B

�/.


1.3. p-Adic torus periodsFrom now on, we also fix an embedding E � Cp and an isomorphism � W Cp

��! C.

Then we have an induced isomorphism E ˝Q R ' C. We assume that the isomor-phism B˝R'Mat2.R/ is chosen such that the induced embedding C'E˝QR!

B ˝R'Mat2.R/ is the standard one sending xC iy to . x y�y x /.

We proceed exactly as in the complex Waldspurger formula. Let � W E�bQ�nbE�!C�p be a finite-order character. Then parallel to (1.1), we can define the p-adic(torus) period integral as

PCp .�;�˙1/ WD

ZE�bQ�nbE� �

��1Œ˙i; t �

��˙1.t/ dt; � 2 �Cp : (1.5)

Here we have used the double coset presentation (1.3) of X.C/ and adopt the Haarmeasure on E�bQ�nbE� of total volume 2. Note that the above integrals are actuallyfinite sums and, respectively, induce elements

PCp .�; �˙1/ 2HombE�.�Cp ˝ �

˙1;Cp/:

Similar to the complex case, both spaces HombE�.�Cp ˝ �˙1;Cp/ have the same

dimension—either 1 or 0. Suppose that they have dimension 1. Now we construct abasis of their tensor product. For �1; �2 2 �Cp , we define

˛0.�1; �2I�/DYv<1

��1˛v.��1; ��2I ��/;

where ˛v is the same as (1.2). Here, by abuse of notation, �� denotes the image of �under the map (1.4). Then ˛0 is a basis of HombE�.�Cp ˝ �;Cp/˝HombE�.�Cp ˝

��1;Cp/. The invariant pairingQv<1.�; �/v we use in the definition of ˛0 is the one

such thatQv<1.��1; ��2/v is equal to the (bilinear) Petersson product of ��1 and

�C..1�1 /1/��2.

Thus, there is a unique constant L.�Cp ; �/ 2Cp , depending only on �Cp and �,such that

PCp .�; �/ �PCp .�; ��1/D L.�Cp ; �/ � ˛

0.�; �I�/:

Our main objective is to give a formula for L.�Cp ; �/, which we call the p-adicWaldspurger formula, under the only assumption that p splits in E .

Thus, from now on we assume that p splits in E . Denote by P the place of Einduced by the default embedding E �Cp , and denote by Pc the other one above p.

1.4. p-Adic charactersPut G D E�bQ�nbE�, which is a profinite group. Denote by OG the continuous dualover Qp . In other words, for every complete (commutative) Qp-algebra R, OG.R/


is the set of all continuous characters from G to R�. Then OG is represented by a(complete) Qp-algebra D.G/. Thus, there is a universal character ı W G!D.G/�

such that composing with ı induces a bijection

Hom�D.G/;R

�' OG.R/ (1.6)

for every complete Qp-algebra R, where Hom is taken in the category of topologicalQp-algebras.

The place P induces an injective homomorphism Z�p ,!G. We say that a char-

acter � 2 OG.R/ has weight w 2 Z if �jV is the wth power homomorphism for somesubgroup V � Z�p of finite index. For a character � W G! C�p of weight w, there isa standard way to attach an automorphic character �.�/ W E�A�nA�E ! C� under �;in fact, �.�/ is the unique automorphic character satisfying (1) �.�/jbE�;p D � ı �jbE�;pand (2) �.�/1.z/D .z= Nz/w for z 2C'E ˝Q R.

A character � 2 OG.Cp/ induces a homomorphism D.G/ ˝Qp Cp ! Cp via(1.6), and we denote its kernel by I�, which is a closed ideal of D.G/˝Qp Cp . Put

D.GI�Cp /DD.G/˝Qp Cp=\

�2„.�Cp /

I�;

where „.�Cp / is the set of all � such that dim HombE�;p .�Cp ˝ �;Cp/D 1. In par-ticular, elements in D.GI�Cp / can be viewed as functions on „.�Cp / valued in Cp .

1.5. p-Adic Waldspurger formulaOur first theorem is about the existence of a p-adic L-function interpolating valuesƒ.�C; �

.�//, which appeared in Theorem 1.1.1, for � of positive weight.

THEOREM 1.5.1There is a unique element L.�Cp / 2D.GI�Cp / such that, for every � 2„.�Cp / ofweight k � 1, we have

�L.�Cp /.�/Dƒ.�C; �.�// � 2�2k�1 �

.1=2; ;…p ˝ �.�/Pc/

L.1=2;…p ˝ �.�/Pc/2

:

Here, W Qp!C� is the standard additive character.

Remark 1.5.2We have the following remarks concerning the above theorem.(1) By the theorem of Saito and Tunnell, a character � 2 OG.Cp/ of an integer

weight belongs to „.�Cp / if and only if, for every finite place v of Q other

than p, we have .1=2;…v; �.�/v /D �v.�1/.Bv/, where .Bv/ is the Hasse

invariant.


(2) The uniqueness part is clear, since the subset of characters in „.�Cp / of pos-itive weight is dense in „.�Cp /.

Using this p-adic L-function, we can answer the question at the end of Sec-tion 1.3 about the ratio L.�Cp ; �/.

THEOREM 1.5.3Let � 2„.�Cp / be a finite-order character, that is, � has weight 0. Then we have

L.�Cp ; �/DL.�Cp /.�/ � ��1� L.1=2;…p ˝ �

.�/Pc/2

.1=2; ;…p ˝ �.�/Pc/

�:

In other words, for every pair of vectors �1; �2 2 �Cp , we have

PCp .�1; �/PCp .�2; ��1/DL.�Cp /.�/ � �

�1� L.1=2;…p ˝ �

.�/Pc/2

.1=2; ;…p ˝ �.�/Pc/

��˛0.�1; �2I�/:

Theorems 1.5.1 and 1.5.3 follow from the more general context of Theorems3.2.10 and 3.4.4. See Remark 3.4.5 for the reduction process.

1.6. Main ideas of the proofsWe now explain the main ideas of our proofs. The same ideas work for the generalcase as well. There are three major steps in the proofs of our main theorems:(1) construct universal torus periods;(2) construct universal matrix coefficient integrals;(3) construct the p-adic L-function.

For (1), by a universal torus period, we mean an element in D.GI�Cp / such thatit specializes to Waldspurger periods at characters of positive weight. A key ingredientin our construction is a Mellin transform for forms on the Shimura curve with theinfinite Iwahori level structure at p. This seems to be new and matches the philosophythat things look more canonical at the infinite level, which has appeared in some otherworks recently. The Mellin transform of a form f has two variables: the Shimuracurve itself and the weight space. If we restrict the Shimura curve to an arbitraryopen disk which reduces to a point on the special fiber, then we recover the (local)Mellin transform on the Lubin–Tate group from [26]. If we restrict to a classical point(a nonnegative integer, actually) on the weight space, then this recovers an iterationof the Atkin–Serre operator on the Shimura curve.

For (2), by a universal matrix coefficient integral, we mean again an element inD.GI�Cp / such that it specializes to classical matrix coefficient integrals at charac-ters of integral weight. In our construction, we need to choose suitable test vectors in


the representation �Cp and show that the classical matrix coefficient integrals form arigid analytic family. Our key idea is to use the Kirillov model to deal with arbitraryramification at p of �Cp and characters in„.�Cp / for the matrix coefficient integrals.

For (3), by the p-adic L-function, we mean an element in D.GI�Cp / such thatit specializes to complex special L-values appearing in the complex Waldspurger for-mula at characters � of positive weight. The p-adic L-function is defined essentiallyas the ratio of a universal torus period to a universal matrix coefficient integral. Thecomplex Waldspurger formula will imply that this ratio is independent of the choiceof the test vectors. In order to show that we have enough universal matrix coefficientintegrals whose nonvanishing loci cover the entire space, we use a classical result ofSaito and Tunnel on the dichotomy of matrix coefficient integrals and some argumentin rigid analytic geometry. In particular, we need our constructions in (1) and (2) tobe applicable to sufficiently many test vectors.

Finally, to obtain the p-adic Waldspurger formula, that is, the special value for-mula for finite-order characters in terms of the p-adic logarithm of Heegner cycles,we use the multiplicity one property, a property from the global Mellin transform, andslight generalization of Coleman’s work from Appendix A.

1.7. A glance at the general caseIn the main body of the article, we will put ourselves in a more general context. Sinceit is a p-adic theory, we fix a CM number field E inside Cp , with the maximal totallyreal subfield F . Let p be the distinguished place of F induced by the inclusion F �Cp . Recall that an Abelian variety A over F is of GL.2/-type if MA WD End.A/˝Q

is a field of the same degree as the dimension of A.Given a modular Abelian variety A over F of GL.2/-type up to isogeny equipped

with an embedding M WDMA ,! Cp , we will construct a p-adic L-function L.A/

and prove a p-adic Waldspurger formula or, rather, a family of p-adic Waldspurgerformulas for all relevant realizations of A via p-adic Maass functions. Note that Ahas a central character !A W F �nbF �!M�.

The space of all locally Fp-analytic and smooth-away-from-p characters � W E�nbE�!K� with a complete field extension K=MFp such that !A � �jF �n bF � D 1 canbe organized into an ind-rigid analytic variety E over MFp. It has a disjoint uniondecomposition E D EC

È� defined by a certain Rankin–Selberg -factor of A. We

denote by D.A;K/ the coordinate algebra of E�bMFpK for every complete field

extension K=MFp. In Theorem 3.2.10, we construct our p-adic L-function for A asan element

L.A/ 2 .LieA˝FM LieA_/˝FM D.A;MF ltp /;

where MF ltp is the complete subfield of Cp generated by M , the maximal unramified

extension of Fp, and the Lubin–Tate period (see Section 1.8); and FM WD F ˝QM ,


which maps to MF ltp naturally. Our p-adic L-function L.A/ interpolates classical

Rankin–Selberg central critical values for algebraic characters � of positive weightwith respect to an arbitrary comparison isomorphism � W Cp

��!C. In other words, we

will not choose an Archimedean place of F as the theory should be entirely p-adic.In Theorem 3.4.4, we prove a p-adic Waldspurger formula computing p-adic torusperiods of p-adic Maass functions coming from A in terms of special values of L.A/

at finite-order characters.

1.8. Notation and conventionsThe article is self-contained from now on in the sense that if readers would like tostudy the general case directly, they can start from here, and no other preliminar-ies will be used. Throughout the article, we fix a prime p and a CM number fieldE � Cp with F the maximal totally real subfield contained in E . Thus, we obtain adistinguished place p (resp., P) of F (resp., E) above p. We introduce the followingkey (and only) assumption of the article.

Assumption 1.8.1We assume that p splits in E; in other words, Fp DEP.

We introduce the following notation and conventions.� Let g be the degree of F .� Let c 2Gal.E=F / be the (nontrivial) Galois involution.� Denote by A (resp., A1) the ring of adèles (resp., finite adèles) of F . Put

AE DA˝F E and A1E DA1˝F E .� Let �D

Q�v W F

�nA�!¹˙1º be the quadratic character associated toE=F .In particular, we have the L-function L.s; �/D

Qv<1L.s; �v/.

� Let dE 2 Z>0 be the absolute value of the discriminant of E .� We denote by Op the ring of integers of Fp. We denote by F nr

p (resp., F abp ) the

completion of the maximal unramified (resp., Abelian) extension of Fp in Cp

and by Onrp (resp., Oab

p ) its ring of integers. Denote by the residue field ofOnr

p , which is an algebraic closure of Fp .� Denote by F � (resp.,E�) the closure of F � (resp.,E�) in A1� (resp., A1�E ).

Put Oantip DO

�Ep=O�p .

� We write elements t 2 Ep D E ˝F Fp in the form .t�; tı/, where t� 2 Fp

(resp., tı 2 Fp) is the component of t at Pc (resp., P). We fix the followingembedding Ep!Mat2.Fp/ of Fp-algebras:

t 7!

�t�

tı

�: (1.7)


� We adopt ND ¹m 2 Z jm� 0º and write elements in A˚m in columns for anobject A (with a well-defined underlying set) in an Abelian category.

� Denote by J the two-by-two matrix . 0 1�1 0 /.

� For m 2 Z, define the p-Iwahori subgroup of level m of GL2.Op/ to be

Up;m D

²g 2GL2.Op/

ˇg�

�1 �

0 1

�mod pm

³if m� 0;

Up;m D

²g 2GL2.Op/

ˇg�

�1 0

� 1

�mod p�m

³if m< 0:

� We adopt the convention that the local or global Artin reciprocity maps senduniformizers to geometric Frobenii.

� If G is a reductive group over F , we always take the Tamagawa measurewhen we integrate on the adèlic group G.A/. In particular, the total volume ofE�A�nA�E is 2.

� For a relative (formal) schemeX=S , we will simply write�1X instead of�1X=S

for the sheaf of relative differentials if the base is clear from the context. Thetensor product of quasicoherent sheaves on X will simply be denoted as ˝,instead of ˝OX , where OX is the structure sheaf of X .

� Denote by bGm (resp., bGa) the multiplicative (resp., additive) formal group.They have the coordinate T . We denote by bGmŒp

1� the induced (formal) p-divisible group of bGm.

� Denote by LT the Lubin–Tate Op-formal group over Onrp , which is unique

up to isomorphism. We denote by LT Œp1� the induced (formal) Op-divisiblegroup of LT (see Section B.1 if not familiar with the terminology).

� Denote by F ltp � Cp the complete field extension of F nr

p generated by the“period” of the Lubin–Tate group LT (see [26, p. 460]). Its valuation is dis-crete only when FpDQp by [26, Lemma 3.9], in which case F lt

p DQnrp .

� In this article, we will only use basic knowledge about rigid analytic varietiesover complete p-adic fields in the sense of Tate. Readers may use the book[2] for a reference. If X is an L-rigid analytic variety for some complete non-Archimedean field L, we denote by O.X;K/ the complete K-algebra of K-valued rigid analytic functions on X for every complete field extension K=L.

Definition 1.8.2 (Abelian Haar measure)We fix the Haar measure dtv on F �v nE

�v for every place v of F determined by the

following conditions:� When v is Archimedean, the total volume of F �v nE

�v 'R�nC� is 1.

� When v is split, the volume of the maximal compact subgroup of F �v nE�v '

F �v is 1.


� When v is nonsplit and unramified, the total volume is 1.� When v is ramified, the total volume is 2.Then the product measure

Qv dtv equals the product of 2�gd�1=2E L.1; �/ and the

Tamagawa measure (cf. [30, Section 1.6]).

Notation 1.8.3In the main part of the article, we will fix the choices of an additive character WFp!C�p of level 0 and a generator � W LT ! bGm in the free Op-module Hom.LT ;bGm/ of rank 1. Then there are unique isomorphisms

�˙ W Fp=Op

��!LT Œp1�.Cp/ (1.8)

such that the induced composite maps

�.Cp/ ı �˙ W Fp=Op! bGmŒp1�.Cp/�C�p

coincide with ˙, respectively, where C D and � D �1.

Definition 1.8.4Recall from [30, Section 1.2.1] that a coherent/incoherent totally definite quaternionalgebra over A is a quaternion algebra B over A such that the ramification set of B,which is a finite set, contains all Archimedean places and has even/odd cardinality.For such B, put B1 D B˝A A1.

An E-embedding of a totally definite quaternion algebra B over A is an embed-ding

eDY0

vev W A1E D

Y0

v<1E ˝F Fv ,! B1 (1.9)

of A1-algebras. We say that B isE-embeddable if there exists anE-embedding of B.

2. Arithmetic of quaternionic Shimura curvesIn this section, we study some p-adic arithmetic properties of quaternionic Shimuracurves over a totally real field. In Section 2.1, we start from the local theory of somep-adic Fourier analysis on Lubin–Tate groups, following the work of [26]. In Sec-tion 2.2, we study the Gauss–Manin connection and the Kodaira–Spencer isomor-phism for quaternionic Shimura curves. This is followed by a discussion of universalconvergent modular forms in Section 2.3. In particular, we prove Theorem 2.3.17,which is one of the most crucial technical results of the article. In Section 2.4, weprove some results involving comparisons with transcendental constructions undera given complex uniformization. Finally, Section 2.5 contains the proofs of the sixclaims in the previous sections, which require the auxiliary use of unitary Shimuracurves.


2.1. Fourier theory on Lubin–Tate groupsWe use [25] for some terminologies from non-Archimedean functional analysis. LetG be a topologically finitely generated Abelian locally Fp-analytic group—for exam-ple, G D Op, which will be studied later. For a complete field K containing Fp,denote by C.G;K/ the locally convex K-vector space of locally (Fp-)analytic K-valued functions on G, and denote by D.G;K/ its strong dual (see Remark 2.1.1),which is a topological K-algebra with the multiplication given by convolution (see[26, Section 1]). We have a natural continuous injective homomorphism

ı W G!D.G;K/�

sending g 2 G to the Dirac distribution ıg . Moreover, we have D.G;K/bKK 0 'D.G;K 0/ for a complete field extension K 0=K .

Remark 2.1.1We briefly recall the notion of strong dual from [25]. Let V be a locally convex K-vector space, like C.G;K/ above. Denote by L.V;K/ theK-vector space of continu-ous K-linear maps from V to K . For every bound subset B of V (i.e., for every openneighborhood U � V of 0, there exists a 2K such that B � aU ) and an ideal I ofOK , the subset L.B; I / WD ¹f 2L.V;K/ j f .B/� I º is a lattice in L.V;K/. Thenthe strong dual of V is the (topological) K-vector space L.V;K/ equipped with thetopology defined by the family of lattices L.B; I / for all bounded subsets B of V andideals I of OK . When G is compact, there is a more explicit description of D.G;K/on [26, p. 451].

Notation 2.1.2Let B be the generic fiber of (the underlying formal scheme of) LT , which is iso-morphic to the open unit disk over F nr

p . We have a map

˛ W B �SpfF nrp

B!B

induced by the formal group law, and we have a map Op � B ! B denoted by.a; z/ 7! a � z coming from the Op-action on LT . Denote by O.B;K/ the set ofall K-valued rigid analytic functions on B, which is a topological K-algebra.

Definition 2.1.3 (Stable function)A function � 2O.B;K/ is stable ifX

z2KerŒp�

��˛.z; �/

�D 0;


where KerŒp��B.Kac/ is the subset of z such that $ � z D 0 for one and hence alluniformizers $ of Op. We denoted by O.B;K/~ the subspace of O.B;K/ of stablefunctions.

From now on, we will assume that K contains F ltp (see Section 1.8). By [26,

Theorems 2.3 and 3.6] (together with the remark after [26, Corollary 3.7]), we have aFourier transform

F W D.Op;K/��!O.B;K/;

which is an isomorphism of topological K-algebras, with respect to the homomor-phism � W LT ! bGm (Notation 1.8.3).

Remark 2.1.4In fact, the pairing .a; z/ 7! �.a � z/ on Op � B identifies B as the rigid analyticspace parameterizing locally analytic characters of O�p ; and the Fourier transform F

is the unique isomorphism satisfying F .ıa/.z/ D �.a � z/ for z 2 B. In particular,the topological K-vector space O.B;K/ is topologically generated by rigid analyticfunctions �a on B defined by �a.z/D �.a � z/, for a 2Op. See [26] for more details.

Remark 2.1.5We have an action of Op on B coming from the Lubin–Tate group and, hence, anaction of Op on D.Op;K/ via F . More precisely, the action of t 2Op on D.Op;K/

is given by the multiplication of the Dirac distribution ıt .

We identify D.O�p ;K/ with the closed subspace of D.Op;K/ consisting of dis-tributions supported on O�p .

LEMMA 2.1.6We have the following.(1) The isomorphism F restricts to an isomorphism F W D.O�p ;K/

��!O.B;K/~

of topological K-vector spaces.(2) The image of ˛�jO.B;K/~ is contained in O.B;K/~bKO.B;K/~.

ProofBy [26, Section 3], for z 2B.K/, we have a locally analytic character z of Op suchthat z.a/ D �.a � z/ for every a 2 Op, and .z/ D F . /.z/ for 2 D.Op;K/.Moreover, the set of z is dense in C.Op;K/. Let e1 (resp., e0) be the characteristicfunction of O�p (resp., OpnO

�p ), viewed as elements in C.Op;K/.


For (1), we have the identityXz2KerŒp�

zf D .#Op=p/e0f:

Thus, if f is supported on OpnO�p , then f D e0f D .#Op=p/

�1Pz2KerŒp� zf . By

[26, Lemma 4.6.5], for � 2O.B;K/~, we have

¹�;f º D°�; .#Op=p/

�1X

z2KerŒp�

zf±D .#Op=p/

�1° Xz2KerŒp�

��˛.z; �/

�; f±D 0;

where ¹�;f º D F �1.�/.f / and the same for the others. This means that F �1O.B;

K/~ �D.O�p ;K/. On the other hand, if � 2 FD.O�p ;K/, then for an arbitrary f 2C.Op;K/, we have

.#Op=p/�1° Xz2KerŒp�

��˛.z; �/

�; f±D°�; .#Op=p/

�1X

z2KerŒp�

zf±D ¹�; e0f º D 0:

This means that FD.O�p ;K/�O.B;K/~.For (2), we consider the map ˛Š, defined as the following composite map:

D.Op;K/F�!O.B;K/

˛�

�!O.B �SpfF nrp

B;K/'O.B;K/bKO.B;K/

F �1 bF �1��!D.Op;K/bKD.Op;K/!

�C.Op;K/˝K C.Op;K/

�_:

In view of (1), it suffices to show that, for every 2D.Op;K/ and f1; f2 2 C.Op;K/,we have the formula

˛Š .f1˝ f2/D .f1f2/: (2.1)

For this, we may assume that fi D zi for some zi 2B.K/ (i D 1; 2) as the imageof D.Op;K/bKD.Op;K/ consists of continuous linear forms. Then we have, for 2D.Op;K/,

˛Š .z1 ˝ z2/D�.F �1˝F �1/

�˛�F . /

��.z1 ˝ z2/D

�˛�F . /

�.z1; z2/

by [26, Lemma 4.6.3]. But�˛�F . /

�.z1; z2/D F . /

�˛.z1; z2/

�D .˛.z1;z2//D .z1z2/

as z1z2 D ˛.z1;z2/. Thus, (2.1) holds, and (2) follows.

Remark 2.1.7Lemma 2.1.6 implies that the function �a in Remark 2.1.4 is stable if and only if a


belongs to O�p . Moreover, the topologicalK-vector space O.B;K/~ is topologicallygenerated by �a for a 2 O�p . The notion of stable functions is a local avatar of thenotation of stabilization in the theory of p-adic modular/automorphic forms.

In a later argument, we will work on the compact Abelian locally Fp-analyticgroup Oanti

p . Note that we have identified Oantip with O�p via t 7! t=t c. Thus, we have

the following definition.

Definition 2.1.8 (Local Mellin transform)We call the following composite map

Mloc W O.B;K/~!O.B;K/~bKO.B;K/~

idbF �1��!O.B;K/~bFp

D.Oantip ;Fp/;

fulfilled by Lemma 2.1.6, the local Mellin transform.

Remark 2.1.9In fact, the composite map

M0 W O.B;K/!O.B;K/bKO.B;K/idbF �1��!O.B;K/bFp

D.Op;Fp/

is more like an analogue of the classical Mellin transform, as we may regard M0 as amap sending a function on B valued in K to a function on B valued in (K-valued)distributions on the Lie group Op. Recall that the classical Mellin transform M sendsa function � on R�C to a function M.�/ on C. In fact, we may regard M as a mapsending a function � on R�C valued in C to a function x 7!M.f .x��// on R�C valuedin (C-valued) distributions on the Lie group Ga.C/. We have analogies between R�Cand B—both are “spaces with Abelian group structure”—and between Ga.C/ andOp—both are commutative Lie groups. Moreover, the properties in [26, Lemma 4.6]are the analogues of those for the classical Mellin transform.

The continuous map M0 is uniquely determined by the formula M0.�a/D �a˝ıafor a 2Op, where �a is the function in Remark 2.1.4.

Notation 2.1.10For every integer k, we have the character hki W Oanti

p 'O�p !O�p �K sending t to.t=t c/k . It is an element in C.Oanti

p ;K/.

LEMMA 2.1.11For every integer N , the topological K-vector space C.Oanti

p ;K/ is topologicallygenerated by hki for all k �N .


ProofWe may assume that N D 0, since for every k 2 Z, the function hki is the limit offunctions hk0i for k0 0. By [26, Theorem 4.7], every function in C.Op;K/ and,hence, C.Oanti

p ;K/ is the limit of finite linear combinations of polynomials on Op.Thus, the lemma holds for N D 0 and then every N .

Definition 2.1.12 (Lubin–Tate differential operator)We define the Lubin–Tate differential operator ‚ on O.B;K/ by the formula

‚� Dd�

�� dTT

;

where we recall that � is as in Notation 1.8.3 and T is the standard coordinate of bGm.

Example 2.1.13For a 2 Op, we have ‚�a D a�a, where �a 2 O.B;K/ is the function in Remark2.1.4.

The following lemma reveals the relation between the local Mellin transform andthe Lubin–Tate differential operator.

LEMMA 2.1.14Let � 2 O.B;K/~ be a stable function. Then Mloc.�/ is the unique element inO.B;K/~bFp

D.Oantip ;Fp/ satisfying

(1) Mloc.�/.hki/D‚k� for every k � 0; and

(2) ‚Mloc.�/.h�1i/D �.Here hki is introduced in Notation 2.1.10.

ProofThis follows from [26, Lemma 4.6.8] and Lemma 2.1.11.

Definition 2.1.15 (Admissible function)We say that a stable function � 2O.B;K/~ is n-admissible for some n 2N if

��˛.�; z/

�D �.z/�

for every z 2KerŒpn��B.Kac/.

LEMMA 2.1.16Let � 2O.B;K/~ be an n-admissible stable function for some n� 1. Then F �1.�/

is supported on 1C pn. In particular, we have


Mloc.�/�hki�DMloc.�/

��hki

�for every k 2 Z and every (locally constant) character � W Oanti

p !K� that is trivialon 1C pn.

ProofThis again follows from [26, Lemma 4.6.5]. In fact, by a similar strategy to the proofof Lemma 2.1.6(1), it suffices to show thatX

z2KerŒpn�

z.a/�.z/�1 D 0

for a 2O�p n.1Cpn/, where KerŒpn��B.Kac/ is the subset of z such that$n �z D 0

for one uniformizer and, hence, all uniformizers $ of Op. This holds as z.a/ D�.a � z/.

Remark 2.1.17Let n� 1 be an integer. Lemma 2.1.16 implies that the function �a in Remark 2.1.4is n-admissible stable if and only if a belongs to 1C pn. Moreover, the topologicalK-vector space of n-admissible stable functions is topologically generated by �a fora 2 1C pn.

2.2. Shimura curves and Kodaira–Spencer isomorphismLet B be a totally definite incoherent quaternion algebra over A equipped with an iso-morphism Bp 'Mat2.Fp/. Then we have the system of (noncompactified) Shimuracurves ¹X.B/U ºU indexed by (sufficiently small) open compact subgroups U of B1�

associated to B over SpecF (see, e.g., [30, Section 1.2.1]). More precisely, X.B/U isthe scheme over SpecF , unique up to isomorphism, such that, for every embedding� W F ,!C, X.B/U ˝F �.F / is the canonical model of the complex Shimura curve

B.�/�nH �B1�=U

over the reflex field �.F /�C, where B.�/ is a nearby quaternion algebra over F withrespect to � (see Definition 2.4.10 for more details).

As projective limits with affine transition morphisms exist in the category ofschemes, we may put X.B/ D lim

�UX.B/U . We will simply write XU and X if B

is clear.

Notation 2.2.1For an element g 2 B1�, we denote by Tg W X ! X the morphism induced by theright translation of g, known as the Hecke morphism.


Denote by U the set of all open compact subgroups of B1p� D .B˝A A1p/�,which is a filtered partially ordered set under inclusion. For U p 2 U and m 2 Z, put

X.m;U p/DXUpUp;m˝F F

nrp ;

where we recall that Up;m is the p-Iwahori subgroup of level m as introduced inSection 1.8. Put

X.˙1;U p/ WD lim �m!1

X.˙m;U p/:

For m 2 N [ ¹1º, if we take the inverse limit over the partially ordered set U, thenwe obtain F nr

p -schemes

X.˙m/D lim �Up2U

X.˙m;U p/:

We have successive surjective morphisms

X.˙1/! � � � !X.˙1/!X.0/;

which are equivariant under the Hecke actions of B1p�. By the work of Carayol [7,Section 6], the F nr

p -scheme X.0/ admits a canonical smooth model (see [19, Defini-tion 2.2] for its meaning) X over SpecOnr

p .

Remark 2.2.2Strictly speaking, Carayol assumed that F ¤Q. But when F DQ, one may take X

to be the model defined by modular interpretation using elliptic curves (resp., Abeliansurfaces with quaternionic actions) when B1 is (resp., is not) the matrix algebra—thisis well known.

We recall the construction in [7, Section 1.4] of an Op-divisible group G on X.We first introduce some notation.

Notation 2.2.3For an integer m� 1, we write(1) U

prp;m WD ¹g 2 Up;0 j g � 1 mod pmº for the principal congruence subgroup

of level pm;(2) X.m/pr! X.0/ for the corresponding covering with respect to the subgroup

Uprp;m; and

(3) O�Ep;mWDO�Ep

\Uprp;m, where Ep is an Fp-subalgebra of Bp'Mat2.Fp/ via

(1.7).


Consider the right action of Up;0=Uprp;m on .p�m=Op/

˚2 such that v:g D g�1vfor g 2 Up;0=U

prp;m 'GL2.Op=p

m/ and v 2 .p�m=Op/˚2. Then the quotient scheme�

X.m/pr � .p�m=Op/˚2�=.Up;0=U

prp;m/

defines a finite flat group scheme Gm over X.0/, with the obvious Op-action. Theinductive system ¹Gmºm�1 defines an Op-divisible group G over X.0/ (which is,however, denoted by E1 in [7, Section 5]). In particular, over X.C1/ (resp.,X.�1/), we have an exact sequence

0 Fp=Op G Fp=Op 0 (2.2)

such that the second arrow is the inclusion into the first (resp., second) factor and thethird arrow is the projection onto the second (resp., first) factor. By [7, Section 6.4],the Op-divisible group G extends uniquely to an Op-divisible group G of dimension1 and height 2 over X, together with an action by B1p� that is compatible with theHecke action on the base.

Put hD ŒFp WQp�. For m� 1, put X.m/ DX ˝OpOp=p

m and G .m/ D G jX.m/ .We have the exact sequence

0 !�.m/p L

.m/p .!

ı.m/p /_ 0; (2.3)

where� L

.m/p is the Dieudonné crystal of G .m/ evaluated at X.m/, which is a locally

free sheaf of rank 2h;� !

�.m/p is the sheaf of invariant differentials of G .m/=X.m/, which is a locally

free sheaf of rank 1; and� !

ı.m/p is the sheaf of invariant differentials of .G .m//_=X.m/, which is a

locally free sheaf of rank 2h� 1.They are equipped with actions ofOp under which (2.3) is equivariant. The projectivesystem of (2.3) for all m� 1 induces the following Op-equivariant exact sequence

0 !�p Lp .!ıp/_ 0 (2.4)

of locally free sheaves over bX, the formal completion of X along its special fiber. LetL (resp., !ı_) be the maximal subsheaf of Lp (resp., .!ıp/

_) where Op acts via thestructure map. Then we have the B1p�-equivariant exact sequence

0 !� L !ı_ 0; (2.5)

where !� D !�p . We call (2.5) the formal Hodge exact sequence.


We have the Gauss–Manin connection

rp W Lp!Lp ˝�1bX ; (2.6)

for the Dieudonné crystal, which is equivariant under the Hecke action of B1p� andthe action of Op. Thus, it induces the Gauss–Manin connection

r W L!L˝�1bX ; (2.7)

which is equivariant under the Hecke action of B1p�.We have the following Lemma 2.2.4 and Proposition 2.2.6 whose proof will be

given in Section 2.5.

LEMMA 2.2.4The formal Hodge exact sequence (2.5) is algebraizable, that is, it is the formal com-pletion of an exact sequence of locally free sheaves

0 !� L !ı_ 0 (2.8)

on X. Here, by abuse of notation we adopt the same symbols for these quasicoherentsheaves. Moreover, the Gauss–Manin connection (2.7) is algebraizable.

We simply call (2.8) the Hodge exact sequence.

Remark 2.2.5For m � 1, one may consider the right action of Up;0=U

prp;m on .Op=p

m/˚2 suchthat v:g D g�1v for g 2 Up;0=U

prp;m ' GL2.Op=p

m/ and v 2 .Op=pm/˚2. Then the

quotient scheme �X.m/pr � .Op=p

m/˚2�=.Up;0=U

prp;m/

defines an Op=pm-local system Lm on X.0/ of rank 2. Denote by L the Op-local

system over X.0/ defined by .Lm/m�1. Then OX.0/ ˝OpL is canonically isomor-

phic to the restriction of L on the generic fiber X.0/. Moreover, the induced connec-tion on OX.0/ ˝Op

L coincides with the restriction of r on X.0/, by the proof ofLemma 2.2.4.

PROPOSITION 2.2.6The composite map

!�!Lr�!L˝�1X! !ı_˝�1X (2.9)

is an isomorphism of locally free sheaves on X, where !ı is the dual sheaf of !ı_.


Definition 2.2.7 (Kodaira–Spencer isomorphism)We call the (B1p�-equivariant) isomorphism

KS W !�˝!ı��!�1X ; (2.10)

induced by the isomorphism (2.9), the Kodaira–Spencer isomorphism.

For w 2N, put LŒw� D Symw L˝ Symw L_. The Gauss–Manin connection r_

on the dual sheaf L_ and the original one r induce a connection

rŒw� W LŒw�!LŒw�˝�1X :

Define ‚Œw� to be the composite map

.�1X/˝w KS�1��! .!�/˝w ˝ .!ı/˝w!LŒw� r

Œw�

��!LŒw�˝�1X ; (2.11)

where KS is the isomorphism (2.10).

Notation 2.2.8Let X.0/ be the (dense) open subscheme of X by removing all points on the spe-cial fiber where G is supersingular. For every integer m � 1, denote by X.m/ thefunctor classifying Op=p

m-equivariant frames over X.0/, that is, homomorphismsLT Œpm�! G Œpm� and G Œpm�! p�m=Op such that the sequence

0 LT Œpm� G Œpm� p�m=Op 0

is exact.

Remark 2.2.9The scheme X.m/ is usually denoted by X.m/ord in the rest of the literature. Butsince we will only work with the ordinary locus, to reduce the burden of notation, wewill omit the superscript.

For m 2 N, the functor X.m/ is representable by a scheme that is finite étaleover X.0/, which we again denote by X.m/. Note that the generic fiber of X.m/

is canonically isomorphic to X.m/. Again as projective limits with affine transitionmorphisms exist in the category of schemes, we may put

X.1/ WD lim �m!1

X.m/:

We define G , rŒw�, KS, ‚Œw�, and the sequence (2.8) for X.m/ (m 2 N [ ¹1º) viarestriction and denote them by the same notation. Over X.1/, we have the universalframe


0 LT Œp1�%univ�

G%univı

Fp=Op 0 : (2.12)

By definition, there is an action of O�Epon the morphism X.1/!X.0/ such that

the pullback of (2.12) along the action of .t�; tı/ 2O�Epis the frame

0 LT Œp1�t�1� ı%

univ�

G%univı ıt

�1ı

Fp=Op 0 : (2.13)

This action is B1p�-equivariant. In what follows, we denote by

�t W X.1/!X.1/ (2.14)

the morphism induced by the action of t 2O�Ep.

Definition 2.2.10We define the transition isomorphisms to be

‡˙ W X.˙1/˝F nrpF abp

��!X.1/˝Onr

pF abp

such that the pullbacks of (2.12) under ‡˙ coincide with (2.2) in terms of the isomor-phisms (1.8), respectively.

LEMMA 2.2.11The Hecke morphism TJ (Notation 2.2.1) descends to an (iso)morphism TJ W

X.C1/!X.�1/, and the following diagram

X.C1/˝F nrpF abp

‡C

TJ

X.�1/˝F nrpF abp

‡�

X.1/˝OnrpF abp

commutes. Here, we regard JD . 0 1�1 0 / as an element in Bp via the fixed isomorphism

Bp 'Mat2.Fp/.Moreover, the isomorphism ‡C (resp., ‡�) is B1p�-equivariant and O�Ep

-equivariant (resp., O�Ep

-conjugate-equivariant).

ProofIt is clear that the Hecke morphism TJ descends as the conjugation of J turns Up;m to


Up;�m. The commutativity of the diagram follows from the fact that T�J GjX.�1/ isisomorphic to GjX.C1/.

The B1p�-equivariant property follows from the construction. By definition, ‡CisO�Ep

-equivariant. The conjugate-equivariant property for‡� follows from the iden-tity

J

�t�

tı

�J�1 D

�tı

t�

�for every t D .t�; tı/ 2O�Ep

.

2.3. Universal convergent modular formsFor m 2N[ ¹1º, denote by X.m/ the formal completion of X.m/ along its specialfiber. It is an affine formal scheme over Onr

p , equipped with an Op-divisible group G

induced from G . In particular, X.1/ is indeed the projective limit lim �m!1

X.m/ inthe category of formal schemes over Onr

p .The action of O�Ep

(2.13) makes itself the Galois group of the B1p�-equivariantpro-étale Galois cover X.1/! X.0/, in which O�Ep;m

(Notation 2.2.3) is the sub-group of O�Ep

that fixes the subcover X.m/!X.0/ for m 2N. By abuse of notation,the formal completion of those quasicoherent sheaves on X.m/ and their maps willbe denoted by the same symbols.

The following lemma will be proved in Section 2.5.

LEMMA 2.3.1There is a unique morphism ˆ W X.0/! X.0/ lifting the Frobenius morphism on thespecial fiber of degree #Op=p such that ˆ�G ' G=G0Œp�, where G0 is the formalpart of G. In particular, ˆ induces an endomorphism ˆ� on L.

Moreover, we have a unique ˆ�-stable splitting

LD !�˚Lı (2.15)

with Lı an invertible quasicoherent formal sheaf on X.0/. In addition, Lı is hori-zontal with respect to the Gauss–Manin connection, that is, rLı �Lı˝�1

X.0/.

Remark 2.3.2The splitting (2.15) is called unit-root splitting. It induces an isomorphism Lı

��!

!ı_. Dually, it induces a splitting L_ D !ı˚L� possessing similar properties as inLemma 2.3.1, with an isomorphism L�

��! !�_.

If we restrict the unit-root splitting in both Lemma 2.3.1 and Remark 2.3.2 toX.m/ for m 2N[ ¹1º, then we obtain a map


�Œw�ord W L

Œw�! .!�/˝w ˝ .!ı/˝wKS�! .�1

X.m//˝w

for allw 2N, where KS is the (formal completion of the restriction of the) map (2.10).

Definition 2.3.3 (Atkin–Serre operator)For m 2N[ ¹1º and w 2N, define the Atkin–Serre operator to be

‚Œw�ord W .�

1X.m//

˝w‚Œw�jX.m/��!LŒw�˝�1

X.m/

�Œw�ord��! .�1

X.m//˝wC1;

where ‚Œw� is defined in (2.11). For k 2N, define the Atkin–Serre operator of degreek to be

‚Œw;k�ord D‚

ŒwCk�1�ord ı � � � ı‚

Œw�ord W .�

1X.m//

˝w! .�1X.m//

˝wCk:

In what follows, w will always be clear from the text; hence, we will suppress w fromnotation. In other words, we simply write ‚ord (resp., ‚kord) instead of ‚Œw�ord (resp.,

‚Œw;k�ord ) for all w 2N.

By using Serre–Tate coordinates (Theorem B.1.1), the formal deformation spaceof theOp-divisible group LT Œp1�˚Fp=Op (over ) is canonically isomorphic to LT .Thus, we have the classifying morphism

c W X.1/!LT

of Onrp -formal schemes. It induces a morphism

c=x W X.1/=x!LT (2.16)

for every closed point x 2 X.1/./, where X.1/=x denotes the formal completionof X.1/ at x. The following Lemma 2.3.4 and Proposition 2.3.5 will be proved inSection 2.5.

LEMMA 2.3.4The morphism c=x is an isomorphism for every x.

By the above lemma, we have, for every closed point x 2X.1/./, a restrictionmap

resx W M0.1;K/!O.B;K/ (2.17)

induced from c=x (see (2.16)).


PROPOSITION 2.3.5There is a morphism ˇ W LT �SpfOnr

pX.1/!X.1/ such that

(1) for every x 2X.1/./, ˇ preserves X.1/=x , and the induced morphism

ˇ=x W LT �SpfOnrpX.1/=x!X.1/=x

is simply the formal group law after identifying X.1/=x with LT via c=x;(2) if we equip LT with the action of O�Ep

� B1p� via the inflation O�Ep!

O�p by t 7! t=t c and trivially on the second factor, then ˇ is O�Ep� B1p�-

equivariant;(3) for every x 2 Fp=Op, the following diagrams

X.1/bOnrpF abp

ˇ�˙.x/

‡�1˙

X.1/bOnrpF abp

‡�1˙

X.˙1/˝F nrpF abp

Tn˙.x/

X.˙1/˝F nrpF abp

commute, where

nC.x/D

�1 x

0 1

�; n�.x/D

�1 0

x 1

�;

respectively; and ˇz is the restriction of ˇ to a point z of B.In particular, LT acts trivially on the special fiber of X.1/, and this ˇ is unique.

Definition 2.3.6We call

!� WD c��

dT

T

the global Lubin–Tate differential, where � is the homomorphism in Notation 1.8.3.It is a nowhere-vanishing global differential form on X.1/bOnr

pOCp , and in fact, it

belongs to H0.X.1/;�1X.1/

/bOnrpF ltp by the definition of F lt

p .

Remark 2.3.7The pullbacks ‡�˙!� depend only on (or rather ˙), not on the choice of �.

In the following definition, we generalize the notion of convergent modular formsfirst introduced by Katz [15] to Shimura curves.


Definition 2.3.8Suppose that we have m 2N[ ¹1º, w 2 Z, and a complete field extension K=F nr

p .(1) Define the space of (K-valued) convergent modular forms of weight w and

p-Iwahori level m to be

Mw.m;K/DH0�X.m/; .�1

X.m//˝w�bOnr

pK;

which is naturally a complete K-vector space.(2) A convergent modular form of weight 0 is simply called a convergent modular

function.(3) Put Mw

[.m;K/D

SUp2U Mw.m;K/U

p

�Mw.m;K/.

Remark 2.3.9For m 2 N [ ¹1º and w 2 Z, the space Mw.m;K/ has a natural action by O�Ep

�

B1p� under which Mw[.m;K/ is stable. Moreover, M0.m;K/ is the complete ten-

sor product of the coordinate ring of X.m/ and the field K , and thus, Mw.m;K/ isnaturally a topological M0.m;K/-module.

In particular for w 2N, we have the Atkin–Serre operator

‚ord W Mw[ .m;K/!MwC1

[.m;K/ (2.18)

induced from the corresponding operator of sheaves (Definition 2.3.3). The operatoris O�Ep

�B1p�-equivariant.

From now on, we suppose that K is a complete field extension of F abp . Then for

every w 2 Z, the multiplication by !w� (Definition 2.3.6) induces a canonical B1p�-

equivariant isomorphism M0.1;K/��!Mw.1;K/.

Definition 2.3.10 (Stable convergent modular forms)A convergent modular function f 2M0.1;K/ is stable ifX

z2KerŒp�B.Kac/

ˇ�zf D 0;

where ˇ�z W M0.1;K/!M0.1;K/ is the map induced by ˇ from Proposition2.3.5. Denote by M0.1;K/~ the subspace of M0.1;K/ of stable convergent mod-ular functions.

For m 2N[ ¹1º and w 2 Z, put

Mw[ .m;K/

~ DMw[ .m;K/\M0.1;K/~ �!w� �Mw.m;K/:


Remark 2.3.11By Proposition 2.3.5, a convergent modular function f is stable if and only if resxffrom (2.17) is stable (Definition 2.1.3) for every x 2X.1/./.

Remark 2.3.12The space Mw

[.m;K/~ does not depend on the choices of or �.

Definition 2.3.13 (Admissible convergent modular forms)Let n � 0 be an integer. We say that a stable convergent modular function f 2M0.1;K/~ is n-admissible if ˇ�zf D �.z/f holds for all z 2 KerŒpn� � B.Kac/.We say that f 2Mw

[.m;K/ is an n-admissible stable convergent modular form if

f!�w� is an n-admissible stable convergent modular function.

Remark 2.3.14By Proposition 2.3.5(1), a stable convergent modular function f is n-admissible ifand only if resxf from (2.17) is n-admissible (in the sense of Definition 2.1.15) forevery x 2X.1/./.

The following lemma is a comparison between the Atkin–Serre operator ‚ord

from (2.18) and the Lubin–Tate differential operator ‚ (Definition 2.1.12).

LEMMA 2.3.15For an element f 2Mw

[.m;K/ for some w;m 2N, we have

resx�.‚ordf /!

�w�1�

�D‚

�resx.f!

�w� /

�for every x 2X.1/./.

ProofIt follows from Lemma 2.3.4, Theorem B.2.3, and the definition of ‚.

Definition 2.3.16 (Universal convergent modular form)A universal convergent modular form of depth m 2 N and tame level U p 2 U is anelement

M 2M0.1;K/bFpD.Oanti

p ;Fp/

such that M is U p-invariant and

��t MD ı�1t �M (2.19)


for t 2O�Ep;m. Here, �t W X.1/! X.1/ is the formal completion of the morphism

(2.14); and we regard ıt as the Dirac distribution of the image of t under the quotienthomomorphism O�Ep

!Oantip .

The next theorem produces universal convergent modular forms from a stableconvergent modular form (of a fixed weight). In this way, the universal convergentmodular forms can be regarded asp-adic families interpolating iterations of the Atkin–Serre operator.

THEOREM 2.3.17Let f 2Mw

[.m;K/~ be a stable convergent modular form for some w;m 2N. Then

there is a unique element

M.f / 2M0.1;K/bFpD.Oanti

p ;Fp/

such that, for every k 2N,

M.f /�hwC ki

�D .‚kordf /!

�w�k� ; (2.20)

where ‚ord is the map (2.18). Moreover, we have the following.(1) If f is fixed by U p 2 U, then so is M.f /.(2) M.f / is a universal convergent modular form of depth m (Definition 2.3.16).(3) If w � 1, then we have

‚ord�M.f /

�hw � 1i

�!w�1�

�D f:

(4) Suppose that f is n-admissible (Definition 2.3.13). Then we have

M.f /�hki�DM.f /

��hki

�for every k 2 Z and every (locally constant) character � W Oanti

p !K� that istrivial on .1C pn/�.

ProofThe uniqueness follows from Lemma 2.1.11. The morphism ˇ in Proposition 2.3.5induces a map

ˇ� W M0.1;K/!M0.1;K/bKO.B;K/:

By Lemma 2.1.6(2) and Remark 2.3.11, it sends M0.1;K/~ into M0.1;K/~bKO.B;K/~. Thus, we may regard ˇ�.f!�w� / as an element in M0.1;K/~bFp

D.Oantip ;Fp/ via the Fourier transform F , since K contains F lt

p . Define a (contin-uous Fp-linear) translation map


�w W D.Oantip ;Fp/!D.Oanti

p ;Fp/

such that .�w�/.g/D �.g � h�wi/ for every g 2 C.Oantip ;Fp/. We take

M.f /D �w�ˇ�.f!�w� /

�:

For the formula (2.20), it suffices to check it after applying resx for every x 2

X.1/./. In fact, we have

resxM.f /�hwC ki

�D resx

�ˇ�.f!�w� /

��hki�DMloc

�resx.f!

�w� /

��hki�

(2.21)

by Definition 2.1.8 and the definition of ˇ. By Lemma 2.1.14, we have that (2.21) isequal to ‚k.resx.f!�w� //. Finally by Lemma 2.3.15, we have ‚k.resx.f!�w� //D

resx..‚kordf /!�w�k� /.

Property (1) follows from Proposition 2.3.5(2). Properties (3) and (4) follow fromLemmas 2.1.14 and 2.1.16, respectively. For property (2), we only need to show that(2.19) holds for M D M.f / and t 2 O�Ep;m

. In fact, since f is fixed by O�Ep;m,

we have M.f /DM.��t f /, which equals ıt � ��t M.f / by Proposition 2.3.5(2) andRemark 2.1.5.

The following definition is suggested by the formula (2.21) in the proof of theabove theorem.

Definition 2.3.18We call M.f / in Theorem 2.3.17 the global Mellin transform of f .

2.4. Comparison with Archimedean differential operatorsNow suppose that B is equipped with an E-embedding as in Definition 1.8.4 suchthat ep coincides with (1.7) under the fixed isomorphism Bp 'Mat2.Fp/.

Definition 2.4.1 (CM-subscheme)We define the CM-subscheme Y to be XE

�, the subscheme of X fixed by the action

of e.E�/ for e as in Definition 1.8.4. Define Y ˙ to be the subschemes of Y suchthat E� acts on the tangent space of points in Y ˙ via the characters t 7! .t=t c/˙1,respectively. See also [30, Section 3.1.2].

In what follows, we will regard Y ˙ as their base change to F nrp ; in particular,

they are closed subschemes of X ˝FpF nrp .

LEMMA 2.4.2We have Y D Y C

`Y �. Moreover, we have the following.


(1) Both Y C.Cp/ and Y �.Cp/ are equipped with the natural profinite topology,isomorphic to E�nA1�E , and admit a transitive action of A1�E via Heckemorphisms (see Section 1.8 for the notation E�).

(2) The projection maps X ˝FpF nrp ! X.˙1/ restrict to isomorphisms from

Y ˙ to their images, respectively. In particular, we may regard Y ˙ as closedsubschemes of X.˙1/.

(3) The closed subschemes Y ˙.1/ WD ‡˙Y ˙ of X.1/ ˝OnrpF abp descend to

closed subschemes of X.1/˝OnrpF nrp , where ‡˙ are the transition isomor-

phisms in Definition 2.2.10.

ProofThe decomposition follows directly from the definition. For the rest, we considerY C.Cp/ without loss of generality.

Part (1) can be seen from the complex uniformization by choosing an arbitraryisomorphism Cp ' C. Part (2) follows from the fact that A1�E does not contain anynontrivial unipotent element. Part (3) follows from the fact that Gal.F ab

p =Fnrp / acts

via local class field theory as the right multiplication of O�p on the double coset pre-sentation X.1/.Cp/ and, hence, preserves the subset Y C.Cp/.

Notation 2.4.3For m 2N\ ¹1º, denote by(1) Y ˙.m/ the image of Y ˙.1/ in X.m/˝Onr

pF nrp ,

(2) Y˙.m/ the Zariski closure of Y ˙.m/ in X.m/, and(3) Y˙.m/ the formal completion of Y˙.m/ along the special fiber.

LEMMA 2.4.4For m 2 N [ ¹1º, we have Y˙.m/.Cp/ D Y˙.m/.F nr

p / D Y˙.m/.Onrp /. Here, for

an Onrp -algebra R, Y˙.m/.R/ are the sets of morphisms from SpecR to Y˙.m/ over

SpecOnrp , respectively.

ProofWithout loss of generality, we only prove the case for YC.m/. We first consider thecase where mD 0. The first identity Y˙.0/.Cp/D Y˙.0/.F nr

p / is well known fromthe class field theory. Take an element x 2 YC.0/.F nr

p /. It induces a unique morphismy W SpecOnr

p !X. Since y is fixed by E�, there are strict actions of E� \O�Epand,

hence, OEpon the Op-divisible group Gy . Therefore, the reduction of Gy is ordinary.

In other words, y factors through X.0/. As YC.0/ is defined as the Zariski closureof Y C.0/ in X.0/, we obtain an element x0 2 YC.0/.Onr

p / uniquely determined by x.Thus, we have Y˙.0/.F nr

p /D Y˙.0/.Onrp /.


The case for general m follows from the case for mD 0 and the following twofacts: (1) YC.m/ is a closed subscheme of YC.0/�X.0/ X.m/; (2) X.m/!X.0/ isa finite étale morphism (resp., a projective limit of finite étale morphisms) for m 2N(resp., mD1).

Notation 2.4.5Let S be a scheme that is locally of finite type over SpecC. We denote by MS the under-lying real analytic space with the complex conjugation automorphism cS W MS! MS .

In what follows, we will sometimes deal with a complex scheme S that is ofthe form lim

�ISi , where I is a filtered partially ordered set and each Si is a smooth

complex scheme, with a sheaf F that is the restriction of a quasicoherent sheaf F0 onsome S0. Then we will write MS D ¹ MSiºi2I for the projective system of the underly-ing real analytic spaces together with the complex conjugation cS , and we will writeMF D ¹ MFiºi�0 for the projective system of real analytification of the restricted sheaf

Fi for i � 0. Moreover, we denote

H0. MS; MF / WD lim�!i�0

H0. MSi ; MFi /:

For an isomorphism � W Cp��!C, put X� DX ˝F;� C and denote by

c� W MX�! MX� (2.22)

the complex conjugation. Denote by .L�;r�/ the restriction of the pair .L;r/˝Onrp ;�C

along �� W X� ! X ˝Onrp ;� C, where .L;r/ appears in Lemma 2.2.4. Applying the

same procedure to the sequence (2.8), we obtain the sequence

0 !�� L� !ı_� 0

of locally free sheaves on X�. Similarly, we have the Kodaira–Spencer isomorphism

KS� W !�� ˝!

ı�

��!�1X� (2.23)

induced by (2.10).

LEMMA 2.4.6The natural map M!�� ˚ c�� M!

�� !

ML� is an isomorphism of sheaves on the real analyticspace MX . Moreover, we have r�.c�� M!

�� /� .c

�� M!�� /˝

M�1X� .

ProofIt follows from Lemma 2.4.12 later in this section.


We have a remark similar to Remark 2.3.2, which together with Lemma 2.4.6induces a map

� Œw�� WMLŒw�� ! . M!�� /

˝w ˝ . M!ı� /˝w

KS�(2.23)��! . M�1X�/

˝w (2.24)

for all w 2N.

Definition 2.4.7Similar to Definition 2.3.3, define the Shimura–Maass operator to be

‚Œw�� W .M�1X�/

˝w(2.11)��! MLŒw�˝ M�1X�

�Œw�� (2.24)��! . M�1X�/

˝wC1:

For k 2N, define the Shimura–Maass operator of degree k to be

‚Œw;k�� D‚ŒwCk�1�� ı � � � ı‚Œw�� W .M�1X�/

˝w! . M�1X�/˝wCk:

As for ‚ord, we will suppress w from the notation and write ‚� (resp., ‚k� ) for ‚Œw��

(resp., ‚Œw;k�� ). In particular, we have the map

‚� W H0�MX�; . M�

1X�/˝w

�!H0

�MX�; . M�

1X�/˝wC1

�: (2.25)

Notation 2.4.8Put

X.m/� DX.m/˝F nrp ;� C; m 2 Z[ ¹˙1º;

Y ˙.m/� D Y˙.m/˝F nr

p ;� C; m 2N[ ¹1º:

Let F abp �K �Cp be a complete intermediate field. Take an element

f 2H0�X.m/; .�1X.m//

˝w�˝F K

with m 2 Z [ ¹˙1º and w 2 N. Then by the transition isomorphism and by therestriction to an ordinary locus, we have an element

ford WD

´‡C�f 2Mw

[.m;K/ for m� 0;

‡��f 2Mw[.�m;K/ for m 0:

(2.26)

By base change, f induces another element

f� 2H0�X.m/�; .�

1X.m/�

/˝w�:

The following lemma shows that the Atkin–Serre operator (2.18) and theShimura–Maass operator (2.25) coincide on CM points. Note that the operator ‚�descends along the projection map X�!X.m/�.


LEMMA 2.4.9Let the notation be as above. We have, for k 2N,

�‡�˙�.‚kordford/jY˙.m/

�D .‚k� f�/jY˙.m/�

as functions on Y ˙.m/�, regarded as closed subschemes ofX.˙m/� via the transitionisomorphisms ‡˙ in Definition 2.2.10, respectively.

ProofGenerally, once we restrict to stalks, we cannot apply differential operators anymore.Therefore, we need alternative descriptions of ‚Œw;k�ord and ‚Œw;k�� . (Here, we retrievethe original notation for clarity.)

We denote by #w the composite map

.!�/˝w ˝ .!ı/˝w!LŒw� rŒw�

��!LŒw�˝�1X.n/

�Œw�ord��! .!�/˝w ˝ .!ı/˝w ˝�1

X.n/

KS�1��! .!�/˝wC1˝ .!ı/˝wC1;

and by ıw the composite map

LŒw� rŒw�

��!LŒw�˝�1X.n/

KS�1��!LŒw�˝ .!�˝!ı/!LŒwC1�:

Since we haverLı �Lı˝�1X

by Lemma 2.3.1(2) and its dual version from Remark2.3.2, the composition #wCk�1 ı � � � ı #w coincides with the map

.!�/˝w ˝ .!ı/˝w!LŒw� ıwCk�1ıııw

��!LŒwCk��ŒwCk�ord��! .!�/˝wCk ˝ .!ı/˝wCk:

Therefore, the map ‚Œw;k�ord coincides with the composite map

.�1X.n//

˝w KS�1��! .!�/˝w ˝ .!ı/˝w!LŒw�

ıwCk�1ıııw

��!LŒwCk�KSı� ŒwCk�ord��! .�1

X.n//˝wCk:

The advantage of the above description is that �ord appears only at the end of thesequence of maps. Since we have r�.c�� M!

�� /� .c

�� M!�� /˝

M�1X� by Lemma 2.4.6, there

is a similar description of ‚Œw;k�� as above. Therefore, to prove the lemma, we onlyneed to show that the splitting in Lemma 2.4.6 coincides with the restriction of thesplitting

L˝Onrp ;� CD .!

�˝Onrp ;� C/˚ .L

ı˝Onrp ;� C/

on Y C� and Y �� .


Pick up an arbitrary point y 2 Y C.m/�.C/ [ Y �.m/�.C/. We have an action ofE� on both the splitting M!�� jy ˚c

�� M!�� jy and .!� ˝Onr

p ;� Cjy/˚ .Lı ˝Onr

p ;� Cjy/. Bydefinition, M!�� jy and !�˝Onr

p ;� Cjy coincide, which is one of the two complex eigen-lines with respect to the E�-action. It follows that c�� M!

�� jy and Lı˝Onr

p ;� Cjy have tocoincide as well, which contributes to the other complex eigenline.

We now study the behavior of the Shimura–Maass operator under complex uni-formization.

Definition 2.4.10 (�-nearby data)Let � W Cp

��!C be an isomorphism. An �-nearby data for B consists of

� a quaternion algebra B.�/ over F such that B.�/v is definite for Archimedeanplaces v other than �jF ,

� an isomorphism B.�/v ' Bv for every finite place v other than p,� an isomorphism B.�/� WDB.�/˝F;� R'Mat2.R/, and� an embedding e.�/ W E ,! B.�/ of F -algebras such that e.�/v coincides with

ev under the isomorphism B.�/v ' Bv for every finite place v other than p,and HE� D ¹˙iº.

We now choose an �-nearby data for B. It induces a complex uniformization

X�.C/'B.�/�nH �B1�=F �;

where H D C nR denotes the union of Poincaré upper and lower half-planes. Let zbe the standard coordinate on H .

LEMMA 2.4.11Denote by L� the C-local system on X� defined by the quotient map

B.�/�nC˚2 �H �B1�=F �!B.�/�nH �B1�=F � 'X�.C/;

where the action of � 2B.�/� is given by the formula

��.a1; a2/

t ; z; gD��.a1; a2/�.�/

�1�t; �.�/.z/; �1g

:

Then we have a canonical isomorphism L� ' OX� ˝C L� under which r� coincideswith the induced connection on OX� ˝C L�.

ProofIt follows from the fact that L� is canonically isomorphic to the restriction of L˝Op;�C

along the natural morphism ��, where L is the Op-local system on X defined inRemark 2.2.5.


The following lemma will be proved in Section 2.5.

LEMMA 2.4.12Under the isomorphism L� 'OX� ˝C L� in Lemma 2.4.11, the subsheaf !�� is gener-ated by the section !�� whose value at z is .z; 1/t .

The following lemma shows that our definition of Shimura–Maass operatorscoincides with the classical one.

LEMMA 2.4.13For every f 2H0. MX�; . M�1X�/

˝w/ with some w 2N, we have

‚�f ˝ dz˝�w�1 D

� @@zC

2w

z � z

�f ˝ dz˝�w :

ProofWe may pass to the universal cover H � B1�=F � and suppress the part B1�=F �

in what follows. Over H , the sheaf L� is trivialized as C˚2, and the subsheaf !�� isgenerated by the section !�� whose value at z is .z; 1/t by Lemma 2.4.12. Dually, thesheaf L_� is trivialized as 2-dimensional complex row vectors, and the subsheaf !ı�is generated by the section !ı� whose value at z is .1;�z/. Then we have KS.!�� ˝!ı� /D dz.

It is easy to see that

‚��.!�� /

˝w ˝ .!ı� /˝w�D

2w

z � z

�.!�� /

˝w ˝ .!ı� /˝w�˝ dz;

since c�� !�� (resp., c�� !

ı� ) is generated by the section .z; 1/t (resp., .1;�z/). The lemma

follows.

We now introduce the notion of automorphic forms.

Notation 2.4.14For every w 2 Z, denote by A.2w/.B.�/�/ (resp., A

.2w/cusp .B.�/

�/) the space of realanalytic (resp., cuspidal) automorphic forms on B.�/�.A/ of weight 2w at �jF andinvariant under the action of B.�/�v at Archimedean places v other than �jF .

The spaces A.2w/.B.�/�/ and A.2w/cusp .B.�/

�/ are representations of B.�/�.A/ bythe right translation R.

LEMMA 2.4.15There is a natural B1�-equivariant map


�� W H0�MX�; . M�

1X�/˝w

�!A.2w/

�B.�/�

�such that, for g� 2B.�/�� DGL2.R/,

��.f /�Œg�; 1�

�j.g�; i /

w D f�g�.i/

�˝ dz˝�w ;

where j.g�; i /D .detg�/�1 � .ci C d/2 is the square of the usual j -factor.

ProofThis is the well-known dictionary between modular forms and automorphic forms.

We denote by H0cusp.MX�; . M�

1X�/˝w/ � H0. MX�; . M�1X�/

˝w/ the inverse image of

A.2w/cusp .B.�/

�/ under ��.

Definition 2.4.16Define �˙ to be the matrices

1

4i

�1 ˙i

˙i �1

�in gl2;C DMat2.C/, respectively. For an isomorphism � W Cp

��!C, define �˙;� to be

the matrices �˙ when regarded as elements in LieC.B.�/˝F;� C/ D gl2;C, respec-tively. Finally, define �k˙;� D�˙;� ı � � � ı�˙;� to be the k-fold composition.

LEMMA 2.4.17For every f 2H0cusp.

MX�; . M�1X�/˝w/ and k 2N, we have

��.‚k� f /D�

kC;��.f /;

where �� is defined in Lemma 2.4.15.

ProofThis follows from Lemma 2.4.13, together with [5, p. 130, p. 143, and Proposi-tion 2.2.5 on p. 155].

2.5. Proofs of claims via unitary Shimura curvesIn this section, we prove the six claims (Lemma 2.2.4, Proposition 2.2.6, Lemma2.3.1, Lemma 2.3.4, Proposition 2.3.5, and Lemma 2.4.12) left from previous sec-tions. Suppose that we are in the case of modular curves, that is, F D Q and B isunramified at every prime; then these statements except for Proposition 2.3.5 are clear.In fact, in this case,


� Lemma 2.2.4 follows from the fact that (2.5) is the formal completion of theHodge sequence coming from the universal elliptic curve;

� Proposition 2.2.6 is well known;� Lemma 2.3.1 is proved by Katz as [16, Theorem 1.11.27];� Lemma 2.3.4 follows from Serre–Tate coordinates in [17];� Proposition 2.3.5 again can be proved via Serre–Tate coordinates (one can

adjust our proof below to the case of modular curves); and� Lemma 2.4.12 is again well known.The main idea is to use the existence of a universal family of elliptic curves withdeformation theory. However, in the general case, X is not a moduli space; therefore,we have to use some auxiliary moduli space to deduce these statements. The readermay skip the rest of this section for the first reading.

Our strategy is to use the unitary Shimura curves considered by Carayol [7]. Thus,we will fix an isomorphism � W Cp

��!C. In particular, F nr

p is a subfield of C. We alsofix an �-nearby data for B (Definition 2.4.10) and put B DB.�/ for short.

Note that when F D Q there is no need to change the Shimura data as X isalready a moduli space. In order to unify the argument, we will choose to do so inthis case as well. We will also assume that we are not in the case of classical modularcurves (i.e., F D Q and B is unramified at every prime) where all these statementsare known, as explained above.

Fix an element 2 C such that Im > 0, � 2 2 N, p splits in Q. / � C, andQ. / is not contained in E . We have subfields F. / and E. / of C, and we identifytheir completion inside C' Cp with Fp. In [7, Section 2] (see also [13, Section 2]),a reductive group G0 over Q is defined as a subgroup of ResF=Q.B� �F � F. /�/(which itself is a subgroup of ResF.�/=Q.B ˝F F. //�) with “rational norms.” Inparticular, we have

G0.Qp/DQ�p �GL2.Fp/� .B�p2� � � � �B�pm/;

where p2; : : : ;pm are primes of F over p other than p. Put

G0pDG0��Yq¤p

Zq

�˝Q

�� .B�p2 � � � � �B

�pm/;

and let U0 be the set of all (sufficiently small) open compact subgroups U 0p of G0p.Then for each U 0p 2 U0, there is a unitary Shimura curve X 0U 0p , smooth and pro-jective over SpecFp, of the level structure Z�p � GL2.Op/ � U

0p. It has a canonicalsmooth model X0U 0p over SpecOnr

p defined via a moduli problem (see [7, Section 6]).In particular, there is a universal Abelian variety � W AU 0p !X0U 0p with a specificp-divisible subgroup G 0U 0p WD .AU 0p Œp

1�/2;11 �AU 0p Œp

1� (it is denoted as E01 in[7, Section 6]), which is an Op-divisible group of dimension 1 and height 2. Here,


for an object M with OF.�/ ˝ Zp-action, we denote by M 2;1 the direct summand

corresponding to the p-adic place F. /�7!��! F. / � Cp . Then .AU 0p Œp

1�/2;1

admits an action by OB ˝OF Op 'Mat2.Op/. Put e1 D . 1 00 0 / and .AU 0p Œp1�/

2;11 D

e1.AU 0p Œp1�/2;1. See [7, Section 2.6] for more details.

We let X0.0/U 0p be the (dense) open subscheme of X0U 0p by removing all pointson the special fiber where G 0 is supersingular. For n 2 N, define X0.n/U 0p to be thefunctor classifying Op-equivariant extensions

0 LT Œpn� G 0Œpn� p�n=Op 0

of G 0 over X0.0/U 0p . The obvious map X0.n/U 0p !X0.0/U 0p is étale. Finally, putX0.1/U 0p D lim

�nX0.n/U 0p .

The construction of Carayol amounts to saying that, for every sufficiently smallU p 2 U and a connected component X

Up of XUp , there exists a member U 0p 2 U0

such that� X.n/

Up WDX

Up �XUp X.n/Up is isomorphic to the neutral connected com-

ponent of X0.n/U 0p for n 2N[ ¹1º; and� under the above isomorphism, GUp j

X.n/�

Upis isomorphic to the restriction of

G 0U 0p to (the neutral connected component of) X0.n/U 0p .In what follows we may and will fix a sufficiently small subgroup U p 2 U, a

connected component XUp of XUp , and a corresponding subgroup U 0p 2 U0. To

simplify notation, we will suppress U p and U 0p and will regard X as a connectedcomponent of X0 as well.

Consider the Hodge exact sequence

0 ��1A=X0

H1dR.A=X

0/ R1��OA 0:

It has a direct summand

0 .��1A=X0

/2;11 H1

dR.A=X0/2;11 .R1��OA/

2;11 0;

(2.27)

which is Op-equivariant, where .�/2;11 is defined similarly as above. Here in (2.27),the three sheaves are locally constant of rank 1, 2h, and 2h � 1, respectively, wherehD ŒFp W Qp�.

We introduce the following notation.

Notation 2.5.1If M is a locally free sheaf on a scheme over SpecOp equipped with an Op-action


Op! EndM , then we denote by MOp the maximal subsheaf on which Op acts viathe structure homomorphism.

In what follows, we denote the sequence (2.27) after applying .�/Op by

0 !0� L0 !0ı_ 0: (2.28)

Proof of Lemma 2.2.4It suffices to consider the problem after the restriction to an arbitrarily chosen con-nected component X of X. By the definition of G 0 in [7, Section 5.4], we know that(2.27) is the Hodge exact sequence for G 0. Since G is isomorphic to G 0 on X, (2.27)is also the Hodge exact sequence for G . Therefore, if we restrict (2.27) to X and

take formal completion, we recover the exact sequence (2.4) (restricted to cX); and ifwe further apply the functor .�/Op , then we recover the exact sequence (2.5). In other

words, the formal completion of (2.28) coincides with (2.5), both restricted to cX.This shows that (2.5) is algebraizable.

For the next assertion, we have the Gauss–Manin connection

r 0A W H1dR.A=X

0/!H1dR.A=X

0/˝�1X0

and the induced connection

r 0p W H1dR.A=X

0/2;11 !H1

dR.A=X0/2;11 ˝�

1X0 : (2.29)

Since applying .�/2;11 commutes with the formation of the Gauss–Manin connection,we know that the formal completion of (2.29) coincides with rp from (2.6) when

restricted to cX. Now applying the functor .�/Op , we know that the formal completionof the induced connection

r 0 W L0!L0˝�1X0 (2.30)

coincides with r from (2.7) when restricted to cX. In other words, r from (2.7) isalgebraizable. Lemma 2.2.4 is proved.

Proof of Proposition 2.2.6Denote by

KS0 W !0�˝!0ı!�1X0 (2.31)

the Kodaira–Spencer map induced from r 0 from (2.30), where !0ı is the dual sheafof !0ı_. We know from the proof of Lemma 2.2.4 that, when restricted to X, (2.28)


coincides with (2.8) under which r 0 from (2.30) coincides with r from (2.7). There-fore, under the previous identification, KS0 coincides with KS from (2.10). ThenProposition 2.2.6 follows from the following analogous statement for X0: (2.31) isan isomorphism.

The proof is similar to [9, Lemma 7], which essentially follows from theGrothendieck–Messing theory. Denote by A_ the dual Abelian variety of A. Then!0ı_ is canonically isomorphic to .Lie.A_=X0/2;11 /Op . We only need to show that,for every closed point t W Speck.t/!X0, the induced map

!0�˝ k.t/!�Lie.A_=X0/2;11

�Op˝�1X0 ˝ k.t/ (2.32)

is surjective, where Lie denotes the sheaf of tangent vectors.Let A=Speck.t/ be the Abelian variety classified by t . Put T D Speck.t/Œ"�=

."2/. The lifts A of A (with other PEL structures) to T correspond to homomor-phisms

� W t�!0�!�Lie.A_=X0/2;11

�Op˝ k.t/:

Since both sides are k.t/-vector spaces of dimension 1, we may choose a homomor-phism � that is surjective. Let t W T !X0 be the morphism that classifies A=T .Compose the isomorphism t�!

0� ˝ k.t/! t�!0� and the surjective map �. By theisomorphism�

Lie.A_=X0/2;11�Op˝ k.t/' t�

�Lie.A_=X0/2;11

�Op˝�1T=k.t/˝ k.t/;

we obtain a surjective map

t�!0�˝ k.t/! t�

�Lie.A_=X0/2;11

�Op˝�1T=k.t/˝ k.t/;

which is the pullback of (2.32) under t . Therefore, (2.32) is surjective.

For n 2 N [ ¹1º, denote by X0.n/ the formal completion of X0.n/ along itsspecial fiber, which is equipped with an Op-divisible group G0 induced from G 0.

Proof of Lemma 2.3.1By the proof of Lemma 2.2.4, it suffices to prove the same statement for X0.0/. Thedesired morphism ˆ0 W X0.0/! X0.0/ is constructed through the moduli interpreta-tion of X0.0/ by “dividing G00Œp�,” which lifts the Frobenius on the special fiber ofdegree #Op=p. The uniqueness of such ˆ0 is ensured by (the proof of) Lemma 2.3.4and Theorem B.1.1.

The proof of the remaining part is similar to [16, Theorem 1.11.27]. The onlymodification we need is to show that the subsheaf L0ı of L0 glues to a formal quasi-coherent sheaf. For this, we adopt the proof of [14, Theorem 4.1] in the case where


Zp is replaced by Op and p is replaced by a uniformizer $ of F . The assumptionsare satisfied because the Newton polygon of the underlying p-divisible group of G0jxfor every x 2X0.0/./ is the one starting with .0; 0/, ending with .2h; 1/, and havingthe unique breaking point at .h; 0/.

Remark 2.5.2In fact, the induced map of ˆ0 constructed in the above proof on the coordinate ringis simply the operator Frob defined in [13, Definition 11.1].

Proof of Lemma 2.3.4We only need to prove a similar statement for X0.1/. By the moduli interpretation ofX0.1/ and the Serre–Tate theorem on deformation of Abelian varieties, we have anisomorphism X0.1/=x 'Mx , where Mx is the formal scheme representing defor-mations of G 0jx . By Theorem B.1.1, we know that Mx is canonically isomorphic toLT , and the induced isomorphism X0.1/=x 'LT is just c=x by definition.

Proof of Proposition 2.3.5Recall that we have a similarly defined formal scheme X0.1/ over SpfOnr

p . Theuniqueness of ˇ is clear. Thus, by comparison, it suffices to construct the morphismˇ0 W LT �SpfOnr

pX0.1/!X0.1/with similar properties to those in Proposition 2.3.5,

since the action of LT is supposed to preserve the special fiber.We use the moduli interpretation of X0.1/. For a scheme S over SpecOnr

p wherep is locally nilpotent, X0.1/.S/ is the set of isomorphism classes of quintuples.A; �; �; kp; p/, where .A; �; �; kp/ is the same data in [7, Section 5.2] but kp is anisomorphism instead of a class, and p is an exact sequence

0 LT Œp1� .Ap1/2;11 Fp=Op 0:

On the other hand, LT .S/ is the set of isomorphism classes of .G; G/ where G isan exact sequence

0 LT Œp1� G Fp=Op 0:

Using the group structure on LT , we may add the above two exact sequences to anew one, denoted by ˛.p; G/, which can be written as

0 LT Œp1� ˛�.Ap1/

2;11 ;G

�Fp=Op 0:

By the Serre–Tate theorem on deformation of Abelian varieties and the fact that étalelevel structures are determined on the special fiber, we canonically associate a quintu-


ple .A0; �0; � 0; k0p; 0p/ with 0p D ˛.p; G/. This defines the morphism ˇ0. The prop-erties of Proposition 2.3.5 for ˇ0 follow directly from the construction.

Proof of Lemma 2.4.12We define X 0� similarly as the projective limit over all level structures over C. Thenwe have the complex uniformization

X 0� .C/'G0.Q/nH �G0.A1/;

where G0.Q/ acts on H via the �-component of G0.R/. We similarly define a C-localsystem L0� on X 0� via the quotient map

G0.Q/nC˚2 �H �G0.A1/!G0.Q/nH �G0.A1/;

where the action of � 2G0.Q/ is given by the formula

��.a1; a2/

t ; z; gD��.a1; a2/�.�/

�1�t; �.�/.z/; �1g

;

where we regard �.�/ as an element in GL2.C/ in the formula .a1; a2/�.�/�1. By thesame reasoning as in Lemma 2.4.11, we have a canonical isomorphism L0� 'OX 0� ˝C

L0�. Here, we regard L0� as the restriction of L0 from (2.28) to X 0� . By the comparisonbetween (2.28) and (2.8) established in the proof of Lemma 2.2.4, it suffices to showthat the subsheaf !0�� is generated by the section !0�� whose value at z is .z; 1/t .

However, the coherent sheaf H1dR.A=X

0/ is obtained from the local system

G0.Q/nC˚2g �H �G0.A1/!G0.Q/nH �G0.A1/;

whereG0.Q/ acts on C˚2g DC˚2˚� � �˚C˚2 diagonally via all Archimedean placesof F . From the Hodge homomorphism in the Shimura data of G0, we see that therestriction of !0� ' .��1A=X0/

2;11 to X 0� is generated by the section !0�� whose value

at z is .z; 1/t . This follows from the same computation for the case of modular curves.Therefore, Lemma 2.4.12 is proved.

3. Statements of main theoremsIn this section, we state our main theorems about p-adic L-functions and the p-adicWaldspurger formula for the general case. We start by recalling some backgroundabout representations of incoherent algebras and Abelian varieties of GL.2/-type inSection 3.1. In Section 3.2, we state the main theorem about p-adic L-functions interms of Heegner cycles on Abelian varieties. In Section 3.3, we state the main the-orem about the p-adic Waldspurger formula in terms of Heegner cycles on Abelianvarieties. In Section 3.4, we provide an alternative formulation of our main theoremsin terms of periods of p-adic Maass functions, in the same spirit as in Section 1, anddeduce them from the previous formulation via Heegner cycles on Abelian varieties.


3.1. Representations for incoherent quaternion algebrasWe recall some materials from [30, Section 3.2]. Let �1; : : : ; �g be all Archimedeanplaces of F . Let B be a totally definite incoherent quaternion algebra over A. Asin Section 2.2, there is an associated projective system of Shimura curves ¹XU DX.B/U ºU over F , and X D lim

�UXU . We recall the following definition from [30,

Section 3.2.2].

Notation 3.1.1Let L be a field embeddable into C. Denote by A.B�;L/ the set of isomorphismclasses of irreducible (admissible) representations … of B1� over L such that, forsome and hence all embeddings L ,! C, the Jacquet–Langlands transfer of …˝LC to GL2.A1/ is a finite direct sum of (finite components of) irreducible cuspidalautomorphic representations of GL2.A/ of parallel weight 2.

Let A be an Abelian variety over F .

Notation 3.1.2Recall from [30, Section 3.2.3] the following notation

….B/A WD lim�!U

Hom�U .X�U ;A/;

where� the colimit is taken over all open compact subgroups U of B1�;� X�U is the smooth compactification of XU (which is simply XU unless in the

case of classical modular curves);� �U is the normalized Hodge class on X�U (see [30, Section 3.1.3]); and� Hom�U .X

�U ;A/ denotes the Q-vector space of modular parameterizations,

that is, the Abelian group of morphisms from X�U to A that send �U to atorsion point, tensoring with Q.

We simply write …A for ….B/A if B is clear from the context.

If we denote by JU the Jacobian of X�U , then �U induces a morphism X�U !

JU . Thus, Hom�U .X�U ;A/ is canonically identified with Hom0.JU ;A/ WDHom.JU ;

A/˝Q.PutMA WD End0.A/ WD End.A/˝Q. It is clear that both….B/A andMA depend

only on A up to isogeny.

Definition 3.1.3We say that A can be parameterized by B if there is a nonconstant morphism from


X D X.B/ to A. Denote by AV0.B/ the set of simple Abelian varieties over F thatcan be parameterized by B up to isogeny.

The set AV0.B/ is stable under duality. Take an element A 2 AV0.B/. Then…A is a nonzero rational irreducible representation of B1�, which is an elementin A.B�;Q/ (Notation 3.1.1). The assignment A 7!…A induces a bijection betweenAV0.B/ and A.B�;Q/. Moreover, MA is a field of degree equal to the dimensionof A, and it acts on the representation …A. Denote by A_ the dual Abelian variety(up to isogeny) of A, and we have …A_ similarly. There is a canonical isomorphismMA_ 'MA as in [30, Section 3.2.4].

Definition 3.1.4 (Canonical pairing, [30, Section 3.2.4])We have a canonical pairing

.�; �/A W …A �…A_!MA

induced by maps

.�; �/U W Hom0.JU ;A/�Hom0.JU ;A_/!MA

defined by the assignment .fC; f�/ 7! vol.XU /�1 ı fC ı f _� 2 End0.A/DMA forall open compact subgroups U of B1�.

Recall that an Abelian variety A (up to isogeny) over F is of GL.2/-type if MA

is a field of degree equal to the dimension of A. Let A be such an Abelian variety (upto isogeny), and denote by

!A W F�nA1�!M�A

the central character associated to A. For a finite place v of F , choose a rational prime` that does not divide v. We have a Galois representation �A;v of Dv , the decomposi-tion group at v, on the `-adic Tate module V`.A/ of A, which is a free module overMA;` WDMA˝Q Q` of rank 2. It is well known that the characteristic polynomial

Pv.T /D detMA;`�1� Frobv

ˇV`.A/

Iv�

belongs to MAŒT � and is independent of `, where Iv �Dv is the inertia subgroup andFrobv 2Dv=Iv is the geometric Frobenius.

Remark 3.1.5We use ! to denote both differential forms and central characters, since both waysare standard. We hope this does not cause any confusion for readers.


Definition 3.1.6 (L-functions and -factors)Let K be a field containing MA.(1) Define the local L-function of A as L.s; �A;v/ WD Pv.N

�s�1=2v /�1 2

MA ˝Q C. In a similar manner, we define the local adjoint L-function of A,which we denote as L.s; �A;v;Ad/; in particular, L.1; �A;v;Ad/ 2MA.

(2) For a locally constant character �v W F �v !K�, we have the twisted local L-function L.s; �A;v ˝ �v/ 2K ˝Q C. If W Fv!K� is a nontrivial additivecharacter, then we have the -factor .1=2; ;�A;v ˝ �v/.

(3) For a locally constant character �v W E�v ! K� such that !A;v � �vjF �v D 1,we have the local Rankin–Selberg L-function L.s; �A;v; �v/ 2 K ˝Q C andthe -factor .1=2; �A;v; �v/. See Remark 3.1.7 for more details.

(4) Let � W K ,! C be an embedding, which induces a homomorphism � W K ˝Q

C! C by abuse of notation. We define the global L-function of A (withrespect to �) to be

L.s; �.�/A / WD

Yv<1

�L.s; �A;v/:

Similarly, we have the global versionL.s; �.�/A ;Ad/ and L.s; �.�/A ; �.�// of other

L-functions as well.(5) We say that A is automorphic if L.s; �.�/A /, for some and hence all �, is (the

finite component of) the L-function of an irreducible cuspidal automorphicrepresentation of GL2.A/.

Remark 3.1.7If v splits into two places v1 and v2 of E , then L.s; �A;v; �v/ is defined to be theproduct L.s; �A;v1 ˝�v1/L.s; �A;v2 ˝�v2/. If v induces a single place w of E , thenwe define L.s; �A;v; �v/ WD L.s; .�A;vjDw /˝ �v/. By choosing a nontrivial additivecharacter W Fv!C�, we have the local Rankin–Selberg -factor .s; ; �A;v; �v/.It is well known that .1=2; v; �A;v; �v/ belongs to ¹˙1º and does not depend onthe choice of . We denote its value by .1=2; �A;v; �v/. The global L-functionsL.s; �

.�/A /, L.s; �

.�/A ;Ad/, and L.s; �

.�/A ; �

.�// are always absolutely convergent forRe s > 1.

Remark 3.1.8It is conjectured that every Abelian variety of GL.2/-type is automorphic. In particu-lar, when F DQ, every Abelian variety of GL.2/-type is parameterized by modularcurves. This follows from Serre’s modularity conjecture (for Q) [22, Theorem 4.4],where the latter has been proved by Khare and Wintenberger [18].


3.2. p-Adic Rankin–Selberg L-functions for Abelian varieties of GL.2/-typeFrom now on, we fix an Abelian variety A of GL.2/-type over F up to isogeny thatis automorphic (Definition 3.1.6(5)) and equipped with an embedding MA �Cp . Forsimplicity, in what follows, we put M WDMA DMA_ regarded as a subfield of Cp ,and we put FM D F ˝Q M , which is naturally equipped with a homomorphism toCp .

Notation 3.2.1Denote by B.A/ the (finite) set of isomorphism classes of totally definite incoherentquaternion algebras B over A that is E-embeddable (Definition 1.8.4) and such thatA can be parameterized by B (Definition 3.1.3).

For each (representative) B 2B.A/, we fix an isomorphism Bp'Mat2.Fp/ andan E-embedding (Definition 1.8.4) under which ep coincides with (1.7). Then wehave the F -scheme X D X.B/ and its closed subscheme Y D Y C

`Y � (Defini-

tion 2.4.1). We also fix an A1�E -equivariant isomorphism

c W Y C.Cp/��! Y �.Cp/; (3.1)

which we call an abstract conjugation for B.

Definition 3.2.2Denote by V the set of open compact subgroups of A1p�

E , which is a filtered partiallyordered set under inclusion. Let K be a complete field extension of Fp.(1) A (K-valued) character

� W E�nA1�E !K�

is a character of weight w 2 Z if the following hold.� � is invariant under some V p 2V.� There is an open compact subgroup Vp of E�p such that �.t/D .tP=

tPc/w for t 2 Vp.We call V p the tame level of �.

(2) For a character � of weight w as above, we define two characters L�P and L�Pc

of F �p by the formulas L�P.t/D t�w�P.t/ and L�Pc.t/D tw�Pc.t/.(3) Suppose thatK is contained in Cp . Let � be aK-valued character of weightw.

Given an isomorphism � W Cp��!C, we define the following local characters:

� �.�/v D 1 if v j1 but is not equal to �jF ;

� �.�/v .z/D .z= Nz/

w for vD �jF , where z 2E ˝F;� R�jE��!C;

� �.�/v D ��v for v <1 but v¤ p;


� �.�/p .t/D �. L�P.tı/ L�Pc.t�// for t 2E�p .

The product �.�/ WDNv �

.�/v W A

�E !C� is called the �-avatar of �.

(4) Suppose that K contains M . Denote by „.A;K/w the set of all K-valuedcharacters of weight w such that(a) !A � �jA1� D 1;(b) #¹v <1; v¤ p j .1=2; �A;v; �v/D�1º � g � 1 mod 2.Put „.A;K/D

SZ„.A;K/w .

Remark 3.2.3The character �.�/ is automorphic, that is, it factors through E�nA�E .

LEMMA 3.2.4For a character � 2„.A;K/, there is a unique element B� 2B.A/ such that .1=2;�A;v; �v/D �v.�1/�v.�1/.B�;v/ for every finite place v¤ p of F .

ProofThe existence of such B� follows from Definition 3.2.2(4.b). The uniqueness is clear,since B� is unramified at p and the .B�;v/’s are prescribed at all other places v.

The following definition generalizes the discussion in [26, Section 1].

Definition 3.2.5 (Distribution algebra)Let K=Fp be a complete field extension that contains M .(1) For a locally constant character ! W F �nA1�!M�, denote by C.!;K/ the

locally convex K-vector space of K-valued locally analytic functions f onthe locally Fp-analytic group E�nA1�E satisfying that� f is invariant under translation by some V p 2V;� f .xt/D !.t/�1f .x/ for all x 2E�nA1�E and t 2 F �nA1�.Let D.!;K/ be the strong dual of C.!;K/ as a topological K-algebra (seeRemarks 2.1.1 and 3.2.6).

(2) Define D.A;K/ to be the quotient K-algebra of D.!A;K/ divided by theclosed ideal generated by elements that vanish on „.A;K/� C.!A;K/.

(3) For B 2B.A/, define D.A;B;K/ to be the quotient K-algebra of D.!A;K/

divided by the closed ideal generated by elements that vanish on � 2„.A;K/with B� ' B for B� as in Lemma 3.2.4.

(4) Define

ı W E�nA1�E !D.!A;K/�!D.A;K/�!D.A;B;K/�

to be various continuous homomorphisms given by Dirac distributions.


Remark 3.2.6The topological K-vector space D.!;K/ is a commutative topological K-algebrawith the multiplication given by convolution (see [26, Section 1]). For a complete fieldextension K 0=K , we have D.!;K/bKK 0 'D.!;K 0/. Moreover, if K is discretelyvalued, then D.!;K/ may be written as a projective limit, indexed by tame levelsV p 2V, of nuclear Fréchet–Stein K-algebras with finite étale transition homomor-phisms (see Remark 4.4.5) and, thus, complete. We have similar remarks for D.A;K/

and D.A;B;K/.

Remark 3.2.7Suppose that F DM D Q and that ! D 1 is the trivial character. Fix a (sufficientlysmall) tame level V p 2 V. Define C.1;Qp; V p/ similarly to Definition 3.2.5 byrequiring that f be invariant under translation by A1�V p , and define D.1;Qp; V p/as the strong dual of C.1;Qp; V p/ as a topological Qp-algebra. Then for every com-plete field extensionK=Qp , there is a natural bijection between continuous charactersD.1;Qp; V p/!K� and continuous characters E�A1�nA1�E =V p!K�. In par-ticular, D.1;Qp; V p/ is isomorphic to the coordinate ring of a finite disjoint unionof open unit disks over Qp (compare with Section 2.1). See Remark 4.4.5 for aninterpretation in the more general case.

For a representative B 2B.A/, put �X;Y˙ D�1X jY˙ . For t 2 E�nA1�E , there

are canonical isomorphisms T�t �X;Y˙ '�X;Y˙ . Put

! ˙ D .‡�˙!�/jY˙˝F nr

pF ltpF

abp; (3.2)

where‡˙ are in Definition 2.2.10 and !� is the global Lubin–Tate differential in Def-inition 2.3.6. Then ! ˙ are sections of �X;Y˙ ˝Fp

F ltpF

abp , respectively, depending

only on the additive character (Remark 2.3.7).Let MF lt

pFabp �K � Cp be a complete intermediate field. Take a character � 2

„.A;K/k with k � 0, and take BD B�. Define �˙� to be theK-subspaces of H0.Y ˙;

�˝�kX;Y˙

/˝F K consisting of ' such that T�t ' D �.t/˙1', respectively. By Lemma

2.4.2(1), both �C� and �� have dimension 1. The abstract conjugation c from (3.1)induces an A1�E -invariant bilinear pairing

.�; �/� W �C� � �

�� !K

by the formula .'C; '�/� D .'C˝!k C/ � c�.'�˝!

k �/, where the right-hand side

is a K-valued constant function on Y C and, hence, can be regarded as an elementin K .

Put AC D A, A� D A_, and …˙ D…A˙ 2 A.B�� ;Q/. We have the canonicalpairing .�; �/A W …C �…�!M �Cp (Definition 3.1.4).


LEMMA 3.2.8Assume that k � 1. For every � W Cp

��! C, we have a unique B1� �A1�E -invariant

bilinear pairing

.�; �/.�/A;� W .…

C˝FM �C� /� .…�˝FM �� /! .LieAC˝FM LieA�/˝FM ;� C

such that, for every f˙ 2…˙, '˙ 2 �˙� , and !˙ 2H0.A˙;�1A˙/, we have˝

!C˝!�; .fC˝ 'C; f�˝ '�/.�/A;�

˛D .�'C˝ c�� '�˝�

k/

ZX�.C/

‚k�1� f �C!C˝ c�� ‚k�1� f ��!�

�kdx; (3.3)

where� h�; �i is the canonical pairing between H0.AC;�1

AC/˝FM H0.A�;�1A�/ and

LieAC˝FM LieA�;� � is an arbitrary Hecke invariant hyperbolic metric on X�.C/;� c� is the complex conjugation on MX� (2.22);� �'C ˝ c�� '�˝�

k is a constant function on Y C� .C/ and, hence, is viewed asa complex number;

� ‚� is the Shimura–Maass operator (Definition 2.4.7); and� dx is the Tamagawa measure on X�.C/.Moreover, there is a unique (nonzero) element P�.A;�/ 2 .LieAC ˝FMLieA�/˝FM ;� C such that

.�; �/.�/A;� D P�.A;�/ � �.�; �/A˝ �.�; �/�:

ProofFor given !˙ 2H0.A˙;�1

A˙/, the formula (3.3) defines a bilinear pairing

.…C˝FM �C� /� .…�˝FM �� /!C;

which is B1� � A1�E -invariant. By duality, all these pairings for different !˙ giverise to a nonzero pairing

.�; �/.�/A;� W .…

C˝FM �C� /� .…�˝FM �� /! .LieAC˝Q LieA�/˝F;� C;

and it is easy to see that .�; �/.�/A;� takes values in .LieAC˝FM LieA�/˝FM ;�C. Theexistence of P�.A;�/ follows from the uniqueness of the Petersson inner product andthe fact that .LieAC˝FM LieA�/˝FM ;�C is a C-vector space of dimension 1.

Remark 3.2.9The element P�.A;�/ can be viewed as a function on the set

Sk�1„.A;K/k valued


in the 1-dimensional C-vector space .LieAC ˝FM LieA�/˝FM ;� C. It depends onthe choices of c and .

THEOREM 3.2.10There is a unique element

L.A/ 2 .LieAC˝FM LieA�/˝FM D.A;MF ltp /

such that, for every character � 2„.A;K/k with k � 1 and MF ltpF

abp �K � Cp a

complete intermediate field and for every � W Cp��!C, we have

�L.A/.�/DL.1=2; �.�/A ; �

.�// �2g�1d1=2E �F .2/P�.A;�/

L.1; �/2L.1; �.�/A ;Ad/

� ��.1=2; ;�A;p˝ L�Pc/

L.1=2; �A;p˝ L�Pc/2

�(3.4)

as an equality in .LieAC˝FM LieA�/˝FM ;� C.

Remark 3.2.11The element L.A/ depends only on the choices of (1) an additive character ofFp of level 0 and (2) the abstract conjugation c from (3.1) for each (representative)B 2B.A/, in an elementary way. More precisely,(1) if we change to a for some a 2 O�p , where a.x/D .ax/ for x 2 Fp,

then L.A/ is multiplied by !�p.a/ � ı2a , where a is regarded at the place Pc in

the Dirac distribution ıa;(2) if we write L.A/D ¹L.A;B/ºB2B.A/ under the canonical isomorphism D.A;

K/'Q

B2B.A/D.A;B;K/ (Remark 4.4.4) and change c (for B) to c0 D Tt ıcfor some t 2 A1�E , then the component L.A;B/ is multiplied by ıt (Defini-tion 3.2.5(4)).

3.3. p-Adic Waldspurger formulaLet K be a complete field extension of M . Consider an element � 2„.A;K/0. Wetake BD B� 2B.A/. Choose a CM point PC 2 Y C.Eab/D Y C.Cp/, and put P� DcPC.

Definition 3.3.1For every f˙ 2…˙, we define the Heegner cycles P˙� .f˙/ on A˙ to be

P˙� .f˙/D

ZE�nA1�

E

f˙.TtP˙/˝M �.t/˙1 dt;


a finite sum in fact. Here, we recall that Tt is the Hecke morphism (Notation 2.2.1),and we adopt the Haar measure dt of total volume 2.

Suppose now that K contains MF abp . We have K-linear maps

logA˙ W A˙.K/˝M K! LieA˙˝FM K

given by p-adic logarithms on A˙ (see, e.g., [3]). As a functional on …C �…�,the product logAC P

C� .fC/ � logA� P

�� .f�/ defines an element in the 1-dimensional

K-vector space

HomA1�E.…C˝ �;K/˝K HomA1�

E.…�˝ ��1;K/˝FM .LieAC˝FM LieA�/:

It depends on the choice of c but not on the choice of PC.

THEOREM 3.3.2 (p-adic Waldspurger formula)There exists a unique element

˛�.�; �/ 2HomA1�E.…C˝ �;K/˝K HomA1�

E.…�˝ ��1;K/

such that, for every � W Cp��!C,

�˛�.fC; f�/D ˛\.fC; f�I�

.�//

for every f˙ 2…˙, where the right-hand side is the (normalized) matrix coefficientintegral appearing in the complex Waldspurger formula (which will be recalled inDefinition 4.1.4). Moreover, for a character � 2„.A;K/0, we have

logAC PC� .fC/ � logA� P

�� .f�/DL.A/.�/ �

L.1=2; �A;p˝ �Pc/2

.1=2; ;�A;p˝ �Pc/� ˛�.fC; f�/

for every f˙ 2…˙.

3.4. p-Adic Maass functions and alternative formulationLet B be an arbitrary totally definite incoherent quaternion algebra over A. As inSection 2.2, we have X.B/D lim

�UX.B/U as the projective limit of Shimura curves

associated to B over SpecF . The following definition generalizes the one in Sec-tion 1.2.

Definition 3.4.1 (p-adic Maass function)We say that a function � W X.B/.Cp/! Cp is a p-adic Maass function on X.B/if it is the pullback of some locally analytic function X.B/U .Cp/! Cp . Denoteby ACp .B

�/ the Cp-vector space of all p-adic Maass functions on X.B/. It is arepresentation of B1�.


We go back to the setting in Section 3.2, where we have fixed an Abelian varietyA of GL.2/-type over F up to isogeny that is automorphic and equipped with anembedding M DMA � Cp . Denote by �.B/rat

A the subspace of ACp .B�/ spanned

by functions of the form

f � log! W X.B/.Cp/f�!A.Cp/

log!��!Cp;

where f W X.B/!A is a nonconstant map, ! is a differential form on A˝QCp , andlog! D hlogA;!i. The subspace �.B/rat

A is a subrepresentation of B1�, which alsoreceives an action of M by acting on A. Denote by �.B/A the subspace of �.B/rat

A onwhichM acts via the default embeddingM �Cp , which is again a subrepresentationof B1�.

LEMMA 3.4.2Suppose that B belongs to B.A/ (Notation 3.2.1). For every nonzero differential form! 2H0.A;�1A/, the map

&! W ….B/A! �.B/A

sending f to f � log! jX.B/.Cp/ is B1�-equivariant and M -linear, and the inducedmap ….B/A˝M Cp! �.B/A is an isomorphism.

ProofIt follows directly from the definition that &! is B1�-equivariant and M -linear. Toshow the isomorphism, it suffices to show that �.B/A is a nonzero irreducible repre-sentation of B1�. Since B belongs to B.A/, the space �.B/rat

A is nonzero and, hence,so is �.B/A.

For the irreducibility, we choose an isomorphism � W Cp��! C. Consider the map

�.B/A ˝Cp ;� C! Acusp.B.�/�/ sending f � log! to �f �! regarded as a weight 2

holomorphic cusp form on B.�/�.A/. The map is well defined, injective, and B1�-equivariant. Its image coincides with the weight 2 subspace of the cuspidal automor-phic representation of B.�/�.A/ determined by A and the embedding � W M �C (see[30, Theorem 3.3.2]). It follows that the image is irreducible as a representation ofB1�. Therefore, �.B/A itself is an irreducible representation of B1�.

From now on, we fix a representative B in B.A/, and we will prove a p-adicWaldspurger formula for p-adic Maass functions on X WDX.B/ contained in �.B/A.Take two nonzero differential forms !˙ 2H0.A˙;�1

A˙/. By Lemma 3.4.2, we have

isomorphisms

&!˙ W ….B/A˙ ˝M Cp��! �.B/A˙ :


Let � 2„.A;Cp/0 be a character such that B� ' B. For �˙ 2 �.B/A˙ , we put

˛!C;!�� .�C; ��/D ˛�.&

�1!C�C; &

�1!��/ 2Cp;

where ˛�.�; �/ is the pairing in Theorem 3.3.2. Then ˛!C;!�� .�; �/ is a basis of the

1-dimensional space

HomA1�E

��.B/AC ˝ �;Cp

�˝Cp HomA1�

E

��.B/A� ˝ �

�1;Cp�: (3.5)

Globally, we have the following definition. Choose a CM point PC 2 Y C.Eab/ D

Y C.Cp/, and put P� D cPC as in Section 3.3.

Definition 3.4.3For �˙ 2 �.B/A˙ and � 2„.A;Cp/0, we define the p-adic torus period to be

PCp .�˙; �˙1/ WD

ZE�nA1�

E

�˙.TtP˙/ � �.t/˙1 dt;

where the Haar measure dt has total volume 2 as in Definition 3.3.1.

The above integrals are, in fact, finite sums valued in Cp . The product PCp .�; �/ �

PCp .�; ��1/ defines another element in (3.5), which depends on the choice of c but

not on the choice of PC. In particular, it is proportional to ˛!C;!�� .�; �/.

The following theorem is the p-adic Waldspurger formula for p-adic Maass func-tions. Recall that we have the p-adic L-function L.A/ from Theorem 3.2.10. Put� D �.B/A as an irreducible subrepresentation of ACp .B

�/.

THEOREM 3.4.4 (p-adic Waldspurger formula for p-adic Maass functions)Put

L!C;!�.�/D˝!C˝!�;L.A/

˛;

regarded as an element in D.A;Cp/. Then for a character � 2„.A;Cp/0, we have

PCp .�C; �/PCp .��; ��1/DL!C;!�.�/ �

L.1=2;�p˝ �Pc/2

.1=2; ;�p˝ �Pc/� ˛!C;!�� .�C; ��/

for every �˙ 2 �.B/A˙ .

ProofIt follows from Theorem 3.3.2, after pairing with !C˝!�.


Remark 3.4.5In this remark, we explain how to deduce Theorems 1.5.1 and 1.5.3. We take A to bean elliptic curve over Q. In particular, we have that F DM DQ, AC DA� DA andthat !A D 1 is the trivial character. We also fix an isomorphism � W Cp

��!C. We have

the indefinite quaternion algebra B over Q. Take B 2B.A/ such that B1 ' B ˝Q

A1. So we may identify B with B.�/ in the �-nearby data for B (Definition 2.4.10).Moreover, we take W Qp ! C�p to be the additive character such that � ı is thestandard one. We choose the abstract conjugation c from (3.1) such that c˝Cp ;� C

coincides with the restriction of the complex conjugation on MX�. We also note thatD.G/ is simply D.1;Cp/; and D.GI�Cp / is simply D.A;B;Cp/ (Definition 3.2.5).

(1) We first deduce Theorem 1.5.1. Take the p-adic L-function L.A/ as in The-orem 3.2.10, regarded as an element in .LieA/˝2 ˝Q D.A;Cp/. Take a basis ! ofH0.A;�1A/. Then there is a unique element P! 2 C�p such that �.P�1! .f1; f2/A/ isequal to the (bilinear) Petersson inner product of ��.f �1 !/ and R.. 1 �1 /1/��.f

�2 !/

for every f1; f2 2…A. Now we define L.�Cp / to be the image of

P! � ��1�d�1=2E L.1; �/

��˝! ˝!;L.A/

˛under the canonical projection D.A;Cp/!D.A;B;Cp/DD.GI�Cp /. It is clearthat L.�Cp / does not depend on the choice of ! and, hence, is well defined. ThenTheorem 1.5.1 follows from Theorem 3.2.10, Remark 1.1.2, and Lemma 3.4.6 below(with r D 2).

(2) Now we deduce Theorem 1.5.3. In Definition 3.4.3, we choose PC suchthat �PC D ŒCi; 1�, and thus, �P� D Œ�i; 1�. Then PCp .�; �

˙1/ in Definition 3.4.3coincide with those in (1.5). Therefore, Theorem 1.5.3 follows from Theorem 3.4.4.

LEMMA 3.4.6Let � be the discrete series representation of weight r � 2 of GL2.R/ with trivial cen-tral character. Fix a nonzero GL2.R/-equivariant bilinear pairing .�; �/ W � ��!C.Let fC 2 � be a generator of weight r , that is, the Archimedean component of holo-morphic modular forms of weight r . Put f� D �.. 1 �1 //fC, which is a generator ofweight �r . Then we have

.�kCfC;�k�f�/

.fC; f�/DkŠ.kC r � 1/Š

4k.r � 1/Š;

where �˙ are as in Definition 2.4.16.

ProofIt is well known that .fC; f�/¤ 0. Put

X˙ D1

2

�1 ˙i

˙i �1

�D 2i�˙; H D�i

�1

�1

�:


Recall from [5, p. 157, (2.39)] the Casimir element

�D�1

4.2XCX�C 2X�XCCH

2/:

It acts on � by the scalar

r WDr

2

�1�

r

2

�:

We say that a vector g 2 � has weight � if HgD �g. For such g, we have

X�XCgD��H 2C 2H

4C�

�gD�

��2C 2�4

C r

�g:

Now for each k � 0, the vector XkCfC is of weight r C 2k. Therefore, we have fork � 1 the formula

X�XkCfC DX�XC.X

k�1C fC/D�

� .r C 2k � 2/.r C 2k/4

C r

�Xk�1C fC:

We prove the identity

.XkCfC;Xk�f�/

.fC; f�/DkŠ.kC r � 1/Š

.r � 1/Š

by induction on k � 0. The case k D 0 is trivial. Suppose that we know this for k� 1.Then we have

.XkCfC;Xk�f�/D�.X�X

kCfC;X

k�1� f�/

D� .r C 2k � 2/.r C 2k/

4C r

�� .Xk�1C fC;X

k�1� f�/

D k.kC r � 1/ � .Xk�1C fC;Xk�1� f�/:

The lemma follows as X˙ D 2i�˙.

4. Proofs of main theoremsThis section is dedicated to the proofs of Theorems 3.2.10 and 3.3.2. In Section 4.1,we construct the distribution interpolating matrix coefficient integrals appearing inthe complex Waldspurger formula. We construct the universal torus period in Sec-tion 4.2, which is a crucial construction toward the p-adic L-function. In Section 4.3,we study the relation between universal torus periods and classical torus periods,based on which we complete the proofs of our main theorems in Section 4.4.


4.1. Distribution of matrix coefficient integralsRecall that we have fixed an Abelian variety A of GL.2/-type over F up to isogenythat is automorphic, equipped with an embedding M DMA �Cp , as in Section 3.2.Let K=MFp be a complete field extension. Take a representative B in B.A/. Asin Section 3.2, we fix an isomorphism Bp 'Mat2.Fp/ and an E-embedding underwhich ep coincides with (1.7). Recall that we put …˙ D …A˙ D ….B/A˙ (Nota-tion 3.1.2).

Definition 4.1.1 (Stable/admissible vector)We say that elements f˙ in …˙ ˝M K or …˙p ˝M K are stable vectors if, respec-tively,(1) f˙ are fixed by N˙.Op/;(2) f˙ satisfy the relation X

g2N˙.p�1/=N˙.Op/

…˙p .g/f˙ D 0:

We denote by .…˙/~K the subsets of …˙˝M K consisting of stable vectors, respec-tively. We denote by .…˙p /

~K the subsets of …˙p ˝M K consisting of stable vectors,

respectively.For n 2 N, we say that stable vectors f˙ in .…˙/~K or .…˙p /

~K are n-admissible

if, respectively,

…˙p�n˙.x/

�f˙ D

˙.x/f˙

for every x 2 p�n=Op, where n˙.x/ are the same as in Proposition 2.3.5.

Remark 4.1.2If we realize…˙p in their Kirillov models with respect to the pair .NC; ˙/, then f˙pbelong to .…˙p /

~K if and only if fCp (resp., …�p .J/f�p) is supported on O�p , and they

are n-admissible if and only if fCp (resp., …�p .J/f�p) is supported on .1C pn/�.

Definition 4.1.3Let w 2 Z, let n 2 N be integers, and let ! W F �nA1�!M� be a locally constantcharacter. We say that a K-valued character � W E�nA1�E ! K� of weight w is ofcentral type ! and depth n if� ! � �jA1� D 1; and� �Pc.t/D t�w for all t 2 .1C pn/�.We denote by „.!;K/nw the set of all K-valued characters of weight w, central type!, and depth n. Moreover, put „.!;K/n D

SZ„.!;K/

nw .


We recall the definition of the classical (normalized) matrix coefficient integral.Suppose that K is contained in Cp . We take a character � 2„.!A;K/. Let � W Cp

��!

C be an isomorphism. Morally speaking, the integral should be defined as

˛\.fC; f�I�.�// “D”

ZA1�nA1�

E

��….t/fC; f�

�A� �.�/.t/ dt:

However, it is not absolutely convergent, so we need regularization recalled as fol-lows.

Definition 4.1.4 (Regularized matrix coefficient integral)Take an arbitrary decomposition �.�; �/A D

Qv<1.�; �/�;v , where .�; �/�;v W …Cv �…

�v !

C is a B�v -invariant bilinear pairing. For f˙ DNv<1 f˙v such that .fCv; f�v/�;v D

1 for all but finitely many v’s, we put

˛.fCv; f�vI�.�/v /D

ZF �v nE

�v

�…v.t/fCv; f�v

��;v�.�/v .t/ dt I

˛\.fCv; f�vI�.�/v /D

��Fv .2/L.1=2; �.�/A;v; �.�/v /L.1; �v/L.1; �

.�/A;v;Ad/

��1˛.fCv; f�vI�

.�/v /:

Here, dt is the measure on F �v nE�v given in Section 1.8, and �.�/A;v is the correspond-

ing admissible complex representation of B�v via �. Then by [29, Section 3] we have˛\.fCv; f�vI�

.�/v /D 1 for all but finitely many v’s, and the product

˛\.fC; f�I�.�// WD

Yv<1

˛\.fCv; f�vI�.�/v /

is well defined. We extend the functional ˛\.�; �I�.�// to all fC; f� by linearity.

Remark 4.1.5The functional ˛\.�; �I�.�// does not depend on the choice of the decomposition of�.�; �/A.

The following proposition is our main result, whose proof will be given at the endof this section. Note that, since „.!;K/n is a subset of C.!;K/, we have a naturalpairing D.!;K/�„.!;K/n!K .

PROPOSITION 4.1.6Let MFp � K � Cp be a complete intermediate field. Let f˙ 2 .…˙/

~K be two n-

admissible stable vectors for some (common) n 2 N. Then there is a unique element


Q.fC; f�/ 2 D.!A;K/ such that, for all K-valued characters � 2 „.!A;K/n ofcentral type !A and depth n and for � W Cp

��!C, we have

�Q.fC; f�/.�/D �� L.1=2; �A;p˝ L�Pc/2

.1=2; ;�A;p˝ L�Pc/

�� ˛\.fC; f�I�

.�//:

Definition 4.1.7The element Q.fC; f�/ is called the (K-valued) local period distribution.

Before giving the proof, we make a convenient choice of a decomposition of.�; �/A. Realize the representations …˙p in their Kirillov models as in Remark 4.1.2.We may assume that f˙ D

Nf˙v , with f˙v 2…˙v ˝M K , are decomposable and are

fixed by some (common) sufficiently small open compact subgroup V p 2V. Choosea decomposition .�; �/A D

Qv<1.�; �/v such that

(1) .fCv; f�v/v D 1 for all but finitely many v’s;(2) .f 0Cv; f

0�v/v 2K for all f 0˙v 2…

˙v ˝M K;

(3) for f 0˙p 2…˙p ˝M K that are compactly supported on F �p ,

.f 0Cp; f0�p/pD

ZF �p

f 0Cp.a/f0�p.a/da;

where da is the Haar measure on F �p such that the volume of O�p is 1.We need two lemmas for the proof of Proposition 4.1.6. For simplicity, write

! D !A. For each finite place v ¤ p and an open compact subgroup Vv of E�v , letD.!v;K;Vv/ be the quotientK-algebra ofD.E�v =Vv;K/ divided by the closed idealgenerated by ¹!v.t/ıt � 1 j t 2 F �v º. Put

D.!v;K/D lim �Vv

D.!v;K;Vv/;

where the limit runs over all Vv’s. Let D.!p;K/ be the quotient of D.E�p ;K/ by theclosed ideal generated by ¹!p.t/ıt � 1 j t 2 F

�p º. For every finite place v, we have a

natural homomorphism D.!v;K/!D.!;K/.

LEMMA 4.1.8Let v¤ p be a finite place of F .(1) There exists a unique element

L�1.�A;v/ 2D.!v;MFp/

such that, for every locally constant character �v W E�v !K� satisfying !v ��vjF �v D 1, we have

L�1.�A;v/.�v/DL.1=2; �A;v; �v/�1:


(2) For f˙v 2…˙v ˝M K , there exists a unique element

Q.fCv; f�v/ 2D.!v;K/

such that, for every locally constant character �v W E�v !K� satisfying !v �

�vjF �v D 1 and for � W Cp��!C, we have

�Q.fCv; f�v/.�v/D ˛\.fCv; f�vI�

.�/v /:

ProofThe uniqueness is clear. In the following proof, we suppress v from the notation, andwe will use the subscript � for all changing of coefficients of representations via �.

To prove (1), we first consider the following situation. Let QF be either F or E ,and let Q… be an irreducible admissible M -representation of GL2. QF /. We claim thatthere is a (unique) element L�1

QF. Q…/ 2D[. QF

�;MFp/, where

D[. QF�;K/ WD lim

�V

D. QF �=V;K/

with V running over all open compact subgroups of QF � such that, for every locallyconstant character � W QF �!K� and � W Cp

��!C,

�L�1QF .Q…/.�/DL.1=2; Q…�˝ ��/

�1:

In fact, for a locally constant character � W QF � ! M�, define L�1QF.�/ 2 D[. QF

�;

MFp/ by the formula

L�1QF .�/.h/D 1�

ZO�QF

�. Q$a/h. Q$a/da

for h 2 lim�!V

C. QF �=V;MFp/. Here, Q$ is an arbitrary uniformizer of QF , and da isthe Haar measure on O�

QFwith total volume 1. Then we have three cases.

� If Q… is supercuspidal, put L�1QF. Q…/D 1.

� If Q… is the unique irreducible subrepresentation of the unnormalized parabolicinduction of .�;�j � j�2/ for a character� W QF �!M�, then we put L�1

QF. Q…/D

L�1QF.�/.

� If Q… is the irreducible unnormalized parabolic induction of .�1;�2j � j�1/for a pair of characters �i W QF � !M� (i D 1; 2), then we put L�1

QF. Q…/ D

L�1QF.�1/ �L�1

QF.�2/.

Here, we adopt the unnormalized induction in order to track the rationality properties.Go back to (1). First, assume that E=F is nonsplit. Then we define L�1.�A/

to be the image of L�1E .…E / in D.!;MFp/, where …E is the base change of … to


GL2.E/, which depends only on �A. Second, assume thatE D F��Fı is split, whereF� D Fı D F . Then we define L�1.�A/ to be the image of L�1F� .…/˝L�1Fı .…/ inD.!;MFp/.

Now we consider (2). First, assume that E=F is nonsplit. Then the torus F �nE�

is compact; hence, the matrix coefficient ˆfC;f�.g/ WD .…C.g/fC; f�/ is finiteunder E�-translation. We may assume that the restrictionˆfC;f� jE� D

Pi ai�i is a

finite K-linear combination of K-valued (locally constant) characters �i of E� suchthat ! � �i jF � D 1. Assigning to every locally constant function h on E� satisfying!.t/h.at/D h.a/ for all a 2E� and t 2 F � the integralX

i

ai

ZF �nE�

�i .t/h.t/ dt;

which is a finite sum, defines an element ˛.fC; f�/ in D.!;K/. Put

Q.fC; f�/D� �F .2/

L.1; �A;Ad/L.1; �/

��1L�1.�A/˛.fC; f�/:

Second, assume that E D F� � Fı is split. We may suppose that the embeddingE!Mat2.F / is given by

.t�; tı/ 7!

�t�

tı

�for t�; tı 2 F . Moreover, a character � of E� is given by a pair .��; �ı/ of charactersof F � such that �..t�; tı//D ��.t�/�ı.tı/.

Now we realize …˙ in their Kirillov models with respect to (nontrivial) additivecharacters ˙ W F ! C� of conductor 0, respectively, where � D . C/�1. More-over, we may assume, for f˙ 2 …˙ ˝M K that are compactly supported on F �,that

.fC; f�/D

ZF �fC.a/f�.a/da;

where da is the Haar measure on F � such that the volume of O�F is c for somec 2M . We have the formula

˛\.fC; f�I��/D��F .2/L.1=2; �.�/A ; ��/L.1; �/L.1; �

.�/A ;Ad/

��1�

ZF ��fC.a/ � ��.a/da

ZF ��f�.b/ � ��.b

�1/ db

D� �F .2/

L.1; �/L.1; �.�/A ;Ad/

��1Z.�fC; ��/Z.�f�; �

�1�� /; (4.1)


where

Z.�f˙; �˙1�� /DL.1=2;…

˙� ˝ �

˙1�� /�1

ZF ��f˙.a/ � �

˙1�� .a/da:

Note that the above integrals are simply local zeta integrals. To conclude, it suf-fices to show that there exist elements Z.f˙/ 2 D[.F

�;K/ such that, for everylocally constant character � W F �!K� and every isomorphism � W Cp

��!C, we have

�Z.f˙/.�/DZ.�f˙; �˙1� /, respectively. Without loss of generality, we only construct

Z.fC/.By enlarging M if necessary to include l1=2, where l is the cardinality of the

residue field of F , there is a subspace …C;c of …C such that …C;c ˝M K is thesubspace of …C ˝M K of functions that are compactly supported on F �. For fC 2…C;c ˝M K , we may define Z.fC/ such that, for every locally constant function hon F �,

Z.fC/.h/DL�1F .…C/.h/�

ZF �fC.a/h.a/da:

Therefore, we may conclude the proof if dim…C=…C;c D 0. There are two casesremaining.

First,…C is a special representation, that is, dim…C=…C;c D 1. We may choosea representative fC D �.a/ � chOF n¹0º.a/ for some character � W F �!M�. ThenZ.�fC; ��/D c (resp., 0) if � � � is unramified (resp., otherwise). Therefore, we maydefine Z.fC/ such that

Z.fC/.h/D

ZO�F

�.a/h.a/da

for every locally constant function h on F �.Second, …C is a principal series, that is, dim…C=…C;c D 2. There are two pos-

sibilities. In the first case, we may choose representatives f iC D �i .a/ � chOF n¹0º.a/

for two different characters �1;�2 W F �!M�. Without loss of generality, we con-sider f 1C. ThenZ.�f 1C; ��/DL.1=2;�

2� ��/

�1 (resp., 0) if �1 �� is unramified (resp.,otherwise). Therefore, we may define Z.f 1C/ such that

Z.f 1C/.h/DL�1F .�1/.h/�

ZO�F

�.a/h.a/da

for every locally constant function h on F �. In the second case, we may chooserepresentatives f 1C D �.a/ � chOF n¹0º.a/ and f 2C D .1 � logl jaj/�.a/ � chOF n¹0º.a/for some character � W F �!M�. The function f 1C has been treated above. For f 2C,we have Z.�f 2C; ��/D c (resp., 0) if � � � is unramified (resp., otherwise). Therefore,we may define Z.fC/ such that


Z.fC/.h/D

ZO�F

�.a/h.a/da

for every locally constant function h on F �.

LEMMA 4.1.9Let f˙p 2 .…˙p /

~K be two n-admissible stable vectors. There exists a unique element

Q.fCp; f�p/ 2D.!p;K/

with the following property: for every character �p W E�p ! K� satisfying !p �

�pjF �p D 1 and �Pc.t/D t�w for t 2 .1Cpn/� and some w 2 Z and for � W Cp��!C,

we have

�Q.fCp; f�p/.�p/D �� L.1=2; �A;p˝ L�Pc/2

.1=2; ;�A;p˝ L�Pc/

�˛\.fCp; f�pI�

.�/p /:

Here, L� is defined similarly as in Definition 3.2.2(2). Moreover, there are n-admissiblestable vectors f˙p 2 .…˙p /

~K such that Q.fCp; f�p/.�p/¤ 0 for every such �p.

ProofThe uniqueness of Q.fCp; f�p/ is clear, as those characters �p in the statement spana dense subspace of C.!p;K/ by Lemma 2.1.11.

For the existence of Q.fCp; f�p/, first note that the formula (4.1) also works forvD p. Moreover, we have the functional equation

Z.�f�p; �.�/�1Pc /D �.1=2; ;�A;p˝ L�Pc/ �Z

��…�p .J/f�p

�; �.�/Pc

�:

By Remark 4.1.2, we only need to show that, for f 2…Cp ˝M K that is supported on.1C pn/�, there exists Q0.f / 2D.!p;K/ such that, for �p as in the statement and �,

�Q0.f /.�p/D

ZO�p

�f .a/ � �.�/Pc.a/da:

Then we may set

Q.fCp; f�p/D� �Fp

.2/

L.1; �A;p;Ad/L.1; �p/

��1Q0.fCp/Q

0�…�p .J/f�p

�:

For the existence of Q0.f /, since �.�/Pc restricts to the trivial character on .1C

pn/�, we have ZO�p

�f .a/ � �.�/Pc.a/daD �

ZO�p

f .a/da:


We may put

Q0.f /D

ZO�p

f .a/da 2K; (4.2)

which is a constant (depending only on f ). The last part of the lemma follows from(4.2).

Proof of Proposition 4.1.6Let f˙ 2 .…˙/

~K be two n-admissible stable vectors. It is clear that Q.fCv; f�v/

constructed in Lemma 4.1.8 is equal to 1 for almost all v’s. Therefore, we may simplydefine Q.fC; f�/ to be the image of

Q.fCp; f�p/˝Ov¤p

Q.fCv; f�v/

in D.!A;K/.

4.2. Universal torus periodsLet B be as in the previous section. As in Section 3.3, we choose a CM point PC 2Y C.Eab/ and put P� D cPC. By Lemma 2.4.2, we regard P˙ as points inX.˙1/.F ab

p /, respectively. By the same lemma, the morphism �t from (2.14) pre-serves Y˙.1/ for t 2O�Ep

, respectively.

Recall that, for m 2 N[ ¹1º, we have the closed formal subscheme Y˙.m/ ofX.m/ as in Section 2.4. For a complete field extension K=F nr

p , put

N ˙.m;K/DH0�Y˙.m/;O

Y˙.m/

�bOnrpK:

LEMMA 4.2.1Suppose that K is a complete field extension of F ab

p . Then the respective maps fromN ˙.1;K/ to the K-algebra of continuous K-valued functions on E�nA1�E thatsend f 2 N ˙.1;K/ to the functions x 7! f .‡˙TxP˙/ are isomorphisms. Werecall that ‡˙ are in Definition 2.2.10. Moreover, the induced actions of t 2 O�Ep

on N ˙.1;K/ are, respectively, given by

.��t f /.x/D

´f .xt/ for f 2N C.1;K/;

f .xt c/ for f 2N �.1;K/;

for x 2E�nA1�E .

ProofThe isomorphism follows from Lemmas 2.4.2 and 2.4.4. The action is a consequenceof Lemma 2.2.11.


Notation 4.2.2Consider a locally constant character

! W F �nA1�!M�:

Let K be a complete field extension of MFp. For every V p 2V on which ! is trivial,denote by D.!;K;V p/ the quotient K-algebra of D.E�nA1�E =V p;K/ divided bythe closed ideal generated by ¹!.t/ıt � 1 j t 2A1�º.

Then by some standard facts from functional analysis (see [27, Propositions 2.11and 2.12]) and Remark 2.1.1, we have a canonical isomorphism

D.!;K/' lim �V p2V

D.!;K;V p/

of topologicalK-algebras, where the former one is in Definition 3.2.5(1). The (unique)continuous homomorphism D.O�Ep

;K/ ! D.E�nA1�E =V p;K/ sending ıt to

!p.tı/ıt for t D .t�; tı/ 2O�Epdescends to a continuous homomorphism w W D.Oanti

p ;

K/!D.!;K;V p/ of K-algebras, which is compatible with respect to the changeof V p. In other words, we have a homomorphism

w W D.Oantip ;K/!D.!;K/: (4.3)

Definition 4.2.3 (Universal character)We respectively define the ˙-universal character to be

�˙univ W E�nA1�E

ı˙1

��!D.!;MFp/�;

where ı is defined in Definition 3.2.5(4).

The universal characters depend on !. Since we will always take ! D !A, wesuppress it from notation.

LEMMA 4.2.4The universal characters �˙univ are elements in N ˙.1;F ab

p /bF abp

D.!;MF abp / satis-

fying

��t �˙univ D ıt � �

˙univ;

respectively, for t 2O�Ep;mif ! is trivial on F �p \O

�Ep;m

.

ProofIt follows from the definition, Lemma 4.2.1, and the observation that conjugation andinversion coincide on Oanti

p .


Suppose that K is a complete field extension of MF ltpF

abp . Given a stable conver-

gent modular form f 2Mw[.m;K/~ for some w;m 2N (Definition 2.3.10), we have

the global Mellin transform M.f / by Theorem 2.3.17, and by (4.3),

wM.f / 2Mw[ .m;K/

~bKD.!;K/:

By restriction, we obtain elements

wM.f /jY˙.1/ 2N ˙.1;K/bKD.!;K/:

By Theorem 2.3.17(2) and Lemma 4.2.4, the product .wM.f /jY˙.1// ��

˙univ descends

to an element in N ˙.m;K/bKD.!;K/ if ! is trivial on F �p \O�Ep;m

.

For every V p 2V under which f is invariant, we regard .wM.f /jY˙.1// ��

˙univ

as elements in N ˙.m;K/bKD.!;K;V p/, respectively. They are invariant under theaction of V p on N ˙.m;K/.

Definition 4.2.5 (Universal torus period)We define the universal torus periods of f to be the elements

P˙! .f / WD2

jE�nA1�E =V pO�Ep;mj

XE�nA1�

E=V pO�

Ep;m

��wM.f /j

Y˙.1/

�� ˙univ

�.t/

in D.!;K;V p/.

Remark 4.2.6We add the factor 2 in the above definition in order to be consistent with the Tamagawameasure we chose in the complex Waldspurger formula recalled in Section 1.1.

By construction, the elements P˙! .f / are independent of m and are compatiblewith respect to the change of V p. Therefore, they are elements in D.!;K/. In fact,for a character � 2„.!;K/, Lemma 4.2.1 allows us to write

P˙! .f /.�/D

ZE�nA1�

E

wM�.f �˙!˙/ord

�.�/.‡˙TtP

˙/ � �.t/˙1 dt:

4.3. Interpolation of universal torus periodsWe keep the setting from the previous section. LetMF lt

pFabp �K �Cp be a complete

intermediate field.By Definition 4.1.1, elements f ˙ 2…˙˝F K can be realized as K-linear com-

binations of morphisms fromXUpUp;˙mto A, respectively, for some (common) U p 2

U and m 2N. We now assume this.Take differential forms !˙ 2 H0.A˙;�1

A˙/. Using the notation in (2.26), we

have convergent modular forms .f �˙!˙/ord 2M2[.1;K/. Then .f �˙!˙/ord are sta-

ble (in the sense of Definition 2.3.10) if and only if f ˙ are stable (in the sense of


Definition 4.1.1), respectively. By Proposition 2.3.5(3), .f �˙!˙/ord are n-admissible(in the sense of Definition 2.3.13) if and only if f ˙ are n-admissible (in the sense ofDefinition 4.1.1), respectively.

Notation 4.3.1For stable vectors f ˙ 2 .…˙/~K , define the elements

P˙univ.f˙/ 2 LieA˙˝FM D.!A;K/

by the formulas ˝!˙;P

˙univ.f˙/

˛DP˙!A

�.f �˙!˙/ord

�:

In this section, we study the relation between

�PCuniv.fC/.�/ � �P�univ.f�/.�/ 2 .LieAC˝FM LieA�/˝FM ;� C

for a given isomorphism � W Cp��! C, with classical torus periods, for f ˙ as above

and a character � 2„.!A;K/nk of weight k � 1 and depth n (Definition 4.1.3). Forthis purpose, we choose an �-nearby data for B (Definition 2.4.10). In particular, wehave

Y ˙� .C/DE�n¹˙iº �A1�E �X�.C/:

Choose elements t˙ 2 A1�E such that �P˙ are represented by Œ˙i; t˙�, respectively.Define �˙� 2C

� such that

dz�Œ˙i; t˙�

�D �˙� � �! ˙jP˙ ; (4.4)

where ! ˙ are defined in (3.2). We also introduce matrices j˙� D . 1 ˙1 / inMat2.R/DB.�/˝F;� R.

Notation 4.3.2For a cusp form ˆ 2 Acusp.B.�/

�/ with central character � ı !˙1A , we respectivelydefine

PC.ˆ;�.�/˙1/ WD

ZE�A�nA�

E

ˆ.t/�.�/.t/˙1 dt

to be the complex torus periods appearing in the complex Waldspurger formula.

LEMMA 4.3.3Let the notation be as above. We have


�˝!C;P

Cuniv.fC/.�/

˛� �˝!�;P

�univ.f�/.�/

˛D .�C� �

�� /k � �.�/.t�1C t�/

�PC

��k�1C;�

�R.jC� /��.f

�C!C/

�; �.�/C1

�PC

��k�1�;�

�R.j�� /��.f

��!�/

�; �.�/�1

�;

where �� is defined in Lemma 2.4.15 and �.�/ is the �-avatar of � as in Defini-tion 3.2.2(3).

ProofTake V p 2 V under which f˙ and � are invariant. By Theorem 2.3.17 and Defini-tion 4.2.5, we have˝!˙;P

˙univ.f˙/.�/

˛D

2


�X

E�nA1�E

=V pO�Ep;m

‚k�1ord .f�˙!˙/ord.‡˙TtP

˙/ �!�k� .‡˙TtP˙/ � �˙1.t/

D2


�X

E�nA1�E

=V pO�Ep;m

‡�˙‚k�1ord .f

�˙!˙/ord.TtP

˙/ �!�k ˙.TtP˙/ � �˙1.t/

for some sufficiently large m� n. By (4.4) and Lemma 2.4.9, we have

�˝!˙;P

˙univ.f˙/.�/

˛D

2.�˙� /k


�X

E�nA1�E

=V pO�Ep;m

‚k�1� f �˙!˙.TtP˙/ dz

�Œ˙i; t˙t �

��k� �.�/.t/˙1

D .�˙� /k

ZE�A�nA�

E

R.j˙� /��.‚k�1� f �˙!˙/.t˙t / � �

.�/.t/˙1 dt

D .�˙� /k � �.�/.t�1˙ /

ZE�A�nA�

E

R.j˙� /��.‚k�1� f �˙!˙/.t/ � �

.�/.t/˙1 dt;

which by Lemma 2.4.17 equals


.�˙� /k � �.�/.t�1˙ /

ZE�A�nA�

E

R.j˙� /��k�1C;� ��.f

�˙!˙/

�.t/ � �.�/.t/˙1 dt

D .�˙� /k � �.�/.t�1˙ /

ZE�A�nA�

E

�k�1˙;��R.j˙� /��.f

�˙!˙/

�.t/ � �.�/.t/˙1 dt:

This completes the proof.

PROPOSITION 4.3.4Given n-admissible stable vectors f˙ 2 .…˙/

~K and a character � 2„.!A;K/nk of

weight k � 1 and depth n, we have

�PCuniv.fC/.�/ � �P�univ.f�/.�/

D �Q.fC; f�/.�/

�L.1=2; �.�/A ; �

.�// �2g�1d1=2E �F .2/P�.A;�/

L.1; �/2L.1; �.�/A ;Ad/

� ��.1=2; ;�A;p˝ L�Pc/

L.1=2; �A;p˝ L�Pc/2

�;

as an equality in .LieAC˝FM LieA�/˝FM ;� C.

ProofIt suffices to show the equality after pairing with !C ˝ !� for an arbitrary pair ofdifferential forms !˙ 2H0.A˙;�1

A˙/.

By the complex Waldspurger formula (see [29] and [30, Theorem 1.4.2]) andProposition 4.1.6, we have

PC

��k�1C;�

�R.jC� /��.f

�C!C/

�; �.�/C1

�PC

��k�1�;�

�R.j�� /��.f

��!�/

�; �.�/�1

�D C�

�F .2/L.1=2; �.�/A ; �

.�//

L.1; �.�/A ;Ad/L.1; �/

2

2�gd�1=2E L.1; �/� ��.1=2; ;�A;p˝ L�Pc/

L.1=2; �A;p˝ L�Pc/2

�� Q.fC; f�/.�/;

where C� is the complex constant such that��k�1C;�

�R.jC� /��.f

�C!C/

�;�k�1�;�

�R.j�� /��.f

��!�/

��Pet D C� � �.fC; f�/A

holds for all fC and f�. Here .�; �/Pet is the bilinear Petersson inner product pairing.By Lemma 4.3.3, the proposition is reduced to the formula˝

!C˝!�;P�.A;�/˛D C� � .�

C� �� /k � �.�/.t�1C t�/: (4.5)

By Lemma 3.2.8, it suffices to show that


�'C˝ c�� '�˝�k

�.'C; '�/�

ZX�.C/

‚k�1� f �C!C˝ c�� ‚k�1� f ��!�

�kdx

D .�C� �� /k � �.�/.t�1C t�/ �

��k�1C;�

�R.jC� /��.f

�C!C/

�;�k�1�;�

�R.j�� /��.f

��!�/

��Pet

for some choice of '˙ 2 �˙� .Cp/. We take elements '˙ such that �'˙.P˙/ Ddz.Œ˙i; t˙�/

k , respectively. Then �.'C; '�/� D .�C� �� /�k by (4.4). Now we take �

to be the standard invariant hyperbolic metric on MX� D B.�/�nH � B1�=F �. Then�'C˝ c�� '�˝�

k is the constant �.�/.t�1C t�/, andZX�.C/

‚k�1� f �C!C˝ c�� ‚k�1� f ��!�

�kdx

D��k�1C;�

�R.jC� /��.f

�C!C/

�;�k�1�;�

�R.j�� /��.f

��!�/

��Pet:

Thus, (4.5) holds, and the proposition follows.

The proposition has the following corollary.

COROLLARY 4.3.5For � 2„.!A;K/nk with k � 1, the ratio

PCuniv.fC/.�/P�univ.f�/.�/

Q.fC; f�/.�/2 .LieAC˝FM LieA�/˝FM K;

if the denominator is nonzero, is independent of the choice of n-admissible stablevectors f˙ 2 .…˙/

~K . Moreover, for � W Cp

��!C, we have

��PCuniv.fC/.�/P

�univ.f�/.�/

Q.fC; f�/.�/

�DL.1=2; �

.�/A ; �

.�// �2g�1d1=2E �F .2/P�.A;�/

L.1; �/2L.1; �.�/A ;Ad/

� ��.1=2; ;�A;p˝ L�Pc/

L.1=2; �A;p˝ L�Pc/2

�:

PROPOSITION 4.3.6For n-admissible stable vectors f˙ 2 .…˙/

~K and a character � 2 „.!A;K/n0 of

weight 0 and depth n, we have

P˙univ.f˙/.�/D logA˙ P˙� .f˙/:

ProofWe may choose a tame level U p 2 U that fixes both fC and f� and such that � isfixed by U p\A1�E . We may realize f˙ asK-linear combinations of morphisms from


XUpUp;˙mtoA˙, respectively, for some sufficiently large integerm� n. By linearity,

we may assume that f˙ are just morphisms from XUpUp;˙mto A˙, respectively.

For !˙ 2H0.A˙;�1A˙/, we have by Theorems 2.3.17(3) and 2.3.17(4) that

dM�.f �˙!˙/ord

�.�jO�

EPc/D‚ordM

�.f �˙!˙/ord

�.�jO�

EPc/D .f �˙!˙/ord:

On the other hand, by Proposition A.0.1, we know that .f �˙ log!˙/ord are Colemanintegrals of .f �˙!˙/ord on (the generic fiber of) X.m;U p/, respectively. Therefore,we have

M�.f �˙!˙/ord

�.�jO�

EPc/D .f �˙ log!˙/ord (4.6)

on X.m;U p/, since both of them are Coleman integrals of f �˙!˙ on X.m;U p/ thatbelong to M0

[.m;K/~, respectively. By Definition 3.3.1, we have

log!˙ P˙� .f˙/D

ZE�nA1�

E

log!˙ f˙.TtP˙/ � �.t/˙1 dt

D

ZE�nA1�

E

f �˙ log!˙.TtP˙/ � �.t/˙1 dt

D

ZE�nA1�

E

.f �˙ log!˙/ord.‡˙TtP˙/ � �.t/˙1 dt;

which by (4.6) is equal toZE�nA1�

E

M�.f �˙!˙/ord

�.�jO�

EPc/.‡˙TtP

˙/ � �.t/˙1 dt

D

ZE�nA1�

E

wM�.f �˙!˙/ord

�.�/.‡˙TtP

˙/ � �.t/˙1 dt;

respectively. Then the proposition follows from Remark 4.2.6.

4.4. Proofs of main theoremsLet K be a complete field extension of MFp. For V p 2V, denote by C.!;K;V p/

the (closed) subspace of C.!;K/ (Definition 3.2.5) of functions that are invariantunder the right translation of V p. It is also a closed subspace of C.E�nA1�E =V p;K/.The strong dual of C.!;K;V p/ is canonically isomorphic to D.!;K;V p/ (Nota-tion 4.2.2).

We consider totally definite (not necessarily incoherent) quaternion algebras B

over A such that, for a finite place v of F , .Bv/D 1 if v is split in E or the Galoisrepresentation �A;v corresponds to a principal series.


For such (a representative in the isomorphism class of) B, we choose an E-embedding as (1.9), which is possible. We define representations

….B/tameA˙DO0

M…v;A˙ ;

where the restricted tensor products (over M ) are taken over all finite places v ¤ p

of F and …v;A˙ are M -representations of B�v determined by �A˙;v , respectively. Inparticular, if B is incoherent (i.e., B 2B.A/ in Notation 3.2.1), then….B/tame

A˙are iso-

morphic to the away-from-p components of ….B/A˙ (Notation 3.1.2), respectively.

Notation 4.4.1Let IC.!A;K;V

p/ be the closed ideal of D.!A;K;Vp/ generated by®

Q.fC; f�/ˇf˙ 2

�….B/tame

A˙

�V p

˝M K;.B/DC1¯;

and let I�.!A;K;Vp/ be the closed ideal of D.!A;K;V

p/ generated by®Q.fC; f�/

ˇf˙ 2

�….B/tame

A˙

�V p

˝M K;.B/D�1¯;

where Q.fC; f�/ is defined as the product of those elements Q.fCv; f�v/ in Lemma4.1.8(2).

Let CC.!A;K;Vp/ (resp., C�.!A;K;V

p/) be the subspace of C.!A;K;Vp/

consisting of functions lying in the kernel of every element in I�.!A;K;Vp/ (resp.,

IC.!A;K;Vp/). Put„.A;K;V p/D„.A;K/\C.!A;K;V

p/ and„.!A;K;V p/D

„.!A;K/\C.!A;K;Vp/, where „.A;K/ and „.!A;K/ are introduced in Defini-

tion 3.2.5.

Remark 4.4.2The ideals I˙.!A;K;V

p/ are topologically finitely generated. The subspacesC˙.!A;K;V

p/ are closed in C.!A;K;Vp/.

The following lemma concerns some algebraic properties of the objects intro-duced above.

LEMMA 4.4.3Suppose that V p 2V is sufficiently small. We have(1) IC.!A;K;V

p/\ I�.!A;K;Vp/D 0;

(2) IC.!A;K;Vp/C I�.!A;K;V

p/DD.!A;K;Vp/;

(3) C.!A;K;Vp/D CC.!A;K;V

p/˚C�.!A;K;Vp/;

(4) the subset „.A;K;V p/ is contained in and generates a dense subspace ofC�.!A;K;V

p/;


(5) IC.!A;K;Vp/ is the closed ideal generated by elements that vanish on „.A;

K;V p/.

ProofWe first realize that „.!A;K;V p/ generates a dense subspace of C.!A;K;V

p/.Thus, (1) follows from the dichotomy theorem of Saito and Tunnell (see [28] and[24]). For (2), assume the converse, and suppose that IC.!A;K;V

p/CI�.!A;K;Vp/

is contained in a (closed) maximal ideal with residue field K 0. Then all local perioddistributions Q.fC; f�/ will vanish on the character

E�nA1�E =V p ı�!D.!A;K;V

p/!K 0;

which contradicts the theorem of Saito and Tunnell. Part (3) is a direct consequenceof (1) and (2). It is clear that „.A;K;V p/ is contained in C�.!A;K;V

p/ and, bythe theorem of Saito and Tunnell, „.!A;K;V p/ n„.A;K;V p/ � CC.!A;K;V

p/,which together imply (4). Finally, (5) follows from (4).

Remark 4.4.4If we put D.A;K;V p/DD.!A;K;V

p/=IC.!A;K;Vp/, then we obtain a canonical

isomorphism

D.A;K/��! lim �V p2V

D.A;K;V p/:

Moreover, we have D.A;K/ 'Q

B2B.A/D.A;B;K/ (Definition 3.2.5). We haveD.A;K;V p/bKK 0 ' D.A;K 0; V p/ and D.A;K/bKK 0 ' D.A;K 0/ for a com-plete field extension K 0=K .

Remark 4.4.5In fact, for sufficiently small V p 2 V, the morphism w from (4.3) is injective withthe quotient which is a finite étale K-algebra. We also have D.Oanti

p ;K/ \ IC.!A;

K;V p/D ¹0º. Thus, if K is discretely valued, then D.A;K;V p/ is a (commutative)nuclear Fréchet–Stein K-algebra (defined, e.g., in [10, Definition 1.2.10]). Moreover,it is not hard to see that the transition homomorphism D.A;K;V 0p/!D.A;K;V p/

is finite étale for V 0p � V p. The rigid analytic variety E�.Vp/ associated to D.A;

MFp; Vp/ is a smooth rigid curve overMFp, which may be regarded as an eigencurve

for the group U.1/E=F of tame level V p, twisted by (the cyclotomic character) !Aand cut off by the condition that .1=2; �A; �/D�1. The ind-rigid analytic variety E�mentioned in Section 1.7 is actually lim

�!V p2VE�.V

p/.

Proof of Theorem 3.2.10For the existence, note that the union

Sk�1„.!A;Cp/

0k

already spans a dense sub-


space of C.!A;Cp/ by Lemma 2.1.11. By Corollary 4.3.5, Lemma 4.4.3, and (thenonvanishing part of) Lemma 4.1.9, the collection of ratios

PCuniv.fC/P�univ.f�/

Q.fC; f�/

for f˙ running over .….B/A˙/~K with .B/D�1 defines an element

L.A/ 2 .LieAC˝FM LieA�/˝FM D.A;Cp/:

It actually belongs to .LieAC˝FM LieA�/˝FM D.A;MF ltp / by the lemma below.

We need to show that the element


introduced in Definition 4.4.7 satisfies (3.4). However, this follows from Corollary4.3.5.

The uniqueness follows from the fact thatSk�1„.!A;Cp/k is dense in C.!A;

Cp/, which we already used in the construction of L.A/.

LEMMA 4.4.6The element L.A/ belongs to .LieAC˝FM LieA�/˝FM D.A;MF lt

p /.

ProofNote that, in the definition of L.A/, we only need to consider f˙ 2 .….B/A˙/

~

MF ltp

such that both fC and ….B/A�.J/f� are invariant under O�Pc . Then the lemma fol-

lows if we can show that, for every � 2Sk�1„.A;Cp/

0k

and � 2 Gal.Cp=MF ltp /,

we have

�˝!˙;P

˙univ.f˙/.�/

˛D˝!˙;P

˙univ.f˙/.� ı �/

˛: (4.7)

Without loss of generality, we consider the one for f C. As in the proof of Lemma4.3.3, we have the equality˝

!C;PCuniv.fC/.�/

˛D C

XE�nA1�

E=V pO�

Ep

‚k�1ord .f�C!C/ord.‡CTtP

C/ �!�k� .‡CTtPC/ � �.t/;

where C is a positive rational constant. However, the product ‚k�1ord .f�C!C/ord � !

�k�

is naturally an element in M0[.1;MF lt

p / (Definition 2.3.8), and � can be viewed as anelement in N C.1;Cp/ (Section 4.2). Thus, (4.7) holds, and the lemma follows.


Definition 4.4.7 (p-adic L-function)We call the element


in the proof of Theorem 3.2.10 the anti-cyclotomic p-adic L-function attached to A.

Proof of Theorem 3.3.2It follows from Propositions 4.3.6 and 4.1.6.

Appendices

A. Compatibility of logarithm and Coleman integralIn this appendix, we generalize a result of Coleman [8] about the compatibility of thep-adic logarithm and Coleman integral. This result will only be used in the proof ofProposition 4.3.6.

Let F be a local field contained in Cp with ring of integers OF and residuefield k. Let X be a quasiprojective scheme over F , and let U � X rig be an affinoiddomain with good reduction. We say a closed rigid analytic 1-form ! on U is Frobe-nius proper if there exist a Frobenius endomorphism � of U and a polynomial P.X/over Cp such that P.��/! is the differential of a rigid analytic function on U andsuch that no root of P.X/ is a root of unity. Therefore, by [8, Theorem 2.1], thereexists a locally analytic function f! on U.Cp/, unique up to an additive constant oneach geometric connected component, such that� df! D !;� P.��/f! is rigid analytic.Such f! is known as a Coleman integral of ! on U , which is independent of thechoice of P (see [8, Corollary 2.1(b)]).

PROPOSITION A.0.1LetX and U be as above. Let A be an Abelian variety over F which has either totallydegenerate reduction or potentially good reduction. Then for a morphism f W X!A

and a differential form ! 2 �1.A=F /, the form f �!jU is Frobenius proper, whichadmits f � log! jU as a Coleman integral, where log! W A.Cp/! Cp is the p-adiclogarithm associated to !.

ProofWe may assume that X is projective. Replacing F by a finite extension, we mayassume that A has good reduction or split totally degenerate reduction (i.e., the con-nected neutral component Aıs of the special fiber As of the Néron model A of A is


isomorphic to Gdm;k

, where d is the dimension of A). The first case follows from [8,Theorem 2.8 and Proposition 2.2].

Now we consider the second case. Denote by Aı� the analytic domain of Arig

of points whose reduction is in Aıs . By a well-known result of uniformization from[20, Section 6], we have Arig ' .Grig

m;F /d=ƒ for a lattice ƒ � Gd

m;F .F /. Moreover,Aı� is isomorphic to SpF hT1; : : : ; Td ; T �11 ; : : : ; T �1

di, the rigid analytic multitorus of

multiradius 1.Choose an admissible covering U of X rig containing U , which determines a for-

mal model XU of X over OF . Since X is projective, we may assume that XU isalgebraic. Let Z be the nonsmooth locus of XU over OF . The set of closed points ofX whose reduction is not inZ forms an analytic domainW ofX rig. Since U has goodreduction, we have U �W . By the Néron mapping property, the morphism f extendsuniquely to a morphism XU � Z!A, which induces a morphism f 0 W U ! Arig.Without loss of generality, we assume that f 0.U / is contained in Aı� . By [8, Propo-sition 2.2], we only need to show that !jAı� is Frobenius proper and log! jAı� is aColeman integral of it.

In fact, we have®!jAı�

ˇ! 2�1.A=F /

¯D SpanF

°dT1T1

; : : : ;dTd

Td

±:

By linearity, we may assume !ı WD !jAı� DdT1T1

. We choose the Frobenius endo-

morphism on Aı� to be given by �..T1; : : : ; Td //D .Tq1 ; : : : ; T

q

d/, where q D jkj. We

have that P.��/!ı D 0 for P.X/DX � q. On the other hand, the p-adic logarithmlog on SpF hT1; T �11 i is also killed by P.��/. Therefore, the function .log; 1; : : : ; 1/on SpF hT1; T �11 i � � � � � SpF hTd ; T �1d i 'Aı� is a Coleman integral of !ı, whichcoincides with the restriction of log! .

B. Serre–Tate local moduli for O-divisible groups (following N. Katz)In this appendix, we describe the Kodaira–Spencer isomorphism for ordinary O-divisible groups in terms of their Serre–Tate coordinates, generalizing a classicalresult of Katz [17, Theorem 3.7.1] which is for ordinary p-divisible groups. OnlyTheorems B.1.1 and B.2.3 will be used in the main part of the article. Some notationin this appendix may be different from that in Section 1.8.

B.1. O-Divisible groups and Serre–Tate coordinatesLet F be a finite field extension of Qp , where p is a rational prime. Denote by MF thecompletion of a maximal unramified extension of F . The ring of integers of F (resp.,MF ) is denoted by O (resp., MO). Let k be the residue field of MO, which is an algebraic

closure of Fp . For a p-divisible group G over SpecR, we denote by �.G=R/ the


R-module of invariant differentials of G over R, which is the dual R-module of thetangent space Lie.G=R/ at the identity.

Let S be an MO-scheme. Recall that an O-divisible group over S is a p-divisiblegroup G over S with an action by O such that the induced action of O on the sheafLie.G=S/ coincides with the natural action as an OS -module (hence, an O-module).Denote by BTO

S the category of O-divisible groups over S , which is an Abelian cate-gory. We omit the superscript O if it is Zp . The height h of G, as a p-divisible group,must be divisible by ŒF W Qp�. We define the O-height of G to be ŒF W Qp��1h. AnO-divisible group G is connected (resp., étale) if its underlying p-divisible group is.We denote by LT the Lubin–Tate O-formal group over Spec MO, which is unique upto isomorphism. We use the same notation for its base change to S .

For an O-divisible group G over S , there exists an O-formal group G0 over S ,unique up to isomorphism, such that its associated p-divisible group G0Œp1� is themaximal connected subgroup of G. In particular, G0Œp1� is an O-divisible group.We define the O-Cartier dual of G to be

GD WD lim�!n

HomO

�GŒpn�;LT Œpn�

�as in [11]. An O-divisible group G is ordinary if .G0Œp1�/D is étale. Denote byTpG D lim

�nGŒpn� the Tate module functor. Denote by Nilp MO the category of MO-

schemes on which p is locally nilpotent.

THEOREM B.1.1 (Serre–Tate coordinates)Let G be an ordinary O-divisible group over k. Consider the moduli functor MG

on Nilp MO such that, for every MO-scheme S on which p is locally nilpotent, MG.S/

is the set of isomorphism classes of pairs .G;'/, where G is an object in BTOS and

' W G�S .S˝ MO k/!G�Speck .S˝ MO k/ is an isomorphism. Then MG is canonically

pro-represented by the MO-formal scheme HomO.TpG.k/˝O TpGD.k/;LT /.In particular, for every Artinian local MO-algebra R with the maximal ideal mR

and G=R a deformation of G, we have a pairing

q.G=RI �; �/ W TpG.k/˝O TpGD.k/!LT .R/D 1CmR:

It satisfies the following.(1) For every ˛ 2 TpG.k/ and ˛D 2 TpGD.k/, we have

q.G=RI˛;˛D/D q.GD=RI˛D; ˛/:

(2) Suppose that we have another ordinary O-divisible group H over k and itsdeformation H over R. Let f W G!H be a homomorphism, and let fD be itsdual. Then f lifts to a (unique) homomorphism f W G!H if and only if


q.G=RI˛; fDˇD/D q.H=RI f˛;ˇD/

for every ˛ 2 TpG.k/ and ˇD 2 TpHD.k/.

By abuse of notation, we will use MG to denote the formal schemeHomO.TpG.k/˝O TpGD.k/;LT /. The proof of the theorem follows exactly thatof [17, Theorem 2.1].

ProofThe fact that MG is pro-presentable is well known. Now we determine the represent-ing formal scheme.

Since G is ordinary, we have a canonical isomorphism

G'G0Œp1�� TpG.k/˝O F=O:

By the definition of O-Cartier duality, we have a morphism

epn W GŒpn��GDŒpn�!LT Œpn�:

The restriction of the first factor to G0Œpn� gives rise to an isomorphism

G0Œpn��!HomO

�GDŒpn�.k/;LT Œpn�

�of group schemes over k preserving O-actions. Passing to the limit, we obtain anisomorphism of O-formal groups over k

G0 ��!HomO

�TpGD.k/;LT

�;

which induces a pairing

EG W G0 � TpGD.k/!LT :

Let G=R be a deformation of G. Then we have an extension

0 G0Œp1� G TpG.k/˝O F=O 0 (B.1)

of O-divisible groups. We have pairings

EG;pn W G0Œpn��GDŒpn�!LT Œpn�;

EG W G0 � TpGD.k/!LT ;

which lift epn and EG, respectively.Similar to the p-divisible group case, the extension (B.1) is obtained from the

extension


0 TpG.k/ TpG.k/˝O F TpG.k/˝O F=O 0

by pushing out along a unique O-linear homomorphism

'G=R W TpG.k/!G0.R/:

The homomorphism 'G=R may be recovered from (B.1) in the way described in [17,p. 151]. It is the composite

TpG.k/! TpGŒpn�.k/hpni��!G0.R/

for every n� 1 such that mnC1R D 0. Therefore, from G=R, we obtain a pairing

q.G=RI �; �/DEG.R/ ı .'G=R; id/ W TpG.k/˝O TpGD.k/!LT .R/D 1CmR:

This shows that the functor MG is canonically pro-represented by the MO-formalscheme HomO.TpG.k/˝O TpGD.k/;LT /.

For (2), if the given homomorphism f W G!H can be lifted to f W G!H , thenwe must have the commutative diagram

0 HomO

�TpGD.k/;LT Œp1�

�ıTp fD.k/

G

f

TpG.k/˝O F=O

Tp f.k/˝OF=O

0

0 HomO

�TpHD.k/;LT Œp1�

�H TpH.k/˝O F=O 0:

Conversely, if we may fill f in the above diagram, then f lifts.The existence of the middle arrow is equivalent to the pushout of the top extension

by the left arrow being isomorphic to the pullback of the lower extension by the rightarrow. The above-mentioned pushout is an element in

ExtBTOR

�TpG.k/˝O F=O;HomO

�TpHD.k/;LT Œp1�

��;

which is isomorphic to

HomO

�TpG.k/˝O TpHD.k/;LT .R/

�by the bilinear pairing

.˛;ˇD/ 7! q.G=RI˛; fDˇD/:

Similarly, the above-mentioned pullback is an element in

HomO

�TpG.k/˝O TpHD.k/;LT .R/

�


defined by the bilinear pairing

.˛;ˇD/ 7! q.H=RI f˛;ˇD/:

It remains to prove (1). Choose n such that mnC1R D 0. Then both G0.R/ and.GD/0.R/ are annihilated by pn. Denote by ˛.n/ the image of ˛ under the canonicalprojection TpG.k/! GŒpn�.k/ and similarly for ˛D.n/. By construction, we have'G=R.˛/D hp

ni˛.n/ 2 G0.R/ and 'GD=R.˛D/D hpni˛D.n/ 2 .G

D/0.R/. There-fore, we have

q.G=RI˛;˛D/DEG;pn�hpni˛.n/;˛D.n/

�:

Similarly, we have q.GD=RI˛D; ˛/DEGD ;pn.hpni˛D.n/; ˛.n//.

The remaining argument is formal, and one only needs to replace bGm (resp.,Abelian varieties) by LT (resp., O-divisible groups) in the proof of [17, Theorem 2.1].In particular, we have the following. Given an integer n � 1 and elements x 2G0Œpn�.R/ and y 2 GDŒpn�.k/, there exist an Artinian local ring R0 that is finiteand flat over R and a point Y 2GDŒpn�.R0/ lifting y. For every such R0 and Y , wehave the equality EG;pn.x; y/D epn.x;Y / inside LT .R0/.

B.2. Main theoremWe fix an ordinary O-divisible group G over k. Denote by R the coordinate ring ofMG, which is a complete MO-algebra. We have the universal pairing

q W TpG.k/˝O TpGD.k/!LT .R/�R�:

Therefore, we may regard q.˛;˛D/ as a regular function on MG. For each O-linearform ` 2HomO.TpG.k/˝O TpGD.k/;O/, denote by D.`/ the translation-invariantcontinuous derivation of R given by

D.`/q.˛;˛D/D `.˛˝ ˛D/ � q.˛;˛D/:

By abuse of notation, we also denote by D.`/ the corresponding map �R= MO!R.

Denote by G the universal O-divisible group over MG. We choose a logarithm log WLT ! bGa over MO ˝ Q such that !0 WD log� dT is a generator of the free rank 1MO-module �.LT = MO/.

Let R be as in Theorem B.1.1, and let G=R be a deformation of G. We have thecanonical isomorphism of O-modules

G W TpGD.k/��!HomBTO

R

�G0Œp1�;LT Œp1�

�:

Define the O-linear map !G W TpGD.k/!�.G=R/ by the formula

!G.˛D/D G.˛D/�!0 2�.G

0=R/D�.G=R/:


Let LG W HomO.TpGD.k/;O/! Lie.G=R/ be the unique O-linear map such that

!G.˛D/ �LG.˛_D/D ˛D � ˛

_D 2O:

In fact, the R-linear extensions

!G W TpGD.k/˝O R!�.G=R/

and

LG W HomO

�TpGD.k/;R

�! Lie.G=R/

are isomorphisms. Similarly, we have an isomorphism

G_ W TpG.k/D TpGKet.k/D TpGKet.R/

��!HomBTR

�.G Ket/_;bGmŒp

1��;

which induces an isomorphism

TpG.k/˝Zp R��!�

�.G Ket/_=R

�by pulling back the differential form dT

Ton bGm. It further induces an isomorphism

!G_ W TpG.k/˝O RD�TpG.k/˝Zp R

�O

��!�

�.G Ket/_=R

�O:

Here, the subscript O denotes the maximal flat quotient on which O acts via thestructure map. By construction, we have the following lemma of functoriality.

LEMMA B.2.1Let f W G! H be as in Theorem B.1.1, and let f W G ! H be a homomorphismlifting f. Then the following hold.(1) We have ..f Ket/_/�.!G_.˛//D !H_.f˛/ for every ˛ 2 TpG.k/, where f Ket W

G Ket!H Ket is the induced homomorphism on the étale quotient.(2) We have f�.LH .˛_D//DLG.˛

_D ı fD/ for every ˛_D 2HomO.TpGD.k/;O/.

Denote by D.G/ the (contravariant) Dieudonné crystal of G. We have the exactsequence

0 �.G_=R/ D.G_/R Lie.G=R/ 0

and the Gauss–Manin connection

r W D.G_/R!D.G_/R˝R �R= MO:

They together define the (universal) Kodaira–Spencer map


KS W �.G_=R/! Lie.G=R/˝R �R= MO;

which factors through the quotient �.G_=R/!�.G_=R/O . The following lemmais immediate.

LEMMA B.2.2The natural map �.G_=R/O!�..GKet/_=R/O is an isomorphism.

In particular, we may regard !G_ as a map from TpG.k/ to �.G_=R/O . Thefollowing result on the compatibility of the Kodaira–Spencer map and the Serre–Tatecoordinate is the main theorem of this appendix.

THEOREM B.2.3We have the equality

!G.˛D/ �KS�!G_.˛/

�D d log

�q.˛;˛D/

�in �

R= MOfor every ˛ 2 TpG.k/ and ˛D 2 TpGD.k/.

Note that the definition of !G, but not !G_ , depends on the choice of log, whichis compatible with the right-hand side.

B.3. FrobeniusDenote by � the Frobenius automorphism of MO such that O D MO D1. Put X DX ˝ MO;

MO for every MO-(formal) scheme X , let †X W X !X be the natural projec-

tion, and let FX W X!X be the relative Frobenius morphism which is MO-linear. Weomit the subscript X if it is MG.

LEMMA B.3.1We have the following results.(1) There is a natural isomorphism

M G��!MG�

under which the regular function q.�.˛/; �.˛D// is mapped to †�q.˛;˛D/.(2) Under the map †.GKet/_ W .G

Ket/_ ' ..G Ket/_/ ! .G Ket/_, we have

†�.GKet/_

!G_.˛/D !.G� /_.�˛/

for every ˛ 2 TpG.k/.(3) Under the map FG W G!G , we have


FG�LG.˛_D/DLG� .˛_D ı �

�1/

for every ˛_D 2HomO.TpGD.k/;O/.

ProofThe proof is the same as that for [17, Lemmas 4.1.1 and 4.1.1.1].

From now on, we choose a uniformizer $ of F , which gives rise to an isomor-phism LT 'LT . In particular, we may identify .GD/ and .G /D . For a deforma-tion G=R of G, we denote by G0=R the quotient of G by the subgroup G0Œ$�. Theinduced projection map

FG W G!G0

lifts the relative Frobenius morphism

FG W G!G :

Define the Verschiebung to be

VG D .FGD /D W G 'GD D!G:

Note that the isomorphism depends on $ .

LEMMA B.3.2For ˛ 2 TpG.k/ and ˛D 2 TpGD.k/, we have formulas(1) FG.˛/D �˛ and VG.�˛/D$˛D; and(2) q.G0=RI�˛;�˛D/D$:q.G=RI˛;˛D/.

ProofThe proof is the same as that of [17, Lemma 4.1.2], with VG ı FG D$ .

LEMMA B.3.3For ˛ 2 TpG.k/ and ˛_D 2HomO.TpGD.k/;O/, we have formulas(1) ..F Ket

G /_/�!G_.˛/D !G0_.�˛/; and

(2) FG�LG.˛_D/D$LG0.˛

_D ı �

�1/.

ProofIt follows from Lemmas B.2.1 and B.3.2.

If we apply the construction to the universal object G, we obtain a formal defor-mation G0=R of G . Its classifying map is the unique morphism


ˆ W MG!MG��!M

G

such that ˆ�G 'G0. Therefore, we may regard FG as a morphism

FG W G!ˆ�G

of O-divisible groups over MG. Taking the dual, we have

F _G W ˆ�G _ ' .ˆ�G /_!G_:

LEMMA B.3.4We have the following statements.(1) The map !G_ W TpG.k/˝O R!�.G_=R/O induces an isomorphism

TpG.k/��!�.G_=R/1O WD

®! 2�.G_=R/O

ˇ.F _G /

�! Dˆ�†�G_!¯

of O-modules.(2) The map LG W HomO.TpGD.k/;R/! Lie.G=R/ induces an isomorphism

HomO

�TpGD.k/;O

��! Lie.G=R/0 WD

®ı 2 Lie.G=R/

ˇFG�ıD$ˆ

�FG�ı¯

of O-modules.

ProofIt can be proved in the same way as [17, Corollary 4.1.5] by using Lemmas B.3.1 andB.3.3.

Consider the following commutative diagram:

0 �.G_=R/Oa

.F _G/�

�D.G_/R

�O

b

D.F_G /

Lie.G=R/

FG�

0

0 �.G0_=R/O�D.G _/R

�O

Lie.G0=R/ 0

0 �.G_=R/O

ˆ�ı†�G_ �

D.G_/R�O

D.†G_ /

Lie.G=R/

ˆ�ıFG�

0

(B.2)

For k 2 Z, we define O-modules


D.G_/kR D®� 2

�D.G_/R

�O

ˇD.F_G/� D$

1�kD.†G_/�¯

D®� 2

�D.G_/R

�O

ˇD.V_G/D.†G_/� D$

k�¯:

LEMMA B.3.5The maps !G_ and a in (B.2) together induce an isomorphism

a1 W TpG.k/��!D.G_/1R

of O-modules. The maps LG and b in (B.2) together induce an isomorphism

b0 W D.G_/0R��!HomO

�TpGD.k/;O

�of O-modules.

ProofFor the first part, by a similar argument to that in [17, Lemma 4.2.1], we know thatb.�/D 0 for � 2D.G_/1

R, that is, � is in the image of a. The conclusion then follows

from Lemma B.3.4(1).For the second part, it is easy to see that Im.a/ \ D.G_/0

GD ¹0º by choos-

ing an O-basis of TpG.k/. Therefore, b restricts to an injective map D.G_/0R!

Lie.G=R/0. We only need to show that this map is also surjective. For every ı 2Lie.G=R/0, choose an element �0 2 .D.G_/R/O . Put �nC1 D D.V_G/D.†G_/�n forn� 0. Then b.�n/D ı, and ¹�nº converge to an element � 2D.G_/0

R.

LEMMA B.3.6For every ` 2HomO.TpG.k/˝O TpGD.k/;O/, the action ofD.`/ under the Gauss–Manin connection on .D.G_/R/O satisfies the formula

D.`/�r�D.V_G/D.†G_/�

��D$D.V_G/D.†G_/

�D.`/.r�/

�for every � 2 .D.G_/R/O .

ProofIt is proved in the same way as [17, Lemma 4.3.3].

LEMMA B.3.7If � 2 .D.G_/R/O satisfies D.V_G/D.†G_/� D � for some 2 MO, then for every` 2HomO.TpG.k/˝O TpGD.k/;O/, the elementD.`/.r�/ 2 .D.G_/R/O satisfies

$D.V_G/D.†G_/�D.`/.r�/

�D D.`/.r�/:


ProofIt follows immediately from Lemma B.3.6.

PROPOSITION B.3.8For ˛ 2 TpG.k/ and ˛D 2 TpGD.k/, there exists a unique character Q.˛;˛D/ ofMG such that

!G.˛D/ �KS�!G_.˛/

�D d logQ.˛;˛D/:

ProofLet ¹˛iº (resp., ¹˛D;j º) be an O-basis of TpG.k/ (resp., TpGD.k/). Let ¹ì;j º bethe basis of HomO.TpG.k/˝O TpGD.k/;O/ dual to ¹˛i ˝ ˛D;j º. Then for everyelement � 2 .D.G_/R/O , we have

r� DXi;j

D.ì;j /.r�/˝ d logq.˛i ; ˛D;j /:

In particular, for � D !G_.˛/, we have

r!G_.˛/DXi;j

D.ì;j /�r!G_.˛/

�˝ d logq.˛i ; ˛D;j /:

By Lemmas B.3.4 and B.3.7, r!G_.˛/ 2 D.G_/0R

. Therefore, there exist uniqueelements ˛_D;i;j 2HomO.TpGD.k/;O/ such that

r!G_.˛/D b�10 .˛_D;i;j /

for every i , j . By definition,

KS�!G_.˛/

�DXi;j

LG.˛_D;i;j /˝ d logq.˛i ; ˛D;j /;

and

!G.˛D/ �KS�!G_.˛/

�D d log

�Yi;j

q.˛i ; ˛D;j /˛D ˛

_D;i;j

�:

The above proposition has the following two corollaries.

COROLLARY B.3.9For elements ˛ 2 TpG.k/, ˛D 2 TpGD.k/, and ` 2 HomO.TpG.k/˝O TpGD.k/;

O/, we have that D.`/.!G.˛D/ �KS.!G_.˛/// is a constant in O.


COROLLARY B.3.10Suppose that, for every integer n� 1, we can find a homomorphism fn W R! MO=pn

such that

fn�D.`/

�!G.˛D/ �KS

�!G_.˛/

��D `.˛˝ ˛D/

holds in MO=pn. Then QD q, and Theorem B.2.3 follows.

The condition of this corollary is fulfilled by Theorem B.4.2. Therefore, we havereduced Theorem B.2.3 to Theorem B.4.2 in the next section.

B.4. Infinitesimal computationLet R be an (Artinian) local MO-algebra with the maximal ideal mR satisfyingmnC1R D 0. Let G=R be the canonical deformation of G. Let QG be a deformation

of G to QR WDRŒ"�=."2/, which gives rise to a map @ W �.G_=R/! Lie.G=R/. Notethat the target Lie.G=R/ may be identified with Ker.G0. QR/!G0.R//.

LEMMA B.4.1The reduction map TpG.R/! TpG.k/ is an isomorphism.

ProofIt follows from the same argument as in [17, Lemma 6.1].

In particular, we may define maps G_ W TpG.k/!HomBTR.G_;bGmŒp

1�/ and

!G_ W TpG.k/!�.G_=R/: (B.3)

THEOREM B.4.2The Serre–Tate coordinate for QG= QR satisfies

q. QG= QRI˛;˛D/D 1C "!G.˛D/ � @�!G_.˛/

�:

LEMMA B.4.3For ˛D 2 TpGD.k/ and ˛ 2Ker.G0. QR/!G0.R//D Lie.G=R/, we have

EG.˛;˛D/D 1C "!G.˛D/˛:

ProofBy functoriality, we only need to prove the lemma for the universal object G=R. Bydefinition,

1C "!G.˛D/˛D 1C "� G.˛D/�˛ �!0

�2LT . QR/:


We also have

G.˛D/�˛ �!LT D�logı G.˛D/

��˛ � dT

in RŒp�1�. Therefore, we have the equality

EG.˛;˛D/D 1C "!G.˛D/˛

in Ker.LT . QRŒp�1�/!LT .RŒp�1�//.

For an integer N > n, denote by ˛N the image of ˛ in GŒpN �.R/. Let QN 2QG. QR/ be an arbitrary lifting of ˛N . Then

pN QN 2Ker�QG. QR/!G.R/

�DKer

�QG0. QR/!G0.R/

�' Lie.G=R/:

This process defines a map 'G W TpG.R/! Lie.G=R/.

PROPOSITION B.4.4We have @!G_.˛/D 'G.˛/ for every ˛ 2 TpG.R/.

Assuming the above proposition, we prove Theorem B.4.2.

Proof of Theorem B.4.2It is clear that G0Œp1� ˝R QR is the unique, up to isomorphism, deformation ofG0Œp1� to QR. Then the deformation QG corresponds to the extension

0 G0Œp1�˝R QR QG TpG.k/˝O F=O 0:

In particular, we may identify QG0 with G0˝R QR. We have

Ker�QG. QR/!G.R/

�DKer

�QG0. QR/!G0.R/

�DKer

�G0. QR/!G0.R/

�D Lie.G0=R/:

For D 2 Lie.G0=R/, we have

E QG.D; �/DEG.D; �/ W TpGD.k/' Tp�G0Œp1�

�D.k/!LT . QR/;

where in the pairing E QG (resp., EG ), we view D as an element in QG0. QR/ (resp.,G0. QR/). For ˛D 2 TpGD.k/, we have

EG.D;˛D/D 1C "!G.!˛/D

by definition. Therefore, Theorem B.4.2 follows from Proposition B.4.4 and the con-struction of q.


The rest of the appendix is devoted to the proof of Proposition B.4.4. We willreduce it to certain statements from [17] about Abelian varieties. It is an interestingproblem to find a proof purely using O-divisible groups.

Recall that ordinary O-divisible groups over k are classified by their dimensionand O-height. Let Gr;s be an O-divisible group of dimension r and O-height r C swith r � 0 and r C s > 0.

Proof of Proposition B.4.4Choose a totally real number field EC such that F ' EC ˝Q Qp , and choose animaginary quadratic field K in which pD pCp� splits. Put E DEC˝QK . Supposethat �1; �2; : : : ; �h are all complex embeddings of EC. Consider the data .Ar;s; �; i/where� Ar;s is an Abelian variety over k;� � W Ar;s!A_r;s is a prime-to-p polarization;� i W OE ! Endk Ar;s is an OE -action which sends the complex conjugation on

OE to the Rosati involution and such that, in the induced decomposition

Ar;sŒp1�DAr;sŒp1�C˚Ar;sŒp1��

of the OE ˝Zp-module Ar;sŒp1�, the summand Ar;sŒp1�C is isomorphic toGr;s as an O-divisible group.

It is clear that the polarization � induces an isomorphism Ar;sŒp1�C��!

.Ar;sŒp1��/_. By the Serre–Tate theorem, MGr;s also parameterizes deformation ofthe triple .Ar;s; �; i/. In what follows, we fix r , s and suppress them from notation.Let R be as in Theorem B.1.1, let A=R be the canonical deformation of A=k, and letQA be a deformation of A to QR such that QG ' QAŒp1�C.

There is a similar map (B.3) for A, and we have !G_.˛/ D !A_.˛/ for ˛ 2TpG.R/� TpA.R/, where we view �.G_=R/ as a submodule of H0.A_;�1

A_=R/.

Moreover, the map 'G W TpG.R/! Lie.G=R/ can be extended in the same way toa map 'A W TpA.R/! Lie.A=R/. Then Proposition B.4.4 follows from [17, Lemma5.4 and Section 6.5], where the argument uses normalized cocycles and does notrequire A to be ordinary in the usual sense.

Acknowledgments. We would like to thank Daniel Disegni for his careful reading ofthe draft version and useful comments. We also thank the anonymous referees fortheir careful reading and helpful comments which, in particular, led us to the newintroduction focusing on the case of rational elliptic curves.

Liu’s work was partially supported by National Science Foundation (NSF) grantDMS-1302000. S. Zhang’s work was partially supported by NSF grants DMS-


0970100 and DMS-1065839. W. Zhang’s work was partially supported by NSF grantsDMS-1301848 and DMS-1601144, and by a Sloan research fellowship.

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Liu

Department of Mathematics, Northwestern University, Evanston, Illinois, USA;

[email protected]

Zhang

Department of Mathematics, Princeton University, Princeton, New Jersey, USA;

[email protected]

Zhang

Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts,

USA; [email protected]

mailto:[email protected]



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