on certain families of special cycles on shimura varieties...l-function is replaced by certain...

On certain families of special cycles on

Shimura varieties

Zhaorong Jin

A Dissertation

Presented to the Faculty

of Princeton University

in Candidacy for the Degree

of Doctor of Philosophy

Recommended for Acceptance

by the Department of

Mathematics

Adviser: Christopher McLean Skinner

June 2020

c© Copyright by Zhaorong Jin, 2020.

All Rights Reserved

Abstract

In this two-part thesis, we study certain families of special cycles on Shimura varieties,

which have interesting arithmetic applications. In the first part, extending the ideas

of Darmon and Rotger, we construct a p-adic family of Hirzebruch-Zagier cycles on

Shimura threefolds obtained from products of modular curves and Hilbert modular

surfaces. This family gives rise to a big cohomology class residing in the Galois

cohomology of a certain Λ-adic Galois representation. We then establish a regulator

formula for the cycles, which allows us to relate the big cohomology class to a twisted

triple product p-adic L-function. As an application, we establish new instances of

the equivariant BSD-conjecture in rank 0 and study the arithmetic of rational elliptic

curves over quintic fields. In the second part, we follow the ideas of Loeffler, Skinner

and Zerbes to set up and conduct local automorphic computations on the algebraic

group GSp4 ×GL1 GL2, establishing technical results that should lead to the norm

relations of an Euler system for GSp4 ×GL1 GL2.

iii

Acknowledgements

First, I would like to express my sincere gratitude to my advisor, Professor Christo-

pher Skinner, for his constant support and encouragement, and years of enlightening

mathematical conversations and guidance. I would also like to thank Professor Shou-

Wu Zhang for reading the thesis, and Professor Nicholas Katz and Professor Peter

Sarnak for serving on my committee.

Special thanks to Michele Fornea for countless fruitful discussions. Work in the

first part of this thesis is in collaboration with Michele, and has been presented at

the number theory seminars at Princeton University and Harvard University. I am

very grateful for his passion and dedication to our joint work.

I would also like to thank Professor David Loeffler and Professor Sarah Zerbes for

proposing the GSp4×GL2-Euler system and answering my questions, thank Chi-Yun

Hsu and Ryotaro Sakamoto for their interest on this project (which is a joint work

in progress building on the results in the second part of the thesis), as well as for

correcting a key mistake in Proposition 12.3.2, and thank Giada Grossi for sharing

her preprint and helping me understand the subtle technical details in [LSZ17].

Many thanks to our wonderful graduate program administrator, Jill LeClair, for

answering all my logistic questions and making my life in Fine Hall so smooth and

easy. Thanks to all my friends and fellow graduate students, among them Weibo

Fu, Shilin Lai, Lue Pan, Congling Qiu, Boya Wen and Ruixiang Zhang, for all the

conversations and cheerfulness.

Finally, I owe my deepest gratitude to my parents, Zhijun Jin and Yanxiao Han,

and my girl friend Weitao Shuai, for their continuous support and love.

iv

To my family and friends.

v

Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

I On generalized Hirzebruch-Zagier cycles 2

1 Introduction 3

1.1 History and background . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 A closer look at the work of Darmon-Rotger . . . . . . . . . . . . . . 5

1.3 Hirzebruch-Zagier cycles . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.4 Notations and conventions . . . . . . . . . . . . . . . . . . . . . . . . 15

2 Preliminaries on Hilbert modular forms 19

2.1 Basic notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2 Hilbert modular forms . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3 Relation to classical Hilbert modular forms . . . . . . . . . . . . . . . 22

2.4 Adelic Fourier expansions . . . . . . . . . . . . . . . . . . . . . . . . 23

2.5 Hecke Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.6 p-adic modular forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.7 Hida families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.8 Diagonal restriction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

vi

2.9 Twists of Hilbert modular forms. . . . . . . . . . . . . . . . . . . . . 34

3 Automorphic p-adic L-functions 41

3.1 Preparations with I-adic forms . . . . . . . . . . . . . . . . . . . . . . 41

3.2 The automorphic p-adic L-function . . . . . . . . . . . . . . . . . . . 45

3.3 Link to the complex L-value in weight one. . . . . . . . . . . . . . . . 46

4 Hilbert modular varieties and Hirzebruch-Zagier classes 48

4.1 Basic notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.2 Hecke correspondences and operators . . . . . . . . . . . . . . . . . . 50

4.3 Atkin-Lehner morphism . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.4 Hirzebruch-Zagier cycles . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.5 Big cohomology classes . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.6 A refinement of specialization formula . . . . . . . . . . . . . . . . . . 69

5 Galois representations 73

5.1 Galois representations at finite level . . . . . . . . . . . . . . . . . . . 73

5.2 Big Galois representations. . . . . . . . . . . . . . . . . . . . . . . . . 77

5.3 Etale cohomology of towers of Hilbert modular surfaces . . . . . . . . 81

5.4 Local cohomology class . . . . . . . . . . . . . . . . . . . . . . . . . . 88

6 Big pairing 96

6.1 Algebra interlude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

6.2 A weak Λ-adic Eichler-Shimura map . . . . . . . . . . . . . . . . . . 100

6.3 Big pairing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

6.4 On Dieudonne modules . . . . . . . . . . . . . . . . . . . . . . . . . . 111

7 Motivic p-adic L-functions 116

vii

7.1 Perrin-Riou’s regulator . . . . . . . . . . . . . . . . . . . . . . . . . . 116

7.2 The motivic p-adic L-function. . . . . . . . . . . . . . . . . . . . . . . 121

8 p-adic Gross-Zagier formulas 123

8.1 P -syntomic cohomology . . . . . . . . . . . . . . . . . . . . . . . . . 123

8.2 Syntomic Abel-Jacobi map . . . . . . . . . . . . . . . . . . . . . . . . 125

8.3 Abel-Jacobi map of Hirzebruch-Zagier cycles . . . . . . . . . . . . . . 128

8.4 Overconvergent Hilbert Modular Forms . . . . . . . . . . . . . . . . . 133

8.5 Relation to p-adic modular forms. . . . . . . . . . . . . . . . . . . . . 142

8.6 The formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

8.7 An explicit reciprocity law . . . . . . . . . . . . . . . . . . . . . . . . 155

9 On the equivariant BSD-conjecture 157

9.1 Artin representations and Selmer groups . . . . . . . . . . . . . . . . 157

9.2 Existence of prime p . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

9.3 Proof of main theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 165

II Local computations for a GSp4 ×GL2 Euler system 168

10 Introduction 169

10.1 Recent progress in Euler systems . . . . . . . . . . . . . . . . . . . . 169

10.2 Outline of the proof and main results . . . . . . . . . . . . . . . . . . 171

10.3 Notations and conventions . . . . . . . . . . . . . . . . . . . . . . . . 175

11 Local representation theory 178

11.1 Principal series of GL2(Ql) . . . . . . . . . . . . . . . . . . . . . . . . 178

11.2 Siegel sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

11.3 Principal series of GSp4(Ql) . . . . . . . . . . . . . . . . . . . . . . . 182

viii

12 Hecke operators and zeta integrals 186

12.1 Local zeta integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

12.2 Double coset computations . . . . . . . . . . . . . . . . . . . . . . . . 189

12.3 Hecke operators and local zeta integrals . . . . . . . . . . . . . . . . . 198

13 Towards norm relations 206

13.1 A local bilinear form . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

13.2 An application of Frobenius reciprocity . . . . . . . . . . . . . . . . . 210

13.3 Main results towards vertical norm relations . . . . . . . . . . . . . . 214

13.4 Main result towards tame norm relations . . . . . . . . . . . . . . . . 222

13.5 Divisibility of mη . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

ix

General notations

Throughout the two parts of the thesis, we will use the following notations.

• For any number field F , we use OF to denote its ring of integers. For any place

v of F , let Fv denote the corresponding local field, and OF,v its ring of integers.

More generally, for any ideal m of OF , let Fm =∏

v|m Fv, and OF,m =∏

v|mOF,v.

• AF is the ring of adeles, AF,f is the ring of finite adeles, and F∞ = F ⊗Q R is

the infinite part.

• For any y ∈ AF , we use y∞ ∈ F∞ to denote its infinite part, and yf ∈ AF,f to

denote its finite part. For any place v of F , yv ∈ Fv is the v-component of y,

and for any ideal m of OF , let ym = (yv)v|m ∈ Fm.

1

Part I

On generalized Hirzebruch-Zagier

cycles

2

Chapter 1

Introduction

1.1 History and background

One of the earliest results towards the Birch and Swinnerton-Dyer conjecture is the

seminal work of Coates and Wiles [CW77], in which they established the algebraic

rank zero case of the BSD conjecture for elliptic curves with complex multiplications.

From a modern perspective, their proof involves interpolating the so-called elliptic

units into a p-adic family (a “big cohomology class”), and then relating its image

under some “big exponential map” to Katz’s two-variable p-adic L-function of an

imaginary quadratic field (an “explicit reciprocity law”), which is done via the so-

called p-adic Kronecker limit formula relating the special values of p-adic L-function

to p-adic logarithms of elliptic units (a “p-adic regulator formula”). In this way, one

can obtain special value formulas outside of the range of interpolation, connecting the

Hasse-Weil L-value of an CM elliptic curve with a global class in the Selmer group,

thereby establishing an instance of the BSD conjecture.

Recently, there have been several works in the spirit of Coates-Wiles, among which

[BDP13] and the sequels [DR14], [DR17] are the most prominent. In [BDP13], the

3

elliptic units are replaced by Heegner points on modular curves (or more generally,

generalized Heegner cycles in the product of Kuga-Sato varieties), and Katz’s p-adic

L-function is replaced by certain p-adic Rankin L-series constructed in op.cit., called

the anticyclotomic p-adic L-function, which interpolates the central values of the

Rankin-Selberg L-function of a modular form and a Hecke character of an imaginary

quadratic field. The main result of op.cit. is a p-adic analogue of the Gross-Zagier

formula, relating the images of the generalized Heegner cycles under the p-adic Abel-

Jacobi map to the special values of anticyclotomic p-adic L-function at critical points

that lie outside the range of classical interpolation. This again can be seen as an

instance of the p-adic regulator formulas, and has many applications to the study

of the BSD conjecture and more generally the Bloch-Kato conjectures, including the

recent work of [Ski20] and [CH18].

The work of Darmon and Rotger focuses on the connection between the so-called

Gross-Kudla-Schoen diagonal cycles and a triple product p-adic L-function interpo-

lating the (square root of) central critical values of the Garrett-Rankin L-function

attached to a triple of modular forms. In [DR14], the authors extended the method

of [HT01] to construct a three-variable triple product p-adic L-function, and proved a

p-adic Gross-Zagier formula relating the special value of the p-adic L-function to the

p-adic Abel-Jacobi image of the diagonal cycles. In [DR17], by modifying the diagonal

cycles in triple products of modular curves, they put the p-adic Abel-Jacobi image of

the diagonal cycles into a one-variable global cohomology class, and established an

explicit reciprocity law, which proved that the image of the global cohomology class

under Perrin-Riou’s big logarithm map is essentially the restriction of the triple prod-

uct p-adic L-function constructed earlier to a one variable line in the three variable

weight space. By specializing at weights outside the range of “geometric interpola-

tion” defining the global cohomology class, they are able to realize the central critical

4

value of certain twisted L-function of an elliptic curve as the obstruction to a coho-

mology class being crystalline at p, thereby establishing instances of the equivariant

BSD conjecture.

1.2 A closer look at the work of Darmon-Rotger

As the present work is greatly inspired by and very closely related to [DR17], it is

worthwhile to discuss the main features of op.cit. in more detail.

Let f be a newform of weight two, level Nf and trivial character, and g and h

two ordinary Hida families of tame levels Ng and Hh with inverse tame characters,

such that Nf and HgNh are coprime, and so are p and N = lcm(Nf , Ng, Nh). For

each α ≥ 1, we have the diagonally embedded curve inside the threefold

X1(Npα) → X0(Np)×X1(Npα)×X1(Npα),

which, after suitably twisting and taking finite quotients, gives rise to a family

∆αα≥1 of null-homologous cycles compatible under the pushforward of the natural

projection down the p-tower as α varies. After taking the inverse limit of the cohomol-

ogy classes AJetp (∆α) obtained under the p-adic etale Abel-Jacobi map and projecting

to suitably isotypic quotients of the etale cohomology of the modular threefold, one

obtains the global cohomology class

κ(f,gh) ∈ H1(Q,Vfgh(N)),

where Vfgh(N) is the direct sum of several copies of the tensor product of Vf , Vg,

Vh, the (classical and Λ-adic) Galois representations associated to f , g and h.

We may specialize the class κ(f,gh) at classical points (y, z) of the weight spaces

5

of the Hida families g and h such that y and z have the same weight ` and character at

p, obtaining a collection of classes κ(f, gy, hz) ∈ H1(Q, Vfgyhz(N)) varying p-adically

analytically as (gy, hz) varies over pairs of specializations of g and h with common

weights and characters at p. In the case of weight two specializations, as they “arise

from geometry”, the work of Saito, later refined by Skinner, combined with a result of

Nekovar, shows k(f, gy, hz) are crystalline after restricting to a decomposition group

at p.

On the other hand, although κ(f,gh) is constructed by interpolating geometric

constructions attached to weight two specializations, it makes sense to consider its

weight one specializations κ(f, g1, h1). These classes, lying outside the range of “ge-

ometric interpolation”, have no a priori reason to be crystalline at p. In the case

where f is a newform associated to an elliptic curve E, and gy and hz are the ordi-

nary p-stabilizations of weight one forms associated to two two-dimensional odd Artin

representations ρ1 and ρ2, it can be shown the classes being non-crystalline at p have

very strong implications for the arithmetic of E. For example, the ρ1 ⊗ ρ2-isotypic

component of the Mordell-Weil group of E must be trivial.

It is now that the decisive input of a p-adic regulator formula comes into play:

using Besser’s theory of finite polynomial cohomology, which is an extension of Cole-

man’s theory of p-adic integration, one is able to compute a formula for AJsyn(∆α)

paired with test vectors, where AJsyn is the syntomic Abel-Jacobi map and is related

to AJetp via the Bloch-Kato logarithm. By comparing the values of AJsyn(∆α) with

the special values of the triple product p-adic L-function at weight two points, one

obtains an explicit reciprocity law: if L is Perrin-Riou’s big logarithm interpolating

the Bloch-Kato logarithms, then L(κ(f,gh)) is essentially the triple product p-adic

L-function. In particular, specializing at weight one expresses the complex L-value

L(E, ρ1 ⊗ ρ2, 1) as the obstruction of κ(f, g1, h1) being crystalline, which is the last

6

missing piece in the proof of an instance of the equivariant BSD conjecture, among

other applications to the arithmetic of elliptic curves.

1.3 Hirzebruch-Zagier cycles

The present work seeks to explore a similar theme in the setting of the diagonal

curve inside the product of a modular curve and a Hilbert modular surface, which

is sometimes referred to as a Hirzebruch-Zagier cycles. Recent work of [BCF] has

constructed the corresponding p-adic L-function in this setting, the twisted triple

product p-adic L-function associated to a Hida family of elliptic modular forms and

a Hida family of Hilbert modular forms over a real quadratic field L, and established

a p-adic Gross-Zagier formula similar to the one in [DR14]. In this part of the thesis,

which is a joint work with Michele Fornea, the second author of [BCF], we establish

results analogous to [DR17] in the case when the prime p splits in L. In particular,

we study the p-adic variation of the Hirzebruch-Zagier cycles and their relations to

the twisted triple product p-adic L-function. We interpolate the Abel-Jacobi image of

the cycles into a big Hirzebruch-Zagier class, and by extending the p-adic regulator

formula in [BCF] to the entire p-tower of modular varieties, we prove an explicit

reciprocity law. As an application, we establish a new instance of the equivariant

BSD conjecture in rank zero and prove arithmetic properties of elliptic curves over

S5-quintic fields.

One can think of the setting of [DR17] as a degenerate case of our work, where

L degenerates to Q ⊕ Q. When L is a real quadratic field in which p splits, the

local behavior of the automorphic representation of ResLQGL2(Q) is very similar to the

tensor product of two automorphic representations of GL2(Q) (except for the presence

of a rank one group of global units), and correspondingly locally at p the Hilbert

7

modular surface behaves similar to a product of two modular curves. Nevertheless,

the main technical difficulties lie in the complexity of the geometry of Hilbert modular

surfaces. Extra work is needed to ensure the cycles are twisted in a way so that they

interpolate p-adically.

Main results

Let L be a real quadratic field and % : GL → GL2(C) a totally odd, irreducible

two-dimensional Artin representation of the absolute Galois group of L. The Asai

representation

As(%) := ⊗-IndQL(%)

is a 4-dimensional complex representation obtained as the tensor induction of % from

GL to GQ. We suppose that % has conductor Q and that the tensor induction of the

determinant det(%) is the trivial character so that As(%) is self-dual. Let H/Q be the

finite extension cut out by the representation As(%).

For any rational elliptic curve E/Q of conductor NE prime to Q, we define the

As(%)-isotypic component of the Mordell-Weil group of E as

E(H)As(%) := HomGal(H/F )(As(%), E(H)⊗ C),

and let the algebraic rank of E twisted by As(%) be defined as

ralg(E,As(%)) = dimCE(H)As(%).

On the analytic side, there is the Hasse-Weil-Artin L-series L(E,As(%), s), which is

the L-function associated to the Galois representation As(%)⊗Vp(E). By the modular-

ity of totally odd Artin representations and rational elliptic curves ([Wil95],[TW95],

8

[PS16]), this is identified with the twisted triple product L-function attached to the

weight two modular form associated to E and the parallel weight one Hilbert modular

form associated to %. We define the analytic rank of E twisted by As(%) to be

ran(E,As(%)) := ords=1L(E,As(%), s).

One of our main results establishes the following new special case of the equivariant

BSD conjecture:

Theorem 1.3.1. Suppose that NE is coprime to Q and split in L. If there is a

rational prime p such that p is coprime to NQ, split over L into narrowly principal

ideals such that the eigenvalues of Frobp on As(%) are all distinct modulo p and there

are no totally positive units in L congruent to −1 modulo p, then

ran

(E,As(%)

)= 0 =⇒ ralg

(E,As(%)

)= 0.

We will see later that using some elementary algebraic number theory we can

always pick such a prime p. Applying the above theorem to the Artin representation

constructed in [For18], we obtain the following result regarding the arithmetic of

rational elliptic curves over S5-quintic fields:

Corollary 1.3.2. Let K/Q be a non-totally real S5-quintic extension of positive

discriminant whose Galois closure K/Q contains a real quadratic field L. Suppose

NE is odd, unramified in K/Q and split in L, then

ran(E/K) = ran(E/Q) =⇒ ralg(E/K) = ralg(E/Q).

9

Overview of the proof

Let g% be the Hilbert cuspform of parallel weight one associated to the Artin repre-

sentation % : GL → GL2(C), and f the elliptic cuspform associated to E/Q.

Assume there exists a rational prime p coprime to the levels of f and g% such that

• f is ordinary and non-Eisenstein at p;

• the eigenvalues of Frobp on As(%) are all distinct modulo p;

• pOL = p1p2 splits into the product of two narrowly principal ideals, and

• there are no totally positive units of L congruent to -1 modulo p.

We will see in the last chapter of Part I that for any real quadratic field L the set

of prime p satisfying the last two conditions has a positive density. which means we

can always choose a p satisfying all the assumptions above.

After choosing a rational prime p and an ordinary p-stabilization g(p)% , one can

find a nearly ordinary Hida family Gn.o. passing through it. In the present work we

focus our attention on the one-variable Hida family G interpolating parallel weight

Hilbert modular forms of certain special level structures (the K,t(pα) level subgroups

defined below), which can be obtained by partially specializing the three variable

Hida family Gn.o. to one variable. The Hida family is equipped with a big Galois

representations, As(VG ), which is of rank 4 over the field of fractions QG of IG . Here

IG is the ring of coefficients of G , which is a finite flat algebra over the standard

Iwasawa algebra Λ = ZpJ1 + pZpK. Let A(IG ) be the set of arithmetic points inside

the weight space WG = Spf(IG )rig associated to IG , and let P ∈ A(IG ) be the weight

one point corresponding to g(p)% , so that

As(VGP) ∼= As(%).

10

On the other hand, there is the Galois representation Vf∼= Vp(E) associated to the

classical modular form f. One can show that there exists a twist V†G ,f of

VG ,f = As(VG )(−1)⊗Vf

interpolating Kummer self-dual representations, and such that the specialization at

P is

V†GP ,f= V%,E.

Refining the construction in [BCF], we construct a rigid analytic function onWG ,

the automorphic p-adic L-function,

L autp (G , f) :WG −→ Cp,

which is essentially the restriction of the three variable twisted triple product p-adic

L-function of op.cit. The value of L autp (G , f) at the weight one point P is, up to a

nonzero constant,

L autp (G , f)(P)

·∼ L(E,As(%), 1).

Furthermore, its value at any arithmetic point P ∈ A(IG ) of weight two is explicitly

given in terms of p-adic modular forms. We will establish a p-adic regulator formula

relating these values to the syntomic Abel-Jacobi images of Hirzebruch-Zagier cycles.

Hirzebruch-Zagier classes.

For α ≥ 1 and compact open K ⊆ GL2(AL,f ) hyperspecial at p, let

K,t(pα) :=

a b

c d

∈ K0(pα)∣∣∣ ap1dp1 ≡ ap2dp2 , dp1dp2 ≡ 1 (mod pα)

11

and denote by S(K,t(pα)) the corresponding Hilbert modular surface. We also set

K ′0(pα) = K0(pα) ∩GL2(AQ,f ),

and denote by Y (K ′0(pα)) the corresponding open modular curve.

Inspired by ideas in [DR17], for every α ≥ 1 we produce a null-homologous codi-

mension 2 cycle, called Hirzebruch-Zagier cycle,

∆α ∈ CH2(Zα(K)

)(Q(ζpα)

)⊗ Zp

on the modular threefold Zα(K) = S(K,t(pα))× Y (K ′0(p)). Moreover, the action of

Gal(Q(ζpα)/Q) is such that ∆α corresponds to a null-homologous rational cycle class

∆α ∈ CH2(Z†α(K)

)(Q)⊗ Zp

on a twisted threefold Z†α(K) with the following property: for every P ∈ A(IG ) of

weight two and level pα there is a natural Galois equivariant surjection

H3et

(Z†α(K)Q,Qp(2)

) V†GP,f

.

The ordinary parts of the Abel-Jacobi images

AJetp (∆α) ∈ H1

(Q,H3

et

(Z†α(K)Q,Zp(2)

))can be made compatible under the degeneracy maps $2 : Z†α+1(K) → Z†α(K) and

packaged together to form a global big cohomology class

κG ,f ∈ H1(Q,V†G ,f

).

12

This class retains information about the Abel-Jacobi image of algebraic cycles at

arithmetic points of weight two and can be specialized at the arithmetic point P of

weight one

κG ,f(P) ∈ H1(Q,V%,E

),

mirroring the automorphic p-adic L-function. In order to make apparent the rela-

tionship between κG ,f and the automorphic p-adic L-function, we use Perrin-Riou’s

machinery to fabricate the motivic p-adic L-function.

The motivic p-adic L-function.

This construction is naturally divided into two steps. First, the localization at p of

the big cohomology class can be projected to a Galois cohomology group

κp := Im(κG ,f) ∈ H1(Qp,U

fG (Θ)

)valued in a subquotient Uf

G (Θ) of the Galois representation V†G ,f on which GQp acts

through explicit characters. Perrin-Riou’s big logarithm ([KLZ17], Theorem 8.2.3)

valued in the big Dieudonne module D(UfG ) gives an element

L(κp) ∈ D(Uf

G

)interpolating the Bloch-Kato logarithm of the specialization of the class at arithmetic

points of weight ≥ 2, and the Bloch-Kato dual exponential at the weight one point

P. The second step requires the construction of a linear map

⟨, ωG ⊗ η

′⟩

: D(Uf

G

)−→ Π⊗Λ IG ,

13

where Π = lim←−αQp[1 + pZp/1 + pαZp], producing functions out of elements of the

Dieudonne module. As we are dealing with Hilbert modular surfaces and Ohta’s Λ-

adic Eichler-Shimura isomorphism [Oht95] is not available in this setting, the linear

map we construct only takes values in the ring of Cp-valued functions with domain

the subset of arithmetic points of weight 2 in A(IG ). The motivic p-adic L-function

is defined as

L motp (G , f) :=

⟨L(κp), ωG ⊗ η

′⟩.

By construction, at a weight two point P of level pα it specializes to (essentially)

AJsyn(∆α)(ωGP⊗ η), (1.1)

where ωGPand η are certain test vectors associated to G and f.

A regulator formula and an explicit reciprocity law

Using Besser’s theory of finite polynomial cohomology, we may evaluate (1.1) explic-

itly (the regulator formula) with a calculation similar in spirit to the ones found in

[DR17], [LSZ16] and [BCF], establishing the following specialization formula for every

arithmetic point P ∈ A(IG ) of weight two:

L motp (G , f)(P)

·∼⟨eordζ

∗(d−12 (w∗pα2 (GP)[P]), f(p)

⟩Pet,

a Petersson inner product involving (ordinary projections of) p-adic modular forms.

This last value is almost exactly the special value of our automorphic p-adic L-function

at the same weight two points, which is enough to show that L autp (G , f) ∈ IG and

L motp (G , f) ∈ Π ⊗ IG coincide (up to a constant in IG which never vanishes at

arithmetic points). This is our explicit reciprocity law.

14

As a side remark, this in particular shows that L motp (G , f) in fact lies in the

smaller, much more manageable fraction field of I×G . It can also be thought of as

saying that the motivic p-adic L-function, which is a priori only defined on arithmetic

points of weight two, can be meromorphically continued to the entire weight space

(and thus justifies the terminology).

The explicit reciprocity law is a bridge between the automorphic and the algebro-

geometric worlds, which can be used to transfer the information about the non-

vanishing of a special L-value into the existence of non-trivial annihilators of the

Mordell-Weil group.

Theorem 1.3.3. Suppose there is a rational prime p satisfying the assumptions as

above. If g(p)% is any ordinary p-stabilization of g%, then

L(E,As(%), 1) 6= 0 ⇐⇒ κG ,f(P) ∈ H1(Q,V%,E

)not crystalline at p.

By assuming that p splits in L and the eigenvalues of Frobp on As(%) are all

distinct, the eigenform g% has four distinct ordinary p-stabilizations. Hence, we obtain

four linearly independent classes by repeatedly applying Theorem 1.3.3. As the self-

dual representation As(%) is four dimensional, these annihilators suffice to prove that

the relevant part of the Mordell-Weil group is trivial.

1.4 Notations and conventions

Throughout this part of the thesis, we will adopt the following general notations and

conventions.

• L is a real quadratic field (except for the preliminary materials of Chapter 3

which applies to all totally real fields L). Let OL be its ring of integers, and dL

15

the different ideal.

• f is an elliptic modular form of weight 2 and level N with trivial character,

and g0 is a Hilbert modular form over L of parallel weight 1 and level Q with

character (χ,1), such that N and Q are coprime, and χ|Q= 1. Put M =

N · NL/Q(Q).

• p is a prime number which splits completely over L into narrowly principal

ideals: pOL = p1p2. We assume that there are no totally positive units of O×L

congruent to −1 modulo p, that p is coprime to MdL, and that p is Eisenstein

for f (in other words, f is not congruent to an Eisenstein series modulo p).

• E is a finite extension of Qp containing all Fourier coefficients of f0 and g0, and

O is its ring of integers.

• Fix a system of compatible p-power roots of unity ζpα .

• Fix an algebraic closure Q of Q. For any number field F inside Q, let GF denote

the absolute Galois group Gal(Q/F ). Similarly, fix an algebraic closure Qp of

Qp. For any E ⊂ Qp, let GE denote the absolute Galois group Gal(Qp/E).

• Fix an embedding Q → Qp which corresponds to a place of Q above p1. In

this way we also identify a decomposition group Dp of GQ at p, which is also a

decomposition group Dp1 of GL at p1, with GQp .

• For any function f : G(AQ)→ C (which in particular includes Hilbert modular

forms) and any A ∈ G(AQ), the slashing operator is given by

f|A(x) = f(xA).

In particular, f|A|B= f|BA.

16

• The normalization of local class field theory is that the local Artin map sends a

uniformizer to an arithmetic Frobenius, so that the p-adic cyclotomic character

Gal(Qp(ζp∞)/Qp) → Z×p is the reciprocal of the map induced by the natural

projection Q×p Z×p via the local Artin map. In particular, for a ∈ Z×p , the

element σa ∈ Gal(Q(ζpα)/Q) associated to a by class field theory sends ζpα to

ζa−1

pα . We will use adelic characters and the corresponding Galois characters

interchangeably.

• For any p-adic representation V and character χ of GQ or GQp , we will use

V (χ) := V ⊗ χ to denote the twisted Galois representation. When χ is the

m-th power of the cyclotomic character for some integer m, we simply write

V (m) for the Tate twist. To avoid overly cumbersome notation, when χ is the

m-th power of the cyclotomic character times another Galois character η, we

simply write V (m+ η) for the twist V (χ).

• G = ResLQGL2 is the algebraic group over Q obtained from restriction of scalars.

• In general we will use K with various subscripts to denote open compact sub-

groups of G(AQ,f ), and use a prime (e.g. K ′) to denote open compact subgroups

of GL2(AQ,f ). We also use S to denote Shimura varieties for G of a certain level

(e.g. S(K)), and Y to denote Shimura varieties for GL2 (e.g. Y (K ′)).

• We will use letters in script font to denote Hida families or Λ-adic forms (e.g.

G ), and use sans serif letters to denote classical modular forms (so we will use

gP to denote the classical specialization of a Hida family G at an arithmetic

point P).

• We will use 〈, 〉 to denote both the Petersson inner product of modular forms

and the de Rham Poincae pairing on modular varieties. In case there is risk

17

of confusion, we will use subscripts (Pet and dR) to differentiate the two. For

the Poincare pairing, if multiple pairings on different varieties are involved, we

will also indicate them using subscripts (e.g. 〈, 〉dR,Yα would mean the de Rham

Poincare pairing on the modular curve Yα).

18

Chapter 2

Preliminaries on Hilbert modular

forms

In this chapter we summarize basic theory of Hilbert modular forms and Hida families.

We follow the account of [Hid91].

2.1 Basic notations

Let L be a totally real number field with ring of integers OL and different dL. Later

on in the paper we will focus on the case of L being a real quadratic field. Let IL

denote the set of embeddings of L into Q. A double digit weight, or simply a weight

of modular forms is a pair of elements (k, w) in the free module Z[IL] generated by the

embeddings in IL. Let tL :=∑

σ∈ILσ ∈ Z[IL] denote the special “parallel weight one”

element. We set ZL(1) := A×L/L× detV (p∞)L×∞,+, the p-adic cyclotomic character

εL : ZL(1) −→ Z×p , y 7→ y−tLp |yf |−1AL .

19

It is clear that the composition of the natural diagonal embedding A×Q → A×L with

εL is precisely ε[L:Q]Q . The canonical isomorphism Z×p ∼= (1 + pZp) × µp−1 induces a

factorization of the character

εL = ηL · θL.

Definition 2.1.1. Let D, G and G∗ be the following algebraic groups over Q:

D = ResLQGm, G = ResLQGL2, G∗ = G×D Gm.

We now identify L∞ := L⊗Q R with RIL and embed L into RIL via the diagonal

map. Then the identity component G(R)+ = GL2(L ⊗Q R)+ of G(R) naturally acts

on HIL where H denotes the Poincar’e upper half plane, which contains a fixed square

root i =√−1 ∈ C of −1. Let i = (i, ..., i) ∈ HIL .

We will also use the following power convention repeatedly: for c = (cσ)σ∈IL ∈ CIL

and s =∑

σ∈IL sσ ·σ ∈ Z[IL], let cs = Πσ∈ILcsσσ . As a slight extension of this notation,

for a ∈ L, we write as = Πσ∈IL(aσ)sσ , where (aσ)σ∈IL is the image of a under the

natural diagonal embedding L → RIL .

2.2 Hilbert modular forms

We are now ready to recall the notion of (adelic) holomorphic Hilbert modular forms.

Definition 2.2.1. Let U ⊆ G(AQ,f ) be an open compact subgroup, and (k, w) ∈

Z[IL]2 be a double digit weight such that k − 2w = mtL for some m ∈ Z. The space

of holomorphic Hilbert modular forms of weight (k, w) and level U , which we denote

by Mk,w(U ;C), is the space of functions f : G(AQ) → C satisfying the following

conditions:

• f(αxu) = f(x)jk,w(u∞, i)−1 for α ∈ G(Q) and u ∈ UC+

∞, where C+∞ denotes the

20

stabilizer of i in G(R)+ and jk,w is the automorphy factor jk,w(

a b

c d

, z) =

(ad− bc)−w(cz + d)k for

a b

c d

∈ G(R) and z ∈ HIL ;

• for every x ∈ G(AQ,f ) the function fx : HIL → C given by fx(z) = f(xu∞)jk,w(u∞, i)

is holomorphic, where for each z ∈ HIL we choose u∞ ∈ G(R)+ such that

u∞i = z.

If in addition to the above f also satisfies

∫L\AL

f(

1 a

0 1

x)dx = 0

for all x ∈ G(A) and all additive Haar measure on L\AL, then we say f is a cuspform,

and we denote by Sk,w(U ;C) the space of Hilbert modular cuspforms of weight (k, w)

and level U .

Let dx be the Tamagawa measure on the quotient space [G(AQ)] := A×LG(Q)\G(AQ).

Then for any f1, f2 ∈ Sk,w(U ;C) with k − 2w = mtL, we define the Petersson inner

product as

〈f1, f2〉 :=

∫[G(AQ)]

f1(x)f2(x)| det(x)|mALdx.

We now introduce the notation for some standard open compact subgroups of

G(Z).

Definition 2.2.2. For any OL ideal N, write

• V0(N) =

a b

c d

∈ G(Z)

∣∣∣∣ c ∈ NOF

,

21

• V1(N) =

a b

c d

∈ V0(N)

∣∣∣∣ d ≡ 1 (mod NOF )

,

• V 1(N) =

a b

c d

∈ V0(N)

∣∣∣∣ a ≡ 1 (mod NOF )

,

• V (N) = V1(N) ∩ V 1(N),

• Γ(N) =

a b

c d

∈ V (N)

∣∣∣∣ b ∈ NOF

.

From now we will focus the exposition on Hilbert modular forms with level of the

form V (N). Any form of level U with U containing some V (N) can be viewed as a

form of level V (N) and thus can be treated similarly.

2.3 Relation to classical Hilbert modular forms

Next we recall the relation of the adelic Hilbert modular forms to classical Hilbert

modular forms. As above let N be any ideal of OL. Consider the narrow class

group Cl+L(N) := L×+\A×L,f/ detV (N) with cardinality h = h+L(N), and fix a set

of representatives t1, ..., th ⊂ A×L,f of Cl+L(N) with t1 = 1. We then obtain a

decomposition of the adelic points of G

G(A) =h∐i=1

G(Q)

t−1i 0

0 1

V (N)G(R)+

via strong approximation.

Given a Hilbert cuspform f ∈ Sk,w(V (N);C), for each i = 1, ..., h, we consider the

22

following holomorphic function fi : HIL → C given by

fi(z) = y−w∞ f

t−1

i y∞ x∞

0 1

where z = x∞ + iy∞, x∞ ∈ RIL and y∞ ∈ RIL+ . Each fi admits a Fourier expansion

fi(z) =∑

ξ∈(tid−1L )+

a(ξ, fi)eL(ξz) (2.1)

where eL(ξz) = exp(2πi∑

σ∈IL σ(ξ)zσ). The h-tuple of classical Hilbert modular

forms (f1, ..., fh) or their Fourier expansions determine f uniquely.

2.4 Adelic Fourier expansions

Fix dL ∈ A×L,f such that dLOL = dL is the absolute different ideal of L. Let LGal be

the Galois closure of L in Q and write V for the ring of integers or a valuation ring

of a finite extension L0 of FGal such that for every ideal a of OL and all σ ∈ IL, the

ideal aσV is principal. Choose a generator qσ ∈ V of qσV for each prime ideal q of

OL and by multiplicativity define av ∈ V for each fractional ideal a of L and each

v ∈ Z[IL].

We may package the collection of classical Fourier expansions in (2.1) for each

component fi together into an Adelic Fourier expansion. As an additional piece of

notation, let A×L,+ = A×L,fL×∞,+.

Theorem 2.4.1. ([Hid91], Theorem 1.1) Let eL : CIL −→ C× be defined as eL(z) =

exp(2πi∑

σ∈IL zτ)

and χL : AL/L −→ C× the additive character of the ideles which

satisfies χL(x) = eL(x∞). Each cuspform f ∈ Sk,w(V (N);C) has an adelic q-expansion

23

of the form

f

y x

0 1

= |y|AL

∑ξ∈L+

a(ξydL, f)(ξydL)tL−w(ξy∞)w−tLeL(iξy∞)χL(ξx)

for y ∈ A×L,+, x ∈ A×L , where a(−, f) : A×L,+ −→ C vanishes outside OLL×∞,+ and

depends only on the coset y∞ detV (N).

Concretely, the function a(−, f) is defined as follows: for every idele y ∈ A×L,+ can

be written as y = ξt−1i du for some 1 ≤ i ≤ h, ξ ∈ L×+ and u ∈ detV (N)L×∞,+, then

a(y, f) := a(ξ, fi)yw−tLξtL−w|ti|AL .

Furthermore, if all a(ξ, fi) belong to Q, then we define the p-adic Fourier coeffi-

cients ap(−, f) : A×L,+ −→ Qp by

ap(y, f) := a(ξ, fi)yw−tLp ξtL−wεL(ai)

−1.

Here εL : ZL(1) := A×L/L× detV (p∞)L×∞,+ → Q×p is the p-adic cyclotomic character

given by y 7→ y−tFp |y∞|−1AL .

Using the Fourier expansions we may define rational and integral structures on

the space of Hilbert modular forms. For any V-algebra A in C, define

Sk,w(U ;A) = f ∈ Sk,w(U ;C) | a(y, f) ∈ A.

24

2.5 Hecke Theory

Let U ⊂ G(AQ,f ) be an open compact subgroup satisfying V0(N) ⊂ V (N). Suppose

V is the valuation ring corresponding to the fixed embedding ιp : LGal → Qp, so that

we may assume ytL−w = 1 whenever the ideal yOL generated by y is prime to pOL.

For every g ∈ G(AQ), we have the following double coset operator [UgU ]. Decom-

posing the double coset into a disjoint union

UgU =∐i

γiU,

we define

f|[UgU ](x) =∑i

f(xγi).

Remark. This is the convention used in [Hid91] (see the discussion immediately fol-

lowing equation (2.1)). However, this differs slightly from the double coset operators

used in [Hid88], [Hid89b] and [Hid04]. However, to keep the exposition consistent

with our main reference [Hid91], we stick with the notation there.

Definition 2.5.1. For every prime ideal q of OL and a uniformizer $q of OL,q, we

define the Hecke operator given by the following double coset operator

T ($q) =[V (N)

$q 0

0 1

V (N)].

We will also write

T0($q) = $w−tLq T ($q).

25

Furthermore, for every a ∈ O×L,N = Πq|NO×L,q, define the double coset operator

T (a, 1) =[V (N)

a 0

0 1

V (N)].

Finally, for z ∈ ZG(AQ,f ) in the center of G(AQ,f ), we have the diamond operator 〈z〉

defined by

〈z〉f(x) = f(xz).

If q is coprime to the level N, then T ($q) (respectively, T0($q)) is independent

of the particular choice of the uniformizer $q, and thus we simply denote it by T (q)

(respectively, T0(q)). Similarly, the diamond operator 〈$q〉 also only depends on the

ideal q and we denote it by 〈q〉. In general, for the V (N) level structure, if q|N,

then T ($q) (respectively, T0($q)) does depend on $q, and we will write it as U($q)

(respectively, U0($q)).

The effects of the Hecke operators on the adelic Fourier coefficients are given by

ap(y, T (q)f) = ap(y$q, f)$tL−wq,p + NL/Q(q)ap(y$

−1q , 〈q〉f)$w−tL

q,p (2.2)

and

a(y, T0(q)f) = ap(y$q, f) + NL/Q(q)q2(w−tL)ap(y$−1q , 〈q〉f), (2.3)

if q - N, and

ap(y, U($q)f) = ap(y$q, f)$tL−wq,p (2.4)

and

a(y, U0($q)f) = a(y$q, f) (2.5)

if q|N. Here $q,p denotes the p-component of $q (in articular it is always trivial

26

unless q|pOL). Furthermore, for a ∈ O×L,N,

ap(y, T (a, 1)f) = ap(ya, f)atL−wp .

Definition 2.5.2. The Hecke algebra hk,w(U ;V) is defined as the V-subalgebra of

EndC(Sk,w(U ;C)) generated by the Hecke operators T0(q) for q outside N, U0($q) for

q|N, and T (a, b) for a, b ∈ O×L,N. For each V-algebra A in C, define

hk,w(U ;A) = hk,w(U ;V)⊗V A.

Theorem 2.5.3. ([Hid91], Theorem 2.2) Let F/LGal be any finite field extension and

A any V-subalgebra of F . Then there is a natural isomorphism

Sk,w(U ;F ) ∼= Sk,w(U ;A)⊗A F.

Moreover, if A is an integrally closed domain containing V , finite flat over either V

or Zp, then Sk,w(U ;A) is stable under the action of hk,w(U ;A), and the pairing

(, ) : Sk,w(U ;A)× hk,w(U ;A) −→ A, (f, h) = a(1, h · f)

induces isomorphisms of A-modules

hk,w(U ;A) ∼= Sk,w(U ;A)∗, Sk,w(U ;A) ∼= hk,w(U ;A)∗,

where (−)∗ denotes the A-linear dual HomA(−, A).

As an extra piece of notation, for any y ∈ O×L , we can factor it as y = auΠq$e(q)q

with a ∈ O×L,N and u ∈ detV (N). Write n for the ideal (Πq-N$e(q)q )OL, then we define

27

the Hecke operator

T (y) = T (a, 1)T (n)Πq|NU($e(q)q ),

which only depends on the idele y. We also put T0(y) = yw−tLT (y).

A cuspform f is said to be an eigenform if it is an eigenvector for all the Hecke

operators, and it is normalized if a(1, f) = 1. Note that for a normalized eigenform

f the T0(y)-eigenvalue is a(y, f) for every idele y. By a result of Shimura ([Shi78],

Proposition 2.2), the Hecke eigenvalues of an eigenform are all algebraic numbers. In

particular, every normalized eigenform f is defined over Q.

We next take a closer look at the behaviors at p.

Definition 2.5.4. Let p|p be a prime ideal of OL and $p ∈ OL,p a uniformizer. A

normalized eigenform f ∈ Sk,w(U ; Q) is said to be nearly ordinary at p if the U0($p)

(or T0(p) if p is away from the level) eigenvalue is a p-adic unit with respect to the

fixed embedding ιp : Q → Qp. Note that this notion is independent of the choice of

the uniformizer. If f is nearly ordinary at all p|p, we say f is p-nearly ordinary.

We also have the following V operator:

Definition 2.5.5. For every finite idele b ∈ A×L,f there is an operator V (b) on the

space of cuspforms defined by

(V (b)f)(x) = NL/Q(bOL)f(x

b−1 0

0 1

).

Remark. In [Hid91] Section 7B this operator is denoted as [b]. The effect on Fourier

coefficients is

ap(y, V (b)f) = bw−tLp ap(yb−1, f).

28

In particular, combined with equation (2.4) it shows

U($q)V ($q) = 1

for all $q a uniformizer of a prime ideal dividing the level N.

2.6 p-adic modular forms

Now fix a valuation ring O in Qp finite flat over Zp containing ιp(V). We also fix

an ideal N of OL which is prime to p and let K be a subgroup of V0(N) containing

V1(N). Put K(pα) = K ∩ V (pα) and K(p∞) = ∩αK(pα).

Consider the limit

Sk,w(K(p∞);O) = lim−→α

Sk,w(K(pα);O)

on which the universal p-adic Hecke algebra

hk,w(K(p∞);O) = lim←−α

hk,w(K(pα);O)

naturally acts. Inside hk,w(K(p∞);O) we have the operators

T(y) = lim←−α

T (y)yw−tLp .

There is a p-adic norm on Sk,w(K(p∞);O) given by |f|p = supy(|ap(y, f)|). We

denote the completion of Sk,w(K(p∞);O) under this norm by Sk,w(K(p∞);O). The

p-adic Fourier coefficients can be regarded as a continuous function

J → O, y 7→ ap(y, f),

29

where J = (OL∩A×L,f )/(K(p∞)∩A×L,f ) = lim←−α(OL∩A×L,f )/(K(pα)∩A×L,f ) is endowed

with profinite topology and is isomorphic toO×p ×IL, where IL denotes the semigroup

of all integral ideals of L. Writing C (X,A) for the space of continuous functions

on a topological space X with values in A, we see that we have an embedding of

Sk,w(K(p∞);O) into C (J , O).

Theorem 2.6.1. The following statements hold regarding the universal p-adic Hecke

algebra:

1. ([Hid89b], Theorem 2.3) for all weights (k, w) ∈ Z[IL]2 satisfying k− 2w = mtL

for some m ∈ Z, there is a canonical algebra isomorphism

hk,w(K(p∞);O) ∼= h2tL,tL(K(p∞);O)

sending T(y) to T(y);

2. ([Hid91], Theorem 3.1) the pairing

(, ) : hk,w(K(p∞);O)× Sk,w(K(p∞);O) −→ O, (h, f) = ap(1, h · f)

induces isomorphisms

hk,w(K(p∞);O)∗ ∼= Sk,w(K(p∞);O) and Sk,w(K(p∞);O)∗ ∼= hk,w(K(p∞);O).

As an immediate corollary, the image of Sk,w(K(p∞);O) inside C (J , O) is inde-

pendent of the weight (k, w), and we will simply denote it by SL(K;O). Similarly, we

will use hL(K;O) to denote hk,w(K(p∞);O). Moreover, if K = V1(N), we will simply

write SL(N;O) and hL(N;O). SL(K;O) is called the space of p-adic cuspforms.

Let ZL(K) = A×L/L×(AL,f ∩K(p∞))L×∞,+. In the case K = V1(N), ZL(K) can

30

be identified with the strict ray class group of L of modulus Np∞. Put GL(K) =

ZL(K)×O×L,p. Then (z, a) ∈ GL(K) acts on Sk,w(K(pα);O) via the operator 〈z, a〉 :=

T (a−1, 1)〈z〉, which we will also call diamond operators. In this way, we have a

natural embedding of GL(K) and hence OJGL(K)K into hL(N;O), allowing us to

view hL(N;O) as an OJGL(K)K-algebra.

Now we normalize this new operator by setting 〈z, a〉k,w = εL(z)k−2w〈z, a〉, which

factors through the finite quotient ClL(Npα)× (OL/pα)× of GL(K).

Definition 2.6.2. For characters ψ : Cl(Npα)→ O× and ψ′ : (OL/pα)× → O×, let

Sk,w(K(pα), ψ, ψ′;O) = f ∈ Sk,w(K(pα);O) : 〈z, a〉f = ψ(z)ψ′(a)f,∀(z, a) ∈ GL(K).

2.7 Hida families

Since hL(K;O) is a compact ring, we can decompose it as a direct sum of algebras

hL(K;O) = hn.o.L (K;O)⊕ hss

L (K;O)

in such a way that T(p) is a unit in hn.o.L (K;O) and is topologically nilpotent in

hssL (K;O). We denote by

en.o. = limn→∞

T(p)n!

the idempotent of the nearly ordinary part. Let Sn.o.

L (K;O) = en.o.SL(K;O) be the

space of nearly ordinary p-adic cuspforms. The ring hn.o.L (K;O) is called the universal

nearly ordinary Hecke algebra.

We have a short exact sequence

1 // E+

Np\(1 + pOL,p

)// ZL(K) // Cl+L(Np) // 1,

31

where Cl+L(Np) is the strict ray class group of modulus Np and E+

Np is the closure of

the totally positive global units of OL congruent to 1 modulo Np. When p is large

enough, the group Cl+L(Np) has order prime to p, in which case there is a canonical

decomposition

ZL(K) ∼= E+

Np\(1 + pOL,p

)× Cl+L(Np)

z 7→(ξz, z

).

(2.6)

Putting IL = E+

Np\(1 + pOL,p

)× (1 + pOL,p), we obtain a short exact sequence

1 // IL // GL(K) // Cl+L(Np)× (OL/p)× // 1

which splits canonically. The group IL is a finitely generated Zp-module of Zp-rank

r = [L : Q] + 1 + δ, where δ is Leopoldt’s defect for L.

Now fix a decomposition of IL = W×IL,tor, where IL,tor is the torsion subgroup

of IL and W is a free Zp-module of rank [L : Q] + 1 + δ. Put ΛL = OJWK.

Theorem 2.7.1. ([Hid89b], Theorem 2.4) The universal nearly ordinary Hecke alge-

bra hn.o.L (K;O) is finite and torsion-free over ΛL.

For each pair of characters ψ : Cl(Npα) → O× and ψ′ : (OL/pα)× → O× and a

weight (k, w) ∈ Z[IL] satisfying k − 2w = mtL for some integer m, let Pk,w,ψ,ψ′ be

the O-algebra homomorphism OJGL(K)K → O induced by the character GL(K) 3

(z, a) 7→ ψ(z)ψ′(a)εmL (z)atL−w.

Let F be the fraction field of ΛL, and fix an algebraic closure F of F. Consider

a Λ-linear map λ : hn.o.L (K;O) → F. Since hn.o.

L (K;O) is finite over ΛL, the image

of λ is contained in the integral closure I of ΛL in a finite extension K of F. If

P : I→ O is an O-algebra homomorphism which coincides with some Pk,w,ψ,ψ′ on ΛL,

32

then by Theorem 2.4 of [Hid89b], the composition λP = Pλ induces an O-linear map

λP : hn.o.k,w(K(pα); Qp)→ Qp for a suitable α > 0, and hence by duality (Theorem 2.5.3)

we obtain a Hilbert cuspform fP ∈ Sk,w(K(pα); Qp) such that ap(y, fP) = λP(T(y)) for

all integral idele y. If λ is a ΛL-algebra homomorphism then fP is an eigenform. This

justifies the following definition:

Definition 2.7.2. The space of nearly ordinary I-adic cuspforms of tame level K is

Sn.o.L (K; I) := HomΛL

(hn.o.L (K(p∞);O), I).

If an I-adic cuspforms is also a ΛL-algebra homomorphism, then we say it is a Hida

family.

We can write OJGL(N)K =⊕

χ ΛL,χ as a direct sum ranging over all the char-

acters of the torsion subgroup GL(N)tor = IL,tor × Cl+L(Np) × (OL/p)×, where each

ΛL,χ is isomorphic to ΛL. Similarly there is a decomposition of the universal nearly

ordinary Hecke algebra hn.o.L (K;O) =

⊕χ hn.o.

L (K;O)χ. Then we may refine the above

definition as follows:

Definition 2.7.3. Let χ : GL(K)tor → O× be a character and I a ΛL,χ-algebra. The

space of nearly ordinary I-adic cuspforms of tame level K and character χ is

Sn.o.L (K,χ; I) := HomΛχ(hn.o.

L (K(p∞)χ;O), I).

Definition 2.7.4. Let I be a ΛL,χ-algebra. The set of arithmetic points AL,χ(I) is

the subset of HomO-alg(I, Qp) consisting of homomorphisms that coincide with some

Pk,w,ψ,ψ′ on ΛL,χ.

33

2.8 Diagonal restriction

If L/F is an extension of totally real fields, there is a restriction map IL → IF which

induces a group homomorphism Z[IL] → Z[IF ] denoted by ` 7→ `|F and satisfies

(tL)|F = [L : F ] · tF . Let N be an ideal of OF , then the natural inclusion ζ :

GL2(AQ,f ) → GL2(AL) defines by composition a diagonal restriction map

ζ∗ : S`,x(V (NOL);L;C)→ S`|F ,x|F (V (N);F ;C).

Lemma 2.8.1. Let g ∈ S`,x(V (NOL);L; Q), then for y ∈ OFF×∞,+ written as y =

ξa−1i dFu where ξ ∈ F×+ and u ∈ detV (N)F×∞,+, we have

ap(y, ζ∗g) = y

x|F−tFp ξtF−x|F εF (ai)

−1∑

TrL/F (η)=ξ

ap(yη, g)(yη)tL−xp ηx−tL

where η ∈ L×+ and yη = ηa−1i dLu.

Proof. A direct computation.

Remark. When p is unramified in L/Q one sees that yp = (ξu)p, (yη)p = (ηu)p and

the formula becomes

ap(y, ζ∗g) = εF (ai)

−1∑

TrL/F (η)=ξ

ap(yη, g)

2.9 Twists of Hilbert modular forms.

Hida defined two kinds of twists for Hilbert modular forms in [Hid91], Section 7F.

We recall these constructions and then modify one of them to get a new one.

Let Ψ : A×L/L× → C× be a Hecke character of conductor C(Ψ) and infinity type

34

m · tL, m ∈ Z. Since Ψ has algebraic values on finite ideles, one has the map

−⊗Ψ : Sk,w(Npα, ψ, ψ′;O

)→ Sk,w+m·tL

(C(Ψ)Npα, ψΨ2, ψ′Ψ−1

p ;O)

f 7→ f ⊗Ψ

where f ⊗Ψ has adelic Fourier coefficients

ap(y, f ⊗Ψ) = Ψ(y∞)ap(y, f)ym·tLp .

In the special case when Ψ = |−|mAL , one finds that

f ⊗ |−|mAL ∈ Sk,w+m·tL(Npα, ψ, ψ′;O

)and

ap(y, f ⊗ |−|mAL) = εF (y)−map(y, f).

Since Ψ is trivial on the principal ideles, it follows immediately from Lemma 2.8.1

we have the following

Lemma 2.9.1. Let L/F be a finite extension of totally real fields as above, and let

ΨF = Ψ|A×F be the restriction of the Hecke character Ψ of L to F . Then for any

Hilbert modular form g over L,

ζ∗(g ⊗Ψ) = ζ∗(g)⊗ΨF .

Another consequence is that this kind of twisting does not affect the classical

Fourier expansion on the identity component of the Hilbert modular surface, given

that we have fixed a set of representatives t1 = 1, t2, ..., th of cl+L(N) in Section 2.3.

More precisely, we have the following

35

Lemma 2.9.2. Let g ∈ Sk,w(Npα, ψ, ψ′;O

)and g1 be the first component of the

corresponding tuple of classical Hilbert modular forms. Then for any Hecke character

Ψ : A×L/L× → C×, we have

(g ⊗Ψ)1 = ψ(dL)g1.

Proof. Since we have chosen t1 = 1 and the idele generator dL of the different dL

to have trivial p-component, by definition the classical Fourier coefficients a(ξ, g1) is

related to the p-adic Fourier coefficients via (1.3 a of [Hid91])

ap(ξdL, g) = a(ξ, g1)

where ξ ∈ L+ and ξdL is understood as an adele.

Suppose Ψ has infinity type m · tL. By construction of the twist we have

ap(ξdL, g ⊗Ψ) = ap(ξdL, g)Ψ((ξdL)∞)ξm·tL = ap(ξdL, g)Ψ(dL)Ψ(ξ) = ap(ξdL, g)Ψ(dL)

where we regard ξ as an adele via diagonal embedding. This implies

a(ξ, (g ⊗Ψ)1) = Ψ(dL)a(ξ, g1)

which finishes the proof.

Now let N,A be integral OL-ideals prime to p and let χ : Cl+L(Apα) → O× be a

finite order Hecke character. Hida defined a function

−|χ : Sk,w(Npα, ψ, ψ′;O

)→ Sk,w

(NpαA2, ψχ2, ψ′;O

)f 7→ f|χ

36

where f|χ is a cuspform whose adelic Fourier coefficients are given by

ap(y, f|χ) =

χ(y)ap(y, f) if yAp ∈ O×L,Ap

0 otherwise

Finally, we define a new kind of twist by slightly modifying Hida’s work. Let p | p

be a OL-prime ideal and χ : O×L,p → O× a finite order character of conductor pc(χ).

Proposition 2.9.3. There is a function

− ? χ : Sk,w(Npα, ψ, ψ′;O

)→ Sk,w

(Npmaxα,c(χ), ψ, ψ′χ−1;O

)f 7→ f ? χ

where the adelic Fourier coefficients of f ? χ are given by

ap(y, f ? χ) =

χ(yp)ap(y, f) if yp ∈ O×L,p;

0 otherwise.

Proof. Define

h(x) :=∑

u∈(OF,p/pc(χ))×

χ−1(u)f

x1 u$

−c(χ)p dL

0 1

.

Looking at the adelic q-expansion we see that

ap(y, h) =

∑u∈(OL,p/pc(χ))

×

χ−1(u)χL(yu$

−c(χ)p

) ap(y, f).

37

Then one notices that

∑u∈(OL,p/pc(χ))

×

χ−1(u)χL(yu$

−c(χ)p

)=

χ(yp)G(χ) if yp ∈ O×L,p;

0 otherwise;

where

G(χ) =∑u

χ−1(u)χL(u$−c(χ)p

)(2.7)

is a Gauss sum. Hence, we define

f ? χ := G(χ)−1h.

Finally, to understand the action of the operators T (a−1, 1) we directly compute that

ap(y, (f ? χ)|T (a−1,1)

)= χ−1(ap)ψ

′(a)ap(y, f ? χ).

Let f ∈ Sk,w(Npα, ψ, ψ′;O) be an eigencuspform. For p a prime ideal of OL, let

τpα ∈ GL2(AL) be defined by

(τpα)p =

0 −1

$αp 0

, (τpα)v = I2 for v 6= p.

Then we have the following operator

f|τ−1pα (x) := f(xτ−1

pα ).

38

Lemma 2.9.4. Let f ∈ S2tF ,tF (Npα, ψ, ψ′;O) be an eigenform, then

ap($p, f)α ·(f|τ−1

pα

)[p]= G(ψp(ψ

′p)

2) ·(f ? ψp(ψ

′p)

2)

the equality taking place in S2tF ,tF

(Npα, ψ, ψ′ · ψ−1

p (ψ′p)−2;O

).

Proof. For u, v ∈ O×L,p such that uv ≡ −1 (mod pα) we have

0 −1

$αp 0

−1$α

p u

0 1

=

1 v$−αp

0 1

v (1 + uv)$−αp

−$αp −u

.

To simplify notation let γu,α =

$αp u

0 1

and δv =

1 v$−αp

0 1

. Note that right

translation by the matrix

v (1 + uv)$−αp

−$αp −u

gives the action of 〈v−1, v−2〉. Thus

we have

f|γu,α|τ−1pα = (〈v−1, v−2〉f)|δv = ψ(v−1)ψ′(v−2) · f|δv. (2.8)

On one hand, summing the right hand side of (2.8) over v ∈ (OF/pα)× gives

∑v∈(OF /pα)×

(ψ(ψ′)2)−1(v)f|δv = G(ψp(ψ′p)

2) ·(f ? ψp(ψ

′p)

2).

On the other hand, summing the left hand side of (2.8) over u ∈ (OF/pα)× gives

∑u∈(OF /pα)×

f|γu,α|τ−1pα =

( ∑u∈(OF /pα)

f|γu,α)|τ−1

pα −( ∑u′∈(pOF /pα)

f|γu′,α−1|τ−1pα

)|

1

$p

= (Uα

p f)|τ−1pα − (Uα−1

p f)|τ−1pα |

1

$p

,

39

where in the first equality we used the following identity for u ∈ p(OF/pα):

0 −1

$αp 0

−1$α

p u

0 1

=

1 0

0 $p

0 −1

$αp 0

−1$α−1

p u/$p

0 1

.

Finally, taking the p-depletion we obtain

ap($p, f)α ·(f|τpα

)[p]= G(ψp(ψ

′p)

2) ·(f ? ψp(ψ

′p)

2)

because

((Uα−1

p f)|τ−1pα |

1

$p

)[p]= 0 and (f ? ψp(ψ

′p)

2)[p] = f ? ψp(ψ′p)

2.

40

Chapter 3

Automorphic p-adic L-functions

3.1 Preparations with I-adic forms

From now on we will assume L is a real quadratic field and write IL = µ, µ′. Let

χ : Cl+L(Q)→ O× be a finite order character satisfying χ|Q = 1 and suppose we are

given a primitive Hilbert cuspform g ∈ StL,tL(Q;χ; Q) over L of level Q, an ideal of

OL, and a primitive elliptic cuspform f ∈ S2,1(N ;1; Q) of level N ∈ Z+ and trivial

character, such that NL/Q(Q) and N are coprime, and we put M = NL/Q(Q) ·N . Let

p be a rational prime which splits completely in L and is coprime to M , and assume

further that g, f are p-ordinary. We write P = p, p′ for the set of prime OL-ideals

dividing p.

Definition 3.1.1. Let Γ denote the p-adic group 1 + pZp and Λ = OJΓK the usual

Iwasawa algebra. For any Λ-algebra I, let W(I) be the weight space W(I) =

HomO−alg(I, Qp), and the subset of arithmetic points A(I) consists of those coinciding

with continuous homomorphisms

w`,χ : Λ −→ Qp, [u] 7→ χ(u)u`−2

41

on Λ, for ` ∈ Z≥1 and χ : Γ→ Q×p a finite order character. We say such an arithmetic

point is of weight ` and character χ.

Let χ be the character

χ : Cl+L(Qp) −→ O×, z 7→ χ(z)θ−1L (z)

AND Consider the surjection

φχ : OJGL(Q)K OJΓK, [(z, a)] 7→ χ(z)[ξ−tLz ]. (3.1)

Then for any arithmetic point P : OJGL(Q)K→ Qp of weight (`tL, tL) and character

(χθ1−`L χ−1,1), where χ : ZL(Q) → O× is the p-power order character factoring

through the norm with χ = χ NL/Q, we have

P`tL,tL,χθ1−`L χ−1,1 = w`,χ φχ (3.2)

Definition 3.1.2. Let Gn.o. ∈ Sn.o.

L (Q,χ; IGn.o.) be the nearly ordinary Hida family

passing through an ordinary p-stabilization g(p) of g, i.e. there exists an arithmetic

point P ∈ AL,χ(IGn.o.) of weight (tL, tL) and character (χ,1) such that Gn.o.(P) =

g(p) . If we set

IG := IGn.o. ⊗φχ Λ and G := (1⊗ φχ) Gn.o.,

then G ∈ Sord

L (Q,χ; IG ) is the ordinary Hida family passing through g(p) ([Wil88],

Theorem 3).

Remark. By (3.2), φχ induces an injection A(IG ) → AL,χ(IGn.o.), and the image

consists of those arithmetic points of weight (`tL, tL) and character (χθ1−`L χ−1,1),

42

where χ = χ NL/Q factors through the norm. In particular, the arithmetic point

P ∈ AL,χ(IGn.o.) of weight (tL, tL) and character (χ,1) in Definition 3.1.2 is the

image of some arithmetic point in A(IG ) of weight 1 and trivial character, which, by

an abuse of notation, we still denote by P.

Let G be the IG -adic cuspform passing through the ordinary p-stabilization of the

test vector g ([DR14] Section 2.6). We define a homomorphism of ΛL,χ-modules

d•µG[P] : hL(MOL;O) −→ IG by

d•µG[P] (〈z〉T(y)) =

G (〈z〉T(y))φχ

([yp, 1]

)y−1p if yp ∈ O×L,p;

0 otherwise.

Using diagonal restriction ζ : hQ(M ;O)→ hL(MOL;O), ζ([z, a]) = [∆(z),∆(a)]a−1,

we define the ordinary IG -adic cuspform

eordζ∗(d•µG [P]

)† ∈ Sord

Q(M,ψ−1

; IG

)(3.3)

by setting

ζ∗(d•µG

[P])†(〈z〉T(y)

)= θQ(y) · φχ

([∆(ξz)

−1∆(ξy)−1/2, 1]

)· d•µG [P]

(ζ [〈z〉T(y)]

),

where ∆ : 1 + pZp → 1 + pOL,p is the diagonal embedding.

Proposition 3.1.3. Let P ∈ A(IG ) be an arithmetic point extending w`,χ . If we

denote by χpθ`−1L,p : O×L,p −→ O× the local character x 7→ χθ

`−1L (x) (χ is regarded as

a character of O×L,p via its natural identification with Z×p ), then the specialization of

the IG -adic form in (3.3) at P is given by


)†(P) = eordζ

∗(d1−`µ (g

[P]P ? χ−1

p θ1−`L,p )

)⊗ θ`−1

Q χ|−|`−2AQ,

43

which is a classical elliptic cuspform of weight (2, 1) and trivial character.

Proof. Let χ = χ NL/Q. A direct computation

P eordζ∗(d•µG [P]

)†([z, a])

=θQ(a−1) · P φχ([∆(ξz)−1∆(ξa)

1/2, 1]) · P d•µG [P]([∆(z),∆(a)]a−1

)=θ−1

Q (a) · χ2(z)η

2(2−`)Q (z)χ−1

(a)η`−2Q (a) · P φχ([∆(z)∆(a)−1

p ,∆(a)])

=θ−1Q (a) · χ2

(z)η2(2−`)Q (z)χ−1

(a)η`−2Q (a) · (χθ1−`

L χ−1)(∆(z)∆(a)−1p )ε`−2

L (∆(z)∆(a)−1p )

=1

shows that eordζ∗(d•µG [P]

)†(P) is a cuspform of weight (2, 1) and trivial character.

Moreover,

ap(y, d•µG

[P](P)) = y1−`p ap(y, g

[P]P ) · χ−1(yp)θ

1−`L (yp),

which implies

ζ∗d•µG[P](P) = ζ∗

(d1−`µ

(g

[P]P ? χ−1

p θ1−`L,p

)).

Finally,

ap(y, ζ∗(d•µG

[P])†(P))

= θQ(y) · χ(y)η2−`Q (y) · ap

(y, ζ∗d•µG

[P](P))

= θ`−1Q (y)χ(y)ε2−`

Q (y) · ap(y, ζ∗d•µG

[P](P))

gives the desired explicit formula


)†(P) = eordζ

∗(d1−`µ (g

[P]P ? χ−1

p θ1−`L,p )

)⊗ θ`−1

Q χ|−|`−2AQ.

44

Corollary 3.1.4. We have


)† ∈ Sord2,1

(Mp;1;O

)⊗O IG .

Proof. By Proposition 3.1.3


)†(P) ∈ Sord

2,1

(Mp;1;O

)for any arithmetic crystalline point P ∈ A(IG ) of weight ` and trivial character, we

have eordζ∗(d•µG [P]

)†(P) ∈ Sord

2,1 (Mp;1;O). By the density of such crystalline points,

the homomorphism eordζ∗(d•µG [P]

)†factors through the reduction to weight (2, 1),

level Mp and trivial character of hordQ (M ;O).

3.2 The automorphic p-adic L-function

Now we are ready to define the p-adic L-function. The construction is an adaption

of the one in Section 3.3 of [BCF], which in turn builds on [DR14]. Let ef(p)

be the

f(p) -isotypical projector, then we can write

ef(p)eordζ

∗(d•µG [P])†

=∑

d|(M/N)

λd · f(p) (qd) for λdd ⊂ IG ,

and define

Lp(G , f) =∑

d|(M/N)

λd ·⟨f(p) (qd), f

(p) (q)

⟩⟨f(p) , f

(p)⟩ ∈ IG .

Definition 3.2.1. The twisted triple product p-adic L-function attached to(G , f

)

45

is the rigid-analytic function

L autp (G , f) : W(IG ) −→ Cp

determined by Lp(G , f) ∈ IG .

For any arithmetic point P ∈ A(IG ) of weight ` and character χ, we have

L autp (G , f)(P) =

⟨eordζ

∗(d1−`µ (g

[P]P ? χ−1

p θ1−`L,p )

)⊗ θ`−1

Q χ|−|`−2AQ, f

(p)

⟩〈f(p) , f

(p) 〉

. (3.4)

Remark. The p-adic L-function L autp (G , f) changes sign with a different choice of µ.

3.3 Link to the complex L-value in weight one.

Recall P is the arithmetic point in A(IG ) of weight 1 and trivial character such that

G (P) = g(p) . The value of the p-adic L-function at P is the complex L-values up to

an explicit term consisting of Euler factors at p. Let αf∗ , βf∗ be the inverses of the

roots of the T ($p)-Hecke polynomial 1− ap($p, f∗)X + ψ−1

(p)pX2. If we set

E(f∗) = 1− βf∗α−1f∗,

then we can rewrite the values of the p-adic L-function at every arithmetic point

P ∈WG as

L autp (G , f)(P) =

1

E(f∗)

⟨eordζ

∗(d1−`µ (g

[P]P ? χ−1

p θ1−`L,p )

)⊗ θ`−1

Q χ|−|`−2AQ, f∗

⟩〈f∗ , f∗〉

. (3.5)

46

Lemma 3.3.1. We have

L autp (G , f)(P) =

E spp (g, f

∗)

E(f∗)E1,p(g, f∗)

⟨eord

(ζ∗(g)⊗ |−|−1

AQ

), f∗

⟩〈f∗ , f∗〉

where

E spp (g, f

∗) =

∏•,?∈α,β

(1− •1 ?2 α

−1f∗

), E1,p(g, f

∗) = 1− α1β1α2β2α

−2f∗.

where αi, βi are the inverses of the roots of the T (pi)-Hecke polynomial for g, i = 1, 2.

Proof. By Proposition 3.1.3, we have


)†(P) = eord

(ζ∗(g[P])⊗ |−|−1

AQ

).

Then the desired claim follows directly from the explicit calculations in Lemmas 3.10

and 3.11 of [BCF].

Corollary 3.3.2. The p-adic L-value L autp (Gord, f)(P) 6= 0 if and only if the complex

L-value L(

12, πg,f

)6= 0.

Proof. By Theorem 3.2 of [BCF] there exists a choice of the test vector g such that

⟨eord

(ζ∗(g)⊗ |−|−1

AQ

), f∗

⟩6= 0

if and only if L(

12, πg,f

)6= 0. Then Lemma 3.3.1 gives the claim after noting that

all the factors appearing in the formula are non-zero. Indeed, the p-adic valuation of

E(f∗) and E1,p(g, f∗) is zero, and E sp

p (g, f∗) is non-zero because αi, βi for i = 1, 2 are

roots of unity while αf∗ is not because of the Weil conjectures.

47

Chapter 4

Hilbert modular varieties and

Hirzebruch-Zagier classes

4.1 Basic notations

We keep the notation that L denotes a real quadratic field with ring of integers OL

and different d. Asumme the prime p splits in L as pOL = p1p2, and assume that p1

(and thus also p2) is narrowly principal in OL, i.e. it is generated by a totally positive

element. There are natural identifications OL,p1 ' Zp and OL,p2 ' Zp, and we will

often identify OL,p = OL,p1×OL,p2 with Zp×Zp and represent its elements with pairs

of the form (a1, a2) with a1, a2 ∈ Zp. We make a further technical assumption that

there is no totally positive units in O×L,+ congruent to −1 modulo p.

Thanks to the excellent exposition in Section 2 of [LLZ16], we will partly follow

the notations there. First recall the definitions

D = ResLQGm, G = ResLQGL2, G∗ = G×D Gm.

48

Note there are natural embeddings GL2 → G∗ → G. Let H be the Poincare upper

half plane. We will consider the following types of Shimura varieties.

Definition 4.1.1. Let K ⊆ G(AQ,f ), K∗ ⊆ G∗(AQ,f ) and K ′ ⊂ GL2(AQ,f ) be open

compact subgroups. We write

S(K)(C) := G(Q)+\H2 ×G(AQ,f )/K,

S∗(K∗)(C) := G∗(Q)+\H2 ×G∗(AQ,f )/K∗,

Y (K ′)(C) := GL2(Q)+\H×GL2(AQ,f )/K′.

IfK (respectively, K∗, K ′) is sufficiently small, then S(K)(C) (respectively, S∗(K∗)(C),

Y (K ′)(C)) has a natural structure of smooth variety defined over Q with the set of

complex points as given above. We will use S(K), S∗(K∗) and Y (K ′) to denote the

resulting Shimura varieties.

Remark. In the literature, both S(K) and S∗(K∗) are referred to as Hilbert modular

surfaces. To distinguish between the two, we will always specify if it is for the algebraic

group G or G∗, unless it is otherwise clear. In this paper we will mostly work with G

as it admits a richer collection of Hecke actions and the corresponding automorphic

representations satisfy the multiplicity one property. We will only use G∗ when we

later relate the computation of syntomic regulators to overconvergent modular forms,

since the Shimura varieties S∗(U∗) are of PEL type, and it is more natural to consider

overconvergent forms there. Note that we will always use S to denote a surface, and

Y to denote a curve.

For every g ∈ G(AQ,f ), we have a map

Tg : S(K) −→ S(gKg−1), [x, h] 7→ [x, hg−1],

49

which is a morphism of varieties defined over Q. The same applies to S∗(K∗) and

Y (K ′), and when there is no risk of confusion, we will use the same notation Tg to

denote the map.

Note that Tg is only defined for S∗(K∗) when g ∈ G∗(AQ,f ), which is a rather

small group compared to G(AQ,f ) = GL2(AL,f ). Indeed, the former only consists of

the elements in the latter with determinant belonging to AQ,f . To partially remedy

this, we enlarge the group of transformations on S∗(K∗) following a construction of

Shimura ([Shi78], page 643; see also [LLZ16], Definition 2.2.4).

Definition 4.1.2. Let G denote the subgroup G(Q)+G∗(AQ,f ) ⊆ G(AQ,f ).

Then for each K∗ there is a bijection Y ∗(K∗) = G(Q)+\H2×G/K∗, and therefore

the maps Tg are defined for all g ∈ G.

4.2 Hecke correspondences and operators

In this section we introduce Hecke correspondences and operators for G. All the

discussions below apply to GL2 as well. As for G∗, the general definitions of Hecke

correspondences and double coset operators also work as long as g ∈ G. We will

revisit this topic for G∗ when we introduce overconvergent Hilbert modular forms

later. Let g ∈ G(AQ,f ) and K be an open compact subgroup of G(AQ,f ). The double

coset KgK defines the following correspondence

S(g−1Kg ∩K)Tg//

pr

S(K ∩ gKg−1)

pr′

S(K) S(K)

(4.1)

and the double coset operator [KgK] acts on the cohomology groups of S(K) by

pr′∗Tg,∗pr∗. A straightforward double coset calculation shows that this geometrically

50

defined operator coincides with the algebraically defined double coset operator used

in [Hid91]. Suppose K contains U(N) for some ideal N of OL. For a, b ∈ O×L,N and

z ∈ A×L,f , we have the operators

T (a, b) = [K

a 0

0 b

K], 〈z〉 = [K

z 0

0 z

K].

Furthermore, we let 〈z, a〉 ∈ GL(N) act as 〈z〉T (a−1, 1) in accordance with definitions

in [Hid91].

Remark. Note that since z · 12 is central, the action of 〈z〉 is precisely Tz,∗ = T∗z−1 .

Moreover, if the matrix

Da,b :=

a 0

0 b

normalizes the compact open subgroup K,

then T (a, b) acts as TDa,b,∗ = T∗D−1a,b

.

The Up-operator.

Let $p1 ∈ A×L,f be the element whose p1-component is p and every other component

is 1, and similarly define $p2 . Set $p = $p1$p1 .

Consider g =

$p 0

0 1

, so that g−1K(pα)g ∩K(pα) = K(pα) ∩K0(pα+1). There

are two natural projections

π1 : S(K(pα)∩K0(pα+1)) −→ S(K(pα)), π2 : S(K(pα)∩K0(pα+1)) −→ S(K(pα)),

where π1 is the natural projection and π2 is induced by Tg. In terms of the diagram

in (4.1), π1 is precisely pr and π2 is just pr′ Tg. Then the Up operator is given by

51

Up = (π2)∗ (π1)∗ and the adjoint Up operator is U∗p = (π1)∗ (π2)∗.

Analogously, we also have a Up operator for each prime above p. Let p be one

of p1 and p2, and gp =

$p 0

0 1

. Then g−1p K(pα)gp = K(pα) ∩ K0(pα+1), and

similar to above we have the natural projection π1,p and the map π2,p induced by Tgp ,

both maps from S(K(pα) ∩ K0(pα+1)) to S(K(pα)). The Up operator is defined as

Up = (π2,p)∗ (π1,p)∗ with its adjoint given by U∗p = (π1,p)∗ (π2,p)

∗. Note the above

discussion of Up and Up operators hold verbatim if we change K(pα) to any other level

subgroup defined in Definitions 4.3.1 and 4.3.2.

Remark. Hida has used two different conventions for the Hecke operator T (x) for

x ∈ AL,f . In [Hid91], [UgU ] is defined by decomposing UgU =∐

γ γU and setting

(f |[UgU ])(h) =∑

γ f(hγ), and T (x) is defined as [U

x 0

0 1

U ]. In [Hid88], [Hid89b]

and [Hid04] (see in particular Section 4.2.6 of [Hid04], end of p.172 and top of p.174),

however, the double coset operator [UgU ]′ (we added a prime to indicate the dif-

ference with the previously defined double coset operator, and similarly for T ′(x)

below) on the automorphic side is defined by decomposing UgU =∐

γ Uγ and set-

ting (f |[UgU ]′)(h) =∑

γ f(hγ−1), at least in the weight (2tL, tL) situation, and T ′(x)

is defined as [U

1 0

0 x

U ]′. The two notions differ by translation by the diagonal

matrix x · 12 so that T (x) = 〈x, 1〉T ′(x), hence the Hida theory is the same. In this

paper we have followed the convention in [Hid91].

4.3 Atkin-Lehner morphism

Our next goal is to define Atkin-Lehner morphisms at a prime above p. Let p be one

of p1 and p2, and p′ the other one.

52

The Tτpα morphism

Consider the matrix τpα ∈ GL2(AL,f ) defined by

(τpα)p =

0 −1

$αp 0

, (τpα)v = I2 for v 6= p

which normalizes K(pα). Then we have the morphism

Tτpα : S(K(pα)) −→ S(K(pα)), [x, h] 7→ [x, hτ−1pα ].

One can check how this operator commute with diamonds operators:

Tτpα ⟨z, a⟩

=⟨z · ap, a · a−2

p

⟩ Tτpα (4.2)

if (z, a) ∈ Zα0,L(K)× (OL/pα)×. Furthermore,

T2τpα

=⟨−$α

p , 1⟩. (4.3)

It will also be useful to look at how Tτpα interacts with the Hecke operators at

p. We keep the notation of the previous section. We have the following commutative

diagrams

S(K(pα) ∩K0(pα+1))

ν1,p

π1

))

S(K(pα) ∩K0(pα+1))

ν2,p

π2

))

S(K(pα) ∩K0(pα+1)) π1,p

// S(K(pα)), S(K(pα) ∩K0(pα+1)) π2,p

// S(K(pα)),

where π1,p and ν1,p are canonical projections, while π2,p and ν2,p are induced by Tgp

53

and Tgp′ respectively. A direct calculation shows we have the following relations

π1,p Tτpα+1 = Tτpα π2,p, π2,p Tτpα+1 = 〈$−1p , 1〉 Tτpα π1,p (4.4)

and

ν1,p′ Tτpα+1 = Tτpα ν2,p′ , ν2,p′ Tτpα+1 = 〈$−1p , 1〉 Tτpα ν1,p′ . (4.5)

It follows that

(Tτpα )∗ Up (Tτpα )∗ = U∗p 〈$p, 1〉. (4.6)

On the other hand, it is clear that Up′ commutes with (Tτpα )∗.

Special level subgroups

We will be interested in the following special types of open compact subgroups of

G(AQ,f ).

Definition 4.3.1. For any α ≥ 1 and any compact open K ⊆ G(AQ,f ) hyperspecial

at p we set

K(pα) :=

a b

c d

∈ K0(pα) : ap1 ≡ ap2 , dp1 ≡ dp2 (mod pα),

K,t(pα) :=

a b

c d

∈ K0(pα) : ap1dp1 ≡ ap2dp2 , dp1dp2 ≡ 1 (mod pα).

KX(pα) :=

a b

c d

∈ K0(pα) : ap1dp1 ≡ ap2dp2 , dp1ap2 ≡ 1 (mod pα),

54

KX,1(pα) :=

a b

c d

∈ K0(pα) : ap1 ≡ dp2 , dp1 ≡ ap2 ≡ 1 (mod pα),

and

K,1(pα) := K(pα) ∩ V1(pα).

Definition 4.3.2. For any α ≥ 1 and any compact open K ⊆ G(AQ,f ) hyperspecial

at p we set

Remark. Note that KX(pα) is precisely the subgroup of K0(pα) generated by KX,1(pα)

and matrices γ of the form

γv = 12 for v 6= p, γp =

d−1

d

with dp1 ≡ dp2 (mod pα).

These level subgroups are related by the following morphisms

Tτpα2: S(K,1(pα)) −→ S(KX,1(pα)), Tτpα2

: S(KX(pα)) −→ S(K,t(pα)).

The subgroups of the form K,t(pα) are related to arithmetic specializations of the

ordinary Hida family G .

The να isomorphism

For a ∈ Z×p , let σa ∈ Gal(Q(ζpα)/Q) be the element corresponding to a by class field

theory. More concretely, σa sends ζpα to ζa−1

pα . Also recall our assumption made at

the beginning of this chapter that there is no totally positive units in O×L,+ congruent

to -1 modulo p.

55

Proposition 4.3.3. There is an isomorphism of Q(ζpα)-varieties

να : S(K(pα))×Q Q(ζpα)

∼−→ S(KX(pα))×Q Q(ζpα)

such that να σa = 〈1, (a, a)〉σa να.

Proof. The component set of S(K(pα)), which is a 0-dimensional Shimura variety for

ResL/QGL1, is given by

π0(S(K(pα))) = L×+\A×L,f/ det(K(pα)).

Similarly we have

π0(S(KX,1(pα))) = L×+\A×L,f/ det(KX,1(pα))

and

π0(S(K,1(pα))) = L×+\A×L,f/ det(K,1(pα)).

Note that these are all defined over Q, and by the reciprocity law, the Galois group

Gal(Q(ζpα)/Q) acts on the points via the Artin map and the diagonal embedding of

A×Q,f inside A×L,f . In particular, since

det(KX,1(pα)) = det(K,1(pα)) = (Zp + pαOL,p)Op,×L ,

the image of Gal(Q(ζpα)/Q) is trivial and hence all points of π0(S(KX,1(pα))) and

π0(S(K,1(pα))) are defined over Q. The natural projection S(K(pα))→ S(K,1(pα))

induces an isomorphism of Q-varieties

S(K(pα)) ' S(K,1(pα))×π0(S(K,1(pα))) π0(S(K(pα))). (4.7)

56

The projection π0(S(K(pα))) → π0(S(K,1(pα))) corresponds to the natural projec-

tion

L×+\A×L,f/(1 + pαOL,p)Op,×L −→ L×+\A×L,f/(Zp + pαOL,p)Op,×L

with fiber of the identity component identified with

(Zp + pαOL,p)Op,×L/L×+ ∩ (1 + pαOL,p)Op,×L ∼= (Z/pαZ)×,

since there is no totally positive units congruent to -1 modulo p. The Galois action of

Gal(Q(ζpα)/Q) is precisely given by the reciprocal of the Artin map, permuting the

fiber. Since the complex points of the two connected components happen to carry

the structure of abelian groups and the projection is a group homomorphism, we see

this is true for all fibers of the projection. This identifies the 0-dimensional variety

π0(S(K(pα))) with π0(S(K,1(pα))) ×Q Q(ζpα), and hence via (4.7), also identifies

S(K(pα)) with S(K,1(pα))×Q Q(ζpα).

Consider the action of

(a−1, a−1) 0

0 1

∈ GL2(OL,p) on S(K(pα)), where (a−1, a−1)

is understood as an element of O×L,p. Through the isomorphism (4.7), it acts as

〈1, (a, a)〉×σa on S(KX,1(pα))×π0(S(KX,1(pα)))π0(S(K11(pα))) ' S(KX,1(pα))×QQ(ζpα).

On the other hand, for S(K,1(pα)) we have completely analogous results. Identifying

S(K(pα)) with S(K,1(pα)) ×Q Q(ζpα), the action of

(a−1, a−1) 0

0 1

is just 1 × σa

by definition.

In summary, we have a canonical isomorphism

να : S(K,1(pα))×Q Q(ζpα)∼−→ S(KX,1(pα))×Q Q(ζpα)

factoring through S(K(pα)), such that να σa = 〈1, (a, a)〉σa να. If we further

57

quotient out the action of

(d−1, d−1)

(d, d)

∈ GL2(OL,p) ⊂ GL2(AL,f ) on

S(K(pα)) (which acts trivially on π0(S(K(pα))) as the determinant is 1), we ob-

tain S(K(pα))×QQ(ζpα) on the source of να and S(KX(pα))×QQ(ζpα) on the target.

Slightly abusing the notation, we still denote by να the resulting isomorphism


∼−→ S(K(pα))∼−→ S(KX(pα))×Q Q(ζpα), (4.8)

where we define K(pα) :=

a b

c d

∈ K0(pα) : ap1 ≡ ap1 ≡ d−1p1≡ d−1

p2(mod pα).

As the diamond operators are defined over Q, this new quotient isomorphism να

satisfies the same twisted Galois equivariance property να σa = 〈1, (a, a)〉σa να.

For future reference we record the following lemma.

Lemma 4.3.4. The identity

Up (να)∗ = (να)∗ Up

holds as maps on cohomology groups H•(S(KX(pα))Q(ζpα )) −→ H•(S(K(pα))Q(ζpα )),

where H• can be either etale cohomology with constant coefficient sheaf, or algebraic

de Rham cohomology.

Proof. Recall the Up operator on S(K?(pα)), ? ∈ ,, X is given by π2,p,∗ π∗1,p,

where

π1,p, π2,p : S(K?(pα) ∩K0(pα+1)) −→ S(K?(p

α))

are given by T12 and Tgp , gp =

$p 0

0 1

, respectively.

58

Repeating the argument in Proposition 4.3.3, we see there is also an isomorphism

να : S(K(pα) ∩K0(pα+1))×Q Q(ζpα)

∼−→ S(K(pα) ∩K0(pα+1))

∼−→ S(KX(pα)) ∩K0(pα+1)×Q Q(ζpα),(4.9)

and both π0(S(K(pα)) and π0(S(K(pα) ∩K0(pα+1)) are identified with

L×+\A×L,f/x ∈ O×L |xp ∈ 1 + pαOL,p.

Since the determinant of both I2 and gp lie in L×+ · x ∈ O×L |xp ∈ 1 + pαOL,p,

both

π1,p, π2,p : S(K(pα) ∩K0(pα+1)) −→ S(K(pα))

induce the identity map on the component group. Therefore via the isomorphisms

in (4.9), they correspond to π1,p × 1 and π2,p × 1, respectively, on both S(K(pα) ∩

K0(pα+1))×Q Q(ζpα) and S(KX(pα)∩K0(pα+1))×Q Q(ζpα). In other words, both π1,p

and π2,p commute with να. This implies

Up (να)∗ =(π2,p)∗ (π1,p)∗ (να)∗ = (π2,p)∗ (να)∗ (π1,p)

∗

=(π2,p)∗ (ν−1α )∗ (π1,p)

∗ = (ν−1α )∗ (π2,p)∗ (π1,p)

∗

=(να)∗ Up

as desired.

Quotients of upper half plane

For any compact open subgroupK ≤ G(Z) the determinant map det : G→ ResLQ(Gm)

induces a bijection between the set of geometric connected components of S(K) and

59

Cl+L(K), the strict class group of K, Cl+L(K) = L×+\A×L,f/ det(K).

As in Section 2.3, we fix a set of representatives t1, ..., th in A×L,f of Cl+L(K) with

t1 = 1 and let gi =

ti 0

0 1

. Write

Γ(gi, K) = giKg−1i ∩G(Q)+.

By strong approximation there is a decomposition

S(K)(C) = G(Q)+\H2 ×G(AQ,f )/K =h∐i=1

Γ(gi, K)\H2, (4.10)

where z mod Γ(gi, K) ∈ Γ(gi, K)\H2 is identified with [z, gi] ∈ S(K)(C).

Remark. This is precisely the geometric counterpart of the decomposition of adelic

Hilbert modular forms into tuples of classical modular forms: every classical form

component is precisely the corresponding section on each connected component in

(4.10).

We are interested in comparing the complex uniformizations for the subgroups

K(pα) and KX(pα). The first observation is that the determinant of these groups are

equal to (Z×p + pαOL,p)Op,×L , thus we can choose the same sets of matrices gi| i =

1, ..., h for both of them.

Lemma 4.3.5. If O×L does not contain a totally positive unit congruent to −1 modulo

p, then for every c and every gi we have

Γ(gi, K(pα)) = Γ(gi, KX(pα)).

Proof. We only prove Γ(gi, K(pα)) ⊆ Γ(gi, KX(pα)) because the other inclusion is

analogous. Let γ be a matrix in Γ(gi, K(pα)), then its entries aγ, bγ, cγ, dγ ∈ L

60

satisfy (aγ)p, (dγ)p ∈ Z×p + pαOL,p, (cγ)p ∈ pαOL,p and its determinant det(γ) ∈ O×L,+

is a totally positive unit. In order to prove the claimed inclusion we need to show

that

(aγdγ)p1 ≡ (aγdγ)p2 (mod pα) and (dγ)p1 ≡ (aγ)p2 (mod pα).

Since NL/Q(

det(γ))

= 1, one sees that either aγdγ − 1 ∈ pαOL or aγdγ + 1 ∈ pαOL.

The second possibility implies that det(γ) ≡ −1 (mod p), which is ruled out by the

hypothesis of the Lemma. Hence we must have aγdγ − 1 ∈ pαOL which implies the

claim.

Corollary 4.3.6. The morphism

να : S(K(pα))(C)

∼−→ S(KX(pα))(C)

is the identity with respect to the complex uniformizations (4.10). Hence, να com-

mutes with the natural projections ν1,p and π1,p for p = p1, p2, since they all are

induced by the identity map on H2. However, να does not in general commute with

ν2,p nor π2,p.

Atkin-Lehner morphism

We are finally ready to define the Atkin-Lehner morphism at a prime aove p.

Definition 4.3.7. The Atkin-Lehner morphism at p2 is defined as

wpα2= Tτpα2


∼−→ S(K,t(pα))×Q Q(ζpα).

61

Since Tτpα2is defined over Q and να σa = 〈1, (a, a)〉σa να, we see that

wpα2 σa = Tτpα2

〈1, (a, a)〉σa να = 〈(1, a−1), (a, a−1)〉σa wpα2. (4.11)

Combining equation (4.6) with Lemma 4.3.4, we obtain the following

Corollary 4.3.8. The identity

Up (wpα2)∗ = (wpα2

)∗ Up1 U∗p2 〈$p2 , 1〉

holds as maps on cohomology groups H•(S(K,t(pα))Q(ζpα )) −→ H•(S(K(p

α))Q(ζpα )),

where H• can be either etale cohomology with constant coefficient sheaf, or algebraic

de Rham cohomology.

Lemma 4.3.9. There is a GQ-equivariant isomorphism

wpα2 ,∗ : Hiet(S(K(p

α))Q,Zp) ' Hiet(S(K,t(p

α))Q,Zp)(δα),

where δα : GQ → O[GαL(K)]× is the character obtained by composing the projection

GQ → Gal(Q(ζpα)/Q) with the homomorphism mapping σa to the group-like element

〈(1, a−1), (a, a−1)〉, where (a−1, 1) and (1, a−1) is understood as elements of O×L,p.

Proof. This follows immediately from equation (4.11).

4.4 Hirzebruch-Zagier cycles

We keep the notations and assumptions made at the beginning of the previous chapter.

Let K ′ := GL2(AQ,f ) ∩ K. Note that K ′0(pα) = GL2(AQ,f ) ∩ K(pα) and there is a

62

cartesian diagram of Shimura varieties over Q

Y (K ′0(pα+1))

π1

ζ(p1)// S(K(p

α) ∩ V0(pα+11 ))

π1,p1

Y (K ′0(pα)) ζ

// S(K(pα))

(4.12)

as the horizontal arrows are closed embeddings and the vertical ones are finite of

degree p.

Definition 4.4.1. For α ≥ 1 we define cycle classes in the respective Chow groups

∆[α := ζ∗ [Y (K ′0(pα))] ∈ CH1

(S(K(p

α))(Q)

and

∆[α(p1) := ζ(p1)∗

[Y (K ′0(pα+1))

]∈ CH1

(S(K(p

α) ∩ V0(pα+11 ))

)(Q).


(π1,p1)∗∆[α = ∆[

α(p1) (4.13)

in CH1(S(K(p

α) ∩ V0(pα+11 ))

)(Q).

Proof. Since the diagram (4.12) is cartesian, the push-pull formula

(π1,p1)∗ ζ∗ = ζ(p1)∗ (π1)∗

holds. The desired conclusion follows immediately.

63

Twisting by Atkin-Lehner.

Definition 4.4.3. Let ∆]α be the following class of the codimension 1 cycle

∆]

α = (wpα2)∗∆

[α ∈ CH1

(S(K,t(p

α)))(Q(ζpα))

defined over Q(ζpα).

Proposition 4.4.4. We have

($2)∗∆]

α+1 = 〈$−1p2, 1〉∗ Up1∆]

α.

Proof. Combining equation (4.5) with Corollary 4.3.6, we have the following commu-

tative diagram

Y (K ′0(pα+1)) ζ

// S(K(pα+1))

ν1,p1µ

wpα+12 // S(K,t(p

α+1))

ν2,p1µ

Y (K ′0(pα+1)) ζ(p1)

// S(K(pα) ∩ U0(pα+1

1 ))〈$−1

p2,1〉wpα2 // S(K,t(p

α) ∩ U0(pα+11 ))

π2,p1

S(K,t(pα)).

(4.14)

By definition, ($2)∗∆]

α+1 is precisely the pushforward of the cycle class [Y (K ′0(pα+1))]

along the top arrows and then the the right arrows, and therefore the commutativity

64

of the diagram shows that

($2)∗∆]

α+1 = (π2,p1)∗ 〈$−1p2, 1〉∗ (wpα2

)∗ (ζ(p1))∗[Y (K ′0(pα+1))

]= 〈$−1

p2, 1〉∗ (π2,p1)∗ (wpα2

)∗∆[α(p1)

= 〈$−1p2, 1〉∗ (π2,p1)∗ (wpα2

)∗ (π1,p1)∗∆[α

= 〈$−1p2, 1〉∗ (π2,p1)∗(π1,p1)∗ (wpα2

)∗∆[α

= 〈$−1p2, 1〉∗ Up1∆]

α,

where the third equality is due to Lemma 4.4.2 and the second to last one from the

fact that wpα2is an isomorphism commuting with π1,p1 as noted in Corollary 4.3.6.

This proves the claim.

Next, we put the cycles in the products of modular curves and Hilbert modular

surfaces.

Definition 4.4.5. We denote by

Zα(K) = S(K,t(pα))× Y (K ′0(p)). (4.15)

The Hirzebruch-Zagier cycles of level α ≥ 1 is defined by

∆α = (〈$αp2, 1〉 wpα2

ια, π1,α)∗[Y (K ′0(pα))] ∈ CH2(Zα(K)

)(Q(ζpα)), (4.16)

where π1,α : Y (K ′0(pα)) −→ Y (K ′0(p)) is the natural projection.

As in Proposition 4.4.4 we find the following relation for the new cycles.

Proposition 4.4.6. The following identity holds in CH2(Zα(K))(Q(ζpα)):

($2, id)∗∆α+1 = (Up1 , id)∆α.

65

Galois twisting

Later on we will apply the p-adic Abel-Jacobi map to the cycles and put them in a

family, so it is desirable to make them defined over Q.

Following equation (4.11), the Galois group GQ acts on ∆α through the finite

quotient Gal(Q(ζpα)/Q) by

σ∗a∆α = (〈(1, a), (a−1, a)〉, id)∆α.

Definition 4.4.7. Let S†(K,t(pα)) to be the twist of the Q-variety S(K,t(p

α)) by

the cocycle

σa ∈ Gal(Q(ζpα)/Q) 7→ (σa 7→ 〈(1, a−1), (a, a−1)〉)

and define Z†α(K) = S†(K,t(pα))× Y (K ′0(p)).

Then ∆α can now be viewed as a codimension 2 cycle in Z†α(K) defined over Q.

Furthermore,

Hiet(S

†(K,t(pα))Q,Zp) ' Hi

et(S(K,t(pα))Q,Zp)(δα), (4.17)

where δα : GQ → O[GαL(K)]× is the character sending σa to the group-like element

〈(1, a−1), (a, a−1)〉 as before.

Null-homologous cycles

Finally, we make the cycles null-homologous by applying suitable modular correspon-

dence. We define a correspondence εf on Y (K ′0(p)) following [DR17] (after equa-

tion (47)). Since p is assumed to be non-Eisenstein for f, i.e. f is not congruent

to an Eisenstein series modulo p, there exists an auxiliary prime ` - Np for which

66

`+ 1− ap(`, f) lies in Z×p . Then the correspondence

εf = (`+ 1− T (`))/(`+ 1− ap(`, f))

has coefficients in Zp, annihilates H0(Y (K ′0(p))) and H2(Y (K ′0(p))) and acts as the

identity on the f-isotypic subspace H1(Y (K ′0(p)))[f].

Definition 4.4.8. The modified Hirzebruch-Zagier cycle is given by

∆α := (id, εf)∗∆α ∈ CH2(Z†α(K))(Q)⊗ Zp.

Proposition 4.4.9. The cycle class ∆α is null-homologous.

Proof. The etale cycle class image clet(∆α) belongs to H4

et(Z†α(K)Q,Zp(2)). As the

integral cohomology of smooth projective curves is torsion free and Hilbert modular

surfaces are simply connected, the integral etale cohomology of Z†α(K) has a Kunneth

decomposition whose every term is annihilated by the correspondence (id, εf).

4.5 Big cohomology classes

For any number field D, the p-adic etale Abel-Jacobi map

AJetp : CH2(Z†α(K))0(D) −→H1

(D,H3

et

(Z†α(K)Q,Zp(2)

))=H1

(D,H3

et

(Zα(K)Q,Zp(2 + δα)

))sends null-homologous cycles to Galois cohomology classes. The modified Hirzebruch-

Zagier cycles ∆α are null-homologous and give rise to cohomology classes in the

67

appropriate Kunneth component

κα := AJetp (∆α) ∈ H1

(Q,H2

et(S(K,t(pα))Q,Zp(2 + ηα))⊗ H1

et(Y (K ′0(p))Q,Zp)).

Lemma 4.5.1. For α ≥ 1 we have

($2, id)∗κα+1 = (Up1 , id)κα.

Proof. It follows immediately from Proposition 4.4.6 combined with the commutativ-

ity of cycle class map and correspondences.

Let

Vn.o.α (K) := en.o.H

2et(S(K,t(p

α))Q,Zp(2 + ηα))⊗ H1et(Y (K ′0(p))Q,Zp)

be the tensor product of the nearly ordinary part of the etale cohomology groups of

S(KX(pα)) and the etale cohomology of Y (K ′0(p)). By a slight abuse of notation we

also denote by

κα ∈ H1(Q,Vn.o.α (K))

the nearly ordinary projection of κα. Since Up1 acts invertibly on the nearly ordinary

part, we may define

κn.o.α :=

(U−αp1

, id)κα ∈ H1(Q,Vn.o.

α (K)). (4.18)

It follows immediately from Lemma 4.5.1 and the commutativity of Up1 with $2,∗

that

($2, id)∗κn.o.α+1 = κn.o.

α , (4.19)

68

which justifies our following definition.

Definition 4.5.2. Let

H2n.o.(K,t(p

∞);Zp(2)) := lim←−α

en.o.H2et(S(K,t(p

α))Q,Zp(2)),

where the inverse limit is taken with respect to ($2)∗, and

Vn.o.∞ (K) := lim←−

α

Vn.o.α (K) = H2

n.o.(K,t(p∞);Zp(2))(δ)⊗ H1

et(Y (K ′(p))Q,Zp)

where the limit is with respect to ($2, id)∗, and δ = lim←−α δα.

Define

κn.o.∞ = lim

←,ακn.o.α ∈ H1(Q,Vn.o.

∞ (K)).

Remark. Vn.o.α (K) has a natural action of GL(K) as diamond operators, and hence is

a module over ZpJGL(K)K. The character δ : GQ → GL(K) is the composition of the

canonical projection GQ → Gal(Q(ζp∞)/Q) with the map Gal(Q(ζp∞)/Q) → GL(K)

sending σa to 〈(1, a−1), (a, a−1)〉.

4.6 A refinement of specialization formula

Our last goal of the section is to further analyze the weight (2, 1) specializations of the

Λ-adic elliptic modular form eordζ∗(d•µG [P]

)†defined before Proposition 3.1.3, so that

we can obtain a refinement of the specialization formula (3.4). To do this we need to

first consider the geometric morphisms corresponding to the various operations used

in defining the Λ-adic form.

69

Let

K ′0,det(pα) :=KX(pα) ∩GL2(AQ,f )

=K ′ ∩

a b

c d

∈ GL2(Z) : apdp ≡ 1, cp ≡ 0 (mod pα).

Then we have the natural diagonal embedding Y (K ′0,det(pα)) → S(Kt

(pα)) of Shimura

varieties induced by the natural diagonal inclusion of algebraic groups GL2 → G =

ResLQGL2.

Now we identify the complex points of the modular curve Y (K ′0,det(pα)) as a

disjoint union of quotients of H. Note that det(K ′0,det(pα)) = a ∈ Z× : ap ∈ 1+pαZp.

Fixing the set of representatives t ∈ A×Q,f : tp = 1, 2, ..., pα, p - tp, tq = 1∀q 6= p for

the class group Q×+\A×Q,f/ det(K ′0,det(pα)) and noting

t−1 0

0 1

K ′0,det(pα)

t 0

0 1

∩SL2(Z) = Γ1(M) ∩ Γ0(pα) independent of t, we identify

Y (K ′0,det(pα))(C) =

∐t

Γ1(M) ∩ Γ0(pα)\H.

On the other hand, clearly det(K ′0(pα)) = Z×, so there is only one connected com-

ponent for Y (K ′0(pα)). Since K ′0(pα) ∩ SL2(Z) = Γ1(M) ∩ Γ0(pα) we may canonically

identify Y (K ′0(pα))(C) with Γ1(M) ∩ Γ0(pα)\H. Therefore we may define a natural

embedding idet : Y (K ′0(pα))(C) → Y (K ′0,det(pα))(C) which is the identity map onto

the identity component (the component indexed by t = 1) of the latter.

By construction we have the following commutative diagram

Y (K ′0(pα))(C) _

ια

idet // Y (K ′0,det(pα))(C) _

ια

S(K(pα))(C) '

να // S(KX(pα))(C)

(4.20)

70

Proposition 4.6.1. Let P ∈ A(IG ) be an arithmetic point of weight 2 and character

χ, a character on 1 + pZp of conductor pα. Then


)†(P) = ap($p, gP)α ·G(θ−1

L,pχ−1p )−1 · eordζ

∗d−1µ w∗pα2 (g

[P]P ).

Proof. In the proof we will work with the classical Fourier coefficients a(ξ,−) on

each components rather than the adelic Fourier coefficients, both for classical Hilbert

modular forms and p-adic Hilbert modular forms, since the former type of coefficients

is more convenient here. Note that these two points of view are completely equivalent

as one determine the other.

By Proposition 3.1.3, the specialization of eordζ∗(d•µG [P]

)†at P is eordζ

∗(d−1µ (g

[P]P ?

θ−1L,pχ

−1p ))⊗ θ−1

Q χ. Lemma 2.9.4 implies that

g[P]P ? θ−1

L,pχ−1p = ap($p, gP)α ·G(θ−1

L,pχ−1p )−1 · (g[P]

P |τ−1pα ).

To simplify notation, write cg for the explicit factor appearing in the right hand

side of the equation above. Then we have


)†(P) = cg ·

(eordζ

∗d−1µ (g

[P]P |τ

−1pα ))⊗ θ−1

Q χ.

Realizing the weight (2tL, tL) Hilbert modular form g[P]P as a differential on the

complex surface S(K,t(pα))(C), we may pull it back via Tτpα and obtain a weight

(2tL, tL) Hilbert modular form on S(KX(pα))(C) which, by definition, is precisely

g[P]P |τ

−1pα . Further pulling back along να we obtain a Hilbert modular form ν∗α(g

[P]P |τ

−1pα )

on S(K(pα))(C), which is by definition w∗pα2 g

[P]P . Since να is the identity map on H2

on each connected component, ν∗α preserves the classical Fourier expansions. Thus

71

for all ξ ∈ L+ and every i = 1, ..., h we have

a(ξ, ((wpα2)∗g

[P]P )i) = a(ξ, (g

[P]P |τ

−1pα )i).

Now we regard (wpα2)∗g

[P]P and g

[P]P |τ

−1pα as p-adic modular forms. In particular,

a(ξ, (g[P]P |τpα)i) is now understood as elements in Qp. It is clear from the definition of

the differential operators that

a(ξ, (d−1

µ (wpα2)∗g

[P]P )i

)= a(ξ, (d−1

µ (g[P]P |τ

−1pα ))i

). (4.21)

Applying diagonal restriction to d−1µ (g

[P]P |τ

−1pα ) we obtain a p-adic elliptic modular

form on Y (K ′0,det(pα)) whose Fourier expansion on the identity component is

∞∑n=1

qn∑

ξ∈d−1L ,NL/Q(ξ)=n

a(ξ, (d−1

µ (g[P]P |τ

−1pα ))1

),

which, by (4.21), is the same as the p-adic modular form ζ∗d−1µ (wpα2

)∗g[P]P on Y (K ′0(pα)).

By Lemma 2.9.2, twisting by θ−1Q χ does not change the classical Fourier expansion

on the identity component. Thus(ζ∗d−1

µ (g[P]P |τpα)

)⊗θ−1

Q χ and ζ∗d−1µ (wpα2

)∗g[P]P , which

are now both modular forms of weight (2, 1) and character (ψ−1 , I) (in particular they

both live on Y (K ′0(pα)) with only one connected component), have the same Fourier

expansions, and thus are the same modular forms. This completes the proof.

Corollary 4.6.2. Let P ∈ A(IG ) be as in Proposition 4.6.1 above. Then

L autp (G , f)(P) = ap($p, gP)α ·G(θ−1

L,pχ−1p )−1 ·

⟨eordζ

∗d−1µ w∗pα2 (g

[P]P ), f

(p)

⟩⟨f(p) , f

(p)

⟩ .

Proof. This follows immediately after combining (3.5) and Proposition 4.6.1.

72

Chapter 5

Galois representations

5.1 Galois representations at finite level

Let f be a p-nearly ordinary eigenform of weight (k, w) and level Npα over a totally

real field F . Then one has a 2-dimensional Galois representation associated to f.

Theorem 5.1.1. (Blasius-Rogawski-Taylor) Let f be as above, and E be a finite

extension of Qp containing its Hecke eigenvalues. Then there is an irreducible 2-

dimensional Galois representation %f over E

%f : GF → GL2(E),

such that for all primes q not dividing Np, %f is unramified at q, and

det(1−X%f(Frobq)) = 1− λ(T (q))X + NF/Q(q)m+1λ(〈q〉)X2.

Definition 5.1.2. The Asai Galois representation associated to g is

As(Vg) = ⊗-IndQF (Vg) : GQ → GL4(E).

73

Lemma 5.1.3. Suppose g ∈ S2tL,tL(K; Q) is a primitive eigenform. If g does not have

complex multiplication nor does it arise from base change, then As(Vg) is irreducible.

Proof. For a choice of element θ ∈ GQ \GL,

As(Vg)|GL∼= Vg ⊗ (Vg)

θ,

where (Vg)θ is the representation given by %θg(γ) = %g(θγθ−1) for any γ ∈ GL. Suppose

by contradiction that As(Vg) is reducible, then it must have a GQ-stable subspace

of dimension one or two. On the one hand, when the Galois stable subspace is 1-

dimensional, let α be the character of GL describing the action. It follows that the

representation

As(Vg)(α−1)|GL ' Hom

(V∨g , (Vg)

θ)(α−1)

has a line with trivial GL-action. From the irreducibility of Vg we deduce that (Vg)θ

is isomorphic to a twist of Vg as GL-representations applying Schur’s lemma. From

the irreducibility of Vg we deduce that (Vg)θ is isomorphic to a twist of Vg as GL-

representations applying Schur’s lemma. Hence by Theorem 2 of [LR98], g arises from

base-change, contradicting our assumption. On the other hand, when As(Vg) has

an irreducible sub-representation W ⊂ As(Vg) of dimension two, the representation∧2As(Vg) has∧2W as a 1-dimensional GQ-stable subspace. We deduce that both

Sym2 Vg and Sym2(Vg)θ have a 1-dimensional GL-stable subspace because

∧2As(Vg)|GL '

(Sym2 Vg ⊗

∧2(Vg)

θ)⊕(

Sym2(Vg)θ ⊗

∧2Vg

).

Moreover, it follows that the GL-representation Hom(V∨g ,Vg

)has two 1-dimensional

74

GL-stable subspaces, one of which is∧2Vg = det Vg as

Hom(V∨g ,Vg

)' Vg ⊗ Vg ' Sym2 Vg ⊕

∧2Vg.

Let β be the character of the stable subspace of Sym2 Vg, then HomGL

(V∨g ,Vg(β

−1))

is 1-dimensional since Vg is irreducible. Therefore β 6= det Vg and Vg is isomorphic

to a twist of itself by a nontrivial character. This means g has complex multiplication

([LR98], Theorem 2), which contradicts our assumptions.

Proposition 5.1.4. (Brylinski-Labesse, Nekovar, c.f. also Theorem 4.4.3 of [LLZ16])

The g-isotypic part H2et(S(K)Q, E(2))[g] is 4-dimensional and isomorphic to As(Vg)

as a GQ-representation.

Hodge-Tate numerology.

If we normalize Hodge-Tate weights by stating that the character NF has negative

weights, then for every OF - prime ideal p | p, the restriction(Vf)|Dp

has a one-

dimensional subrepresentation with Hodge-Tate weights −kτ2− m

2τ∈IF,p and a one-

dimensional quotient with Hodge-Tate weights kτ−22− m

2τ∈IF,p ([Hid91], Introduc-

tion). Therefore, if we twist by the −m/2-th power of the character ηF , the restriction

at Dp of the representation

V†f = Vf

(−m

2

)has Hodge-Tate weights −kτ

2, kτ−2

2τ∈IF,p .

Let g be a nearly ordinary Hilbert eigenform of weight (`, x) over a real quadratic

field L. The Galois representation V†g is related to the cohomology of Hilbert modular

varieties via tensor induction, which is why we are interested in the Asai representa-

tion.

When pOL = p1p2 splits in the real quadratic field L, As(V†g)|Dp = (V†g)|Dp1⊗

75

(V†g)|Dp2, so the restriction to the decomposition group at p has a 3-step filtration

As(V†g) ⊃ Fil1As(V†g) ⊃ Fil2As(V†g) ⊃ 0

in which the graded pieces have dimension 1, 2 and 1. We can compute the Hodge-

Tate weights as follows.

Graded piece Hodge-Tate weights

Gr0As(V†g)|`|2− 2

Gr1As(V†g) ( `1−`22− 1, `2−`1

2− 1)

Gr2As(V†g) − |`|2

Let f be an elliptic nearly ordinary eigenform of weight (k, w) (+ other conditions).

We are interested in the GQ-representation V †g,f = As(V†g)(−1)⊗V†f whose restriction

at the decomposition group at p has a 4-step filtration

V †g,f ⊃ Fil1V †g,f ⊃ Fil2V †g,f ⊃ Fil3V †g,f ⊃ 0

with graded pieces of dimension 1, 3, 3 and 1 given by

Gr0V †g,f = Gr0(As(V†g))⊗Gr0V†f (−1)

Gr1V †g,f = [Gr0(As(V†g))⊗Gr1V†f ](−1)⊕ [Gr1(As(V†g))⊗Gr0V†f ](−1)

Gr2V †g,f = [Gr1(As(V†g))⊗Gr1V†f ](−1)⊕ [Gr2(As(V†g))⊗Gr0V†f ](−1)

Gr3V †g,f = Gr2(As(V†g))⊗Gr1V†f (−1).

The Hodge-Tate weights of the graded pieces are computed in the following table.

76

Graded piece Hodge-Tate weights

Gr0V †g,f|`|+k

2− 2

Gr1V †g,f ( |`|−k2− 1, `1−`2+k

2− 1, `2−`1+k

2− 1)

Gr2V †g,f ( `1−`2−k2

, `2−`1−k2

, k−|`|2

)

Gr3V †g,f 1− |`|+k2

Corollary 5.1.5. The Hodge-Tate weights of Fil2V †g,f are all strictly negative if and

only if the triple of weights (`, k) is balanced.

5.2 Big Galois representations.

The ordinary family G passing through g(p) has an associated big Galois representa-

tion %G : GL → GL2(QG ) acting on a two dimensional QG -vector space VG ([Hid89a],

Theorem 1), where QG is the fraction field of IG . The representation %G is unramified

outside Qp with determinant

det(%G )(z) = φχ([z, 1])εL(z) = χ(z)ηL(z)[ξ−tLz ], ∀ z ∈ A×L,f . (5.1)

The characteristic polynomial at any prime q - Qp is given by

det(1− %G (Frobq)X) = 1− G (T(q))X + φχ([$q, 1])NL/Q(q)X2,

and for every prime p above p, fixing a decomposition group Dp in GL, there is

an unramified character ΨG ,p : Dp → I×G , Frobp 7→ G (T($p)), such that ([Hid89a],

77

Proposition 2.3)

(%G )|Dp ∼

Ψ−1G ,p · det(%G )|Dp ∗

0 ΨG ,p

. (5.2)

Definition 5.2.1. We let

As(VG ) := ⊗-IndQL (VG )

denote the tensor induction of VG .

Definition 5.2.2. We define

ηQ : GQ −→ OJΓK×, z 7→[ηQ(z)

]=[ξ−1z

]for any z ∈ A×Q. Then (ηQ)|Dp is the character associated by local class field theory

to the homomorphism Q×p → Γ→ OJΓK×, x 7→[〈x〉−1

].

Note that since pOL = p1p2 splits in the real quadratic field L, the decomposition

group Dp ⊂ GQ at p is contained in GL. Hence we can consider the characters ΨG ,p

as characters of Dp and we can define ΨG ,p = ΨG ,p1ΨG ,p2 . Furthermore, the fixed

θ ∈ GQ \ GL interchanges p1 and p2, inducing an isomorphism Lp1 ' Lp2 . Thanks

to equation (5.2), we have a concrete description of the action of the decomposition

group on As(VG ).

First, by (5.1) we have

As(det(%G )) = η2Qη

2Q.

Furthermore, we have the following filtration of As(VG ) as a local Galois representa-

tion.

Proposition 5.2.3. The restriction of As(VG ) to Dp admits a three step Dp-stable

78

filtration

As(VG ) ⊃ Fil1As(VG ) ⊃ Fil2As(VG ) ⊃ 0

with graded pieces

Gr0As(VG ) = QG

(ΨG ,p

), Gr2As(VG ) = QG

(Ψ−1

G ,p · (η2Qη

2Q)|Dp

)and

Gr1As(VG ) = QG

(Ψ−1

G ,p1ΨG ,p2 · (χ,p1ηQηQ)|Dp

)⊕QG

(ΨG ,p1Ψ

−1G ,p2· (χ,p2ηQηQ)|Dp

)where, for any prime p | p, we let χ,p : Dp → O× be the unramified character

determined by χ,p(Frobp) = χ(Frobp). In particular, χ,p1 · χ,p2 = 1.

Proof. Let V+G denote the QG -rank 1 submodule of VG coming from the upper left

corner in equation (5.2), and V−G = VG /V+G the rank 1 quotient. If we define

Fil2As(VG ) := V+G ⊗ (V+

G )θ and Fil1As(VG ) := V+G ⊗ (VG )θ + VG ⊗ (V+

G )θ,

then Fil2As(VG ) has rank 1 over QG while Fil1As(VG ) has rank 3. By the description

in (5.2), Dp acts on Fil2As(VG ) through the character Ψ−1G ,p ·

(η2Qη

2Q)|Dp

, while it acts

on the zero-th graded piece Gr0As(VG ) = V−G ⊗ (V−G )θ through ΨG ,p. Finally, the

first graded piece is

Gr1As(VG ) = QG

(Ψ−1

G ,p1Ψθ

G ,p1· det(%G )|Dp

)⊕QG

(ΨG ,p1(Ψθ

G ,p1)−1 · det(%G )θ|Dp

).

It is clear that ΨθG ,p1

= ΨG ,p2 and that

det(%G )|Dp =(χ,p1ηQηQ

)|Dp, det(%G )θ|Dp =

(χ,p2ηQηQ

)|Dp.

79

Definition 5.2.4. We define the following GQ-representation

As(VG )† := As(VG )(θQη−1Q ).

Let Vf denote the representation attached to the ordinary elliptic cuspform f. Then

V†G ,f := As(VG )†(−1)⊗ Vf .

interpolates Kummer self-dual Galois representations.

The explicit realization of that Galois representation in the cohomology of a tower

of certain threefolds with increasing level at p plays an important role in the construc-

tion of big Hirzebruch-Zagier classes.

Lemma 5.2.5. The restriction to a decomposition group at p of the Galois repre-

sentation V†G ,f is endowed with a 4-steps filtration by GQp-stable submodules with

graded pieces given by

Gr0V†G ,f = QG

(ΨG ,pδp(f) · (η−1

Q η−1Q )|Dp

),

Gr1V†G ,f =QG

(ΨG ,pδp(f)

−1 · (θQη−1Q )|Dp

)⊕QG

(Ψ−1

G ,p1ΨG ,p2δp(f) · (χ,p1)|Dp

)⊕QG

(ΨG ,p1Ψ

−1G ,p2

δp(f) · (χ,p2)|Dp),

Gr2V†G ,f =QG

(Ψ−1

G ,p1ΨG ,p2δp(f)

−1(χ−1,p2

εQ)|Dp)⊕QG

(ΨG ,p1Ψ

−1G ,p2

δp(f)−1 · (χ−1

,p1εQ)|Dp

)⊕QG

(Ψ−1

G ,pδp(f) · (ηQηQ)|Dp),

Gr3V†G ,f = QG

(Ψ−1

G ,pδp(f)−1 · (θQη2

QηQ)|Dp)

80

Proof. As V†G ,f = As(VG )(η−1Q η

−1Q )⊗ Vf , its graded pieces are given by

Gr0V†G ,f = Gr0As(VG )(η−1Q η

−1Q )⊗Gr0Vf ,

Gr1V†G ,f = [Gr0As(VG )(η−1Q η

−1Q )⊗Gr1Vf ]⊕ [Gr1As(VG )(η−1

Q η−1Q )⊗Gr0Vf ],

Gr2V†G ,f = [Gr1As(VG )(η−1Q η

−1Q )⊗Gr1Vf ]⊕ [Gr2As(VG )(η−1

Q η−1Q )⊗Gr0Vf ],

Gr3V†G ,f = Gr2As(VG )(η−1Q η

−1Q )⊗Gr1Vf .

Hence, the statement follows from Proposition 5.2.3 and a direct computation.

Definition 5.2.6. We define the direct summand VfG of Gr2V†G ,f by

VfG := QG

(Ψ−1

G ,pδp(f) · (ηQηQ)|Dp).

5.3 Etale cohomology of towers of Hilbert modular

surfaces

We first show the ordinary Hida family introduced in Definition 3.1.2 is of K,t(p∞)

level.

Let h2tL,tL(K,t(pα);Zp) be the Hecke algebra acting on S2tL,tL(K,t(p

α);Zp), which

is a subspace of S2tL,tL(K(pα);Zp). Write

GαL,0(K) = O×LK0(pα)/O×LK(pα), GL,0(K) = lim←−

α

GαL,0(K)

and

GαL,0(K,t) = O×LK0(pα)/O×LK,t(p

α), GL,0(K,t) = lim←−α

GαL,0(K,t).

81

Recall the group GαL,0(K) acts as diamond operators on S2tL,tL(K(pα);Zp), and

S2tL,tL(K,t(pα);Zp) is the subspace fixed by the subgroup O×LK,t(pα)/O×LK(pα).

Therefore

h2tL,tL(K,t(pα);Zp) = h2tL,tL(K(pα);Zp)⊗Zp[GαL,0(K)] Zp[Gα

L,0(K,t)].

Passing to inverse limit, we define as before

hL(K,t;Zp) := lim←−α

h2tL,tL(K,t(pα);Zp)

and

hn.o.L (K,t;Zp) := en.o.hL(K,t;Zp).

Then we have

hn.o.L (K,t;Zp) = hn.o.

L (K;Zp)⊗ZpJGL,0(K)K ZpJGL,0(K,t)K. (5.3)

Lemma 5.3.1. The Hida family G : hn.o.L (K;Zp)→ IG factors through hn.o.

L (K,t;Zp).

Proof. By the equation (5.3) above, it suffices to check that φχ factors through

ZpJGL(Q)K⊗ZpJGL,0(K)KZpJGL,0(K,t)K, which is straightforward from the definition of

K,t(pα).

Of course, the same also applies to our test vector G .

Definition 5.3.2. Let K = V1(MOL) be a compact open subgroup of G(AQ,f ). We

define the anemic Hecke algebra hn.o.L (K,t;Zp) to be the subalgebra of hn.o.

L (K,t;Zp)

generated by the Hecke operators T(y) with yM = 1. Moreover, the algebra homo-

morphism G : hn.o.L (K,t;Zp) → IG is the restriction of the Hida family G to the

anemic Hecke algebra.

82

Recall the modules H2n.o.(K,t(p

∞);Zp(2)) and Vn.o.∞ (K) and the big cohomology

class κn.o.∞ introduced in Definition 4.5.2. We would like to consider their projections to

suitably defined G -isotypic quotients. We note that H2n.o.(K,t(p

∞);Zp(2)) is naturally

a module over hn.o.L (K,t;Zp) and the module structure commutes with Galois action.


VG (M) := H2n.o.(K,t(p

∞), O(2))(δ)⊗G IG (5.4)

using the homomorphism G : hn.o.L (K;O) → IG , where δ is as defined in Definition

4.5.2.

Note that

VG (M) = H2n.o.(K,t(p

∞), O(2))⊗G IG (θQη−1Q )

because

G δ(σa) = φχ δ(σa) = θQ(a)η−1Q (a) ∈ I×G .

The anemic Hecke algebra hn.o.L (K;O) is finite and torsion-free over the complete

Noetherian local ring Λ, hence it decomposes as the product of the localizations at

its maximal ideals. If we denote by mG the maximal ideal containing ker(G ) then

VG (M) = H2n.o.(K,t(p

∞);O(2))mG⊗G IG (θQη

−1Q ).

Specializations

Let P ∈ A(IG ) be an arithmetic point of weight 2 and character χ, a p-power

order character of conductor pα. Then by Theorem 2.4 of [Hid89b], the composition

83

GP = P G factors through a Zp-linear map of finite level Hecke algebra

GP : hn.o.L (K,t;Zp) −→ O

and similarly for the map on anemic Hecke algebra

GP : hn.o.L (K,t;Zp) −→ O.

Definition 5.3.4. The specialization of VG (M) at P is

VGP(M) := VG (M)⊗IG ,P O.

Combining the above discussion with equation (5.3.3) we see that

VGP(M) ' H2

n.o.(K,t(pα), O(2))(δα)⊗GP

O = H2n.o.(K,t(p

α), O(2))⊗GPO(χθQ)

as representations of GQ.

Lemma 5.3.5. Let M be a finitely generated IG -module. If there is an infinite

collection Σ ⊂ A(IG ) of arithmetic points such that for all P ∈ Σ with image OP ⊂ Qp,

the specialization M⊗IG ,POP is OP-free of constant rank r, then M is also IG -free of

rank r.

Proof. Since M is finitely generated, it fits in a short exact sequence

0 //N // I⊕rG//M // 0

84

of IG -modules. For every P ∈ Σ we obtain an induced exact sequence

N⊗P OP// O⊕rP

∼ //M⊗P OP// 0,

where the map O⊕rP M⊗POP, a priori only a surjection, is actually an isomorphism

being a surjective map between OP-free modules of the same rank. Therefore,

N ⊆⋂P∈Σ

ker P · (IG )⊕r = 0

because the intersection of the kernels of infinitely many arithmetic points of IG is

trivial ([LV11], Proposition 9.4).

Proposition 5.3.6. Let ω(M/Q) be the number of OL-ideals dividing MOL/Q. The

big Galois module VG (M) is a finite free IG -module of rank 4 ·ω(M/Q) and it satisfies

exact control:

for any arithmetic point P ∈ A(IG ) of weight 2 there is a natural Galois equiv-

ariant surjection

VG (M) AsO(VGP)(χθQ)⊕ω(M/Q),

where AsO(VGP) is isomorphic to a GQ-stable O-lattice in the 4-dimensional

Asai representation attached to GP in Definition 5.1.2.

Proof. For any α ≥ 1 there is a Hecke equivariant surjection

H2et(K,t(p

∞);O(2))mG H2

et(S(K,t(pα))Q;O(2))mG

(5.5)

onto a finite free O-module ([Dim13], Theorem 3.8(iii) & Corollary 3.9). Let P ∈

A(IG ) be an arithmetic point of weight 2 and level pα > 1, then by Proposition 5.1.4,

the GP-isotypic submodule H2et(S(K,t(p

α))Q, O(2))[GP] of H2et(S(K,t(p

α))Q;O(2))mG

85

is free over O of rank 4 · ω(M/Q) and isomorphic to a GQ-stable lattice in the GQ-

representation As(VGP)⊕ω(M/Q). Moreover, Dimitrov’s result (5.5) together with the

perfectness of the Hecke equivariant twisted Poincare duality imply that

H2et(K,t(p

∞);O(2))mG⊗GP

O ∼= H2et(S(K,t(p

α))Q;O(2))mG⊗GP

O

is a free O-module of rank 4 · ω(M/Q). Hence, if we set

AsO(VGP) := H2

et(S(K,t(pα))Q;O(2))mG

⊗GPO,

then AsO(VGP) is a GQ-stable O-lattice in the 4-dimensional Asai representation at-

tached to GP and

VG (M)⊗GPO ∼= AsO(VGP

)(χθQ)⊕ω(M/Q).

Finally, we showed that for every arithmetic point P ∈ A(IG ) of weight 2, the quotient

VG (M)⊗GPO is O-free of rank 4 ·ω(M/Q). By Lemma 5.3.5, the IG -module VG (M)

is free of rank 4 · ω(M/Q).

Proposition 5.3.7. There is an isomorphism of GQ-representations

VG (M)⊗IGQG∼=(As(VG )†

)⊕ω(M/Q). (5.6)

Proof. For every prime q -Mp the traces of Frobq acting the Galois modules VG (M)

and As(VG )⊕ω(M/Q) belong to IG . Proposition 5.3.6 show that these traces become

equal modulo every arithmetic point P ∈ A(IG ) of weight 2, hence they must agree as

elements of IG . Since G does not have complex multiplication nor does it arise from

base change, the two GQ-representations in (5.6) are semisimple by Lemma 5.1.3.

86

The theorem of Brauer-Nesbitt then implies they are isomorphic.

Corollary 5.3.7 suggests that under some mild technical conditions we may transfer

the filtration of the GQp-representation As(VG )⊕ω(M/Q) given by Proposition 5.2.3 to

the free IG -module VG (M). The following lemma shows that VG (M) inherits such a

filtration when the Jordan-Holder factors of the residual representation As(VG )⊗IGFp

are all distinct.

Lemma 5.3.8. Let (R,m) be a local ring, G a profinite group and M a finite free

R-module with a continuous action of G. Suppose that M = M ⊗R Frac(R) has a

G-stable filtration with graded pieces GriM = Frac(R)(χi), for characters χi : G −→

R×, i = 0, . . . , r − 1.

If the Jordan-Holder factors of the R/m[G]-module M/mM are all distinct then

there is a filtration

M ⊃ Fil1M ⊃ · · · ⊃ FilrM = 0

of free R-submodules with graded pieces GriM = R(χi) of rank 1 over R.

Proof. Consider the natural projection M → Gr0M and define Gr0M = Im(M →

Gr0M), Fil1M = ker(M → Gr0M), so that we have an exact sequence of R[G]-

modules

0→ Fil1M →M → Gr0M → 0.

We claim that Gr0M is free of rank 1 over R. If that is true than Fil1M is a projective

R-module, so free since R is local, and we conclude by descending induction.

Clearly Gr0M is torsion-free, so it suffices to show it is generated by one element.

The reduction Gr0M ⊗R R/m is non-zero because Gr0M ⊗R FracR = Gr0M and it is

of dimension one over the residue field because the Jordan-Holder factors of M/mM

are all distinct.

87

Remark. Recall the primitive eigenform g ∈ StL,tL(Q;χ;O) and denote by αi, βi

the eigenvalues of %g(Frobpi) for p1, p2 the OL-prime ideals above p. Thanks to

Proposition 5.2.3, the Jordan-Holder factors of the residual representation As(VG )⊗IG

Fp are all distinct if and only if the products α1α2, α1β2, β1α2, β1β2 are all distinct

in Fp.

Proposition 5.3.9. Let K/Q be a non-totally real S5-quintic extension whose Galois

closure contain a real quadratic field L. Recall there is a parallel weight one Hilbert

eigenform gK over L such that As(%gK ) ∼= IndQK1−1 ([For18], Corollary 4.2). If p 6= 5

is a rational prime unramified in K whose Frobenius conjugacy class is that of 5-cycles

in S5, then p splits in L and the residual GQp-representation(As(%gK )⊗ Fp

)|Dp

has

distinct Jordan-Holder factors.

Proof. The representation As(%gK ) : GQ → GL2(O) factors through the Galois group

of the Galois closure of K. As a representation of the symmetric group S5 it is

isomorphic to the irreducible 4-dimensional direct summand of the permutation rep-

resentation of S5 acting on 5 elements. If p 6= 5 is a rational prime unramified in

K whose Frobenius conjugacy class is that of 5-cycles in S5, then the decomposition

group Dp is cyclic of order 5 and we can conclude by noting that(As(%gK )⊗Fp

)(Frobp)

has four distinct eigenvalues given by the non-trivial 5-th roots of unity.

5.4 Local cohomology class

Let Vf(M) the f-isotypic quotient of H1et(Y (K ′0(p))Q, O(1)) which comes equipped

with a natural projection H1et(Y (K ′0(p))Q, O(1)) Vf(M).

Definition 5.4.1. We define the Galois module

VG ,f(M) := VG (M)⊗ Vf(M)(−1).

88

There is a natural projection prG ,f : Vn.o.∞ (M)→ VG ,f(M).

Definition 5.4.2. The IG -adic cohomology class attached to G and f is the class

κG ,f = prG ,f(κn.o.∞ ) ∈ H1(Q,VG ,f(M)).

Let

κp(G , f) := locp(κG ,f) ∈ H1(Qp,VG ,f(M))

denote the local cohomology class obtained by restricting to the subgroup Dp ' ΓQp

of GQ.

By Corollary 5.3.7,

VG ,f(M)⊗IGFrac(IG ) ∼=

(V†G ,f ⊗IG

Frac(IG ))⊕ω(M/Q)

.

Thanks to Lemma 5.3.8, we have a three step Dp-invariant filtration on VG ,f(M)

coming from the filtration on V†G ,f as in Lemma 5.2.5.

Lemma 5.4.3. The natural map

H1(Qp,Fil2VG ,f(M)) −→ H1(Qp,VG ,f(M))

induced by the Dp-stable filtration is an injection.

Proof. Lemma 5.2.5 implies that

H0(Qp,Gr1VG ,f(M)) = 0, H0(Qp,Gr0VG ,f(M)) = 0.

Taking the long exact sequence in Galois cohomology associated with the short exact

89

sequence of Dp-modules

0 // Gr1VG ,f(M) // VG ,f(M)/Fil2 // Gr0VG ,f(M) // 0,

we deduce that H0(Qp,VG ,f(M)/Fil2) = 0 and the lemma follows.

Proposition 5.4.4. Consider the following element of the ring IG

ξG ,f =(1− αfχ(p1)G

(T($p1)−1T($p2)

))(1− αfχ(p2)G

(T($p1)T($p2)−1

)).

(5.7)

We have

ξG ,f · κp(G , f) ∈ H1(Qp,Fil2VG ,f(M)).

Proof. We follow the argument in ([DR17], Proposition 2.2). By exact control,

VG ,f(M)⊗Λ Λα is realized as a quotient of

H2et(S(K,t(p

α))Q, O(1))(δα)⊗ H1et(Y (K ′(p))Q, O(1)),

a direct summand of H3et(Z

†α(K)Q, O(2)). Let

H1f (Qp,VG ,f(M)⊗Λ Λα) ⊂ H1

g(Qp,VG ,f(M)⊗Λ Λα) ⊂ H1(Qp,VG ,f(M)⊗Λ Λα)

denote the finite part and geometric part of the Galois cohomology as in [BK90],

Section 3. By the work of Saito ([Sai09]), later refined by Skinner ([Ski09]), the

purity conjecture for the monodromy filtration holds for the middle cohomology of

Hilbert modular varieties and hence for the middle cohomology of Z†α(K). Therefore,

by Theorem 3.1 of [Nek00], the natural image of locp(κG ,f) in H1(Qp,VG ,f(M)⊗ΛΛα)

lies in H1g(Qp,VG ,f(M)⊗Λ Λα), which coincides with H1

f (Qp,VG ,f(M)⊗Λ Λα) in this

90

case.

By Lemma 5.2.5, VG ,f(M)⊗Λ Λα is an ordinary ΓQp-representation in the sense

of [Wes05], i.e. there is a decreasing exhaustive and separated GQp-stable filtration

such that an open subgroup of the inertia group Ip acts on the j-th graded piece

GrjVG ,f(M)⊗Λ Λα via the j-the power of the cyclotomic character εjQ. According to

Lemma 2 of [Fla90], which assumes ordinarity in the slightly more restrictive sense

of [Gre89] but the same argument nonetheless applies in the setting of [Wes05],

H1f (Qp,VG ,f(M)⊗Λ Λα) = ker

(H1(Qp,VG ,f(M)⊗Λ Λα)

→ H1(Ip, (VG ,f(M)/Fil2)⊗Λ Λα)).

Hence, the natural image of locp(κG ,f) in lim←−α H1(Ip,(VG ,f(M)/Fil2

)⊗Λ Λα

)is zero.

Since

Fil2(VG ,f(M)

)= lim←−

α

(Fil2VG ,f(M)

)⊗Λ Λα,

VG ,f(M) = lim←−α

(VG ,f(M)

)⊗Λ Λα,

and the Mittag-Leffler condition is satisfied, we conclude

VG ,f(M)/Fil2 = lim←,α

(VG ,f(M)/Fil2

)⊗Λ Λα,

and

lim←,α

H1(Ip,(VG ,f(M)/Fil2

)⊗Λ Λα

)' H1

(Ip,VG ,f(M)/Fil2

).

We deduce that

locp(κG ,f) ∈ ker(H1(Qp,VG ,f(M)

)→ H1

(Ip,VG ,f(M)/Fil2

)).

91

Therefore the image of locp(κG ,f) in H1(Qp,VG ,f(M)/Fil2

)lies in

H1(Qurp /Qp,

(VG ,f(M)/Fil2

)Ip) ∼= ker(H1(Qp,VG ,f(M)/Fil2)

→ H1(Ip,VG ,f(M)/Fil2)).

The choice of the arithmetic Frobenius Frobp as a topological generator gives the

identification Gal(Qurp /Qp) ∼= Z and allows us to compute that

H1(Qurp /Qp,

(VG ,f(M)/Fil2

)Ip) ∼= (VG ,f(M)/Fil2)Ip/(Frobp − 1).

Since VG ,f(M)/Fil2 sits in the short exact sequence

0→ Gr1VG ,f(M)→ VG ,f(M)/Fil2 → Gr0VG ,f(M)→ 0

and Lemma 5.2.5 implies (Gr0VG ,f(M)

)Ip= 0,

we see

(VG ,f(M)/Fil2

)Ip=(Gr1VG ,f(M)

)Ip

=IG

(Ψ−1

G ,p1ΨG ,p2δp(f)χ,p1

)⊕ IG

(ΨG ,p1Ψ

−1G ,p2

δp(f)χ,p2

).

Therefore if we define

ξG ,f =(1− αfχ(p1)G

(T($p1)−1T($p2)

))(1− αfχ(p2)G

(T($p1)T($p2)−1

))then

ξG ,f · H1(Qurp /Qp,

(VG ,f(M)/Fil2

)Ip)= 0.

92

Hence the image of ξG ,f · κp(G , f) in H1(Qp,VG ,f(M)/Fil2

)vanishes and the claim

follows.

In light of Proposition 5.4.4, from now to the end of this article, we replace the

ring IG and the various modules over it with their respective localizations at the

multiplicative set generated by ξG ,f . Note that the arithmetic specializations of ξG ,f

never vanish: for any P ∈ A(IG ), the specialization of G(T($p1)−1T($p2)

)at P is

an algebraic integer with complex absolute value 1, whereas αf has complex absolute

value p1/2.

Corollary 5.4.5. With the above convention in place,

κp(G , f) ∈ H1(Qp,Fil2VG ,f(M)).


V fG (M) := Fil2VG (M)⊗Gr0Vf(M)(−1).

We denote by

κfp (G ) ∈ H1

(Qp,V f

G (M))

the image of κp(G , f) under the natural surjection Fil2VG ,f(M) V fG (M).

Note that V fG (M) is isomorphic a direct sum of copies of the GQp-representation

V fG (M) ∼=

(Vf

G

)⊕ω(M/N).

93

Specialization in parallel weight one.

The specialization at arithmetic point P ∈ Aχ(IG ) of the big Galois representation

VG ,f(M) can be written as

VGP ,f(M) = As(%g)(M)⊗ Vf(Mp).

We denote by

κp(g(p) , f) ∈ H1

(Qp,VGP ,f

(M))

(5.8)

the specialization of the big cohomology class κp(G , f). Consider the map

∂p : H1(Qp,VGP ,f

(M))−→ H1

(Qp,As(%g)(M)⊗Gr0Vf(Mp)

)induced by the projection VGP ,f

(M)→ As(%g)⊗Gr0Vf(Mp), then the local Selmer

group at p arises as the kernel of ∂p,

H1f

(Qp,VGP ,f

(M))

= ker(∂p),

because of the Hodge-Tate weights.

Proposition 5.4.7. The class κp(g(p) , f) is not crystalline at p if and only if the

specialization of κfp (G ) at P is non-trivial:

∂p(κp(g

(p) , f)

)6= 0 ⇐⇒ κf

p (G )(P) 6= 0.

Furthermore

∂p(κp(g

(p) , f)

)∈ H1

(Qp,As(%g)(M)βp ⊗ Vf(Mp)

)

94

where As(%g)βp is the subspace where Frobp acts as multiplication by βp = β1β2.

Proof. By Corollary 5.4.5,

κp(g(p) , f) ∈ H1

(Qp,Fil2VGP ,f

(M)).

As

V fGP

(M) = Im(Fil2VGP ,f

(M) −→ As(%g)(M)⊗Gr0Vf(Mp))

the two classes ∂p(κp(g

(p) , f)

),κf

p (G )(P) coincide. For the last claim we invoke

Proposition 5.2.3 which implies that

As(%g)βp = Fil2As(%g).

95

Chapter 6

Big pairing

We now generalize some of the results in [Oht95], which will be essential in the

construction of motivic p-adic L-functions later. Note that due to the complicated

geometry of Hilbert modular surfaces, we are not able to generalize the so-called Λ-

adic Eichler-Shimura isomorphism; instead, we can only obtain a very crude form of

the map.

6.1 Algebra interlude

Recall Γ = 1 + pZp, Γα = Γ/Γpα

and Λ = lim←−αO[Γα]. The arithmetic points of the

weight space W are those corresponding to the continuous homomorphisms

w`,χ : Λ −→ Qp, [u] 7→ χ(u)u`−2

for ` ∈ Z≥1 and χ : Γ→ Q×p a finite order character.

Definition 6.1.1. We define

Π := lim←−α

E[Γα].

96

For any Λ-module M, we write MΠ = M⊗Λ Π.

Let P : Λ → Qp be any weight point in W . Then we have a specialization map

P : M → M ⊗Λ,P Qp, which, by a slight abuse of notation, will again be denoted

by P. Now assume M = lim←−αMα is given by a projective system (Mα)α≥1 of flat

O[Γα]-modules, such that it satisfies M ⊗Λ O[Γα] ' Mα for all α ≥ 1. If P ∈ W

is a weight 2 arithmetic point w2,χ induced by a finite order character χ : Γ → Q×p

factoring through Γα, then the specialization map at w2,χ becomes the canonical

projection w2,χ : M → Mα ⊗O[Γα],χ Qp. Note that here again we are slightly abusing

the notation to use χ to denote the O-algebra homomorphism O[Γα] → Qp induced

by the character with the same name.

Lemma 6.1.2. Let M = lim←−αMα be a Λ-module satisfying M ⊗Λ O[Γα] ' Mα as

above. Let x = (xα)α ∈M. Then x = 0 if and only if w2,χ(x) = 0 for all finite order

character χ : Γ→ Q×p .

Proof. The forward implication is trivial. We look at the other direction. The injec-

tivity of the homomorphism ⊕χ : O[Γα] → ⊕χQp where χ runs over all characters

of Γα implies the injectivity of ⊕w2,χ : Mα → ⊕χ(Mα ⊗χ Qp). In particular, if

w2,χ(x) = 0 for all finite order characters, then xα = 0 for all α ≥ 1.

Note that since Π⊗ΛO[Γα] ' E[Γα], the above lemma in particular applies to Π.

Moreover, the next lemma shows it will also apply to any finitely generated Λ-module.

Lemma 6.1.3. Suppose M is a finitely generated Λ-module. Then we have a canon-

ical isomorphism

M ' lim←−α

M⊗Λ O[Γα].

Proof. This is exactly Lemma 2.4.4 of the Iwasawa Theory notes by Sharifi.

97

Lemma 6.1.4. If A B is a surjective morphism of Λ-modules then

lim←−α

(A⊗Πα) lim←−α

(B ⊗Πα).

Proof. Let Q = ker(A→ B), and κα = ker(A⊗Πα → B⊗Πα) for every α ≥ 1, then

there is a commutative diagram

Q⊗Πα

%% %% **

κα //

A⊗Πα//

B ⊗Πα//

0

Q⊗Πα−1

%% %% **

κα−1 // A⊗Πα−1

// B ⊗Πα−1// 0

Since tensor product is right exact, Q ⊗ Πα κα is surjective for every α ≥ 1.

Therefore the transition map κα → κα−1 is surjective for every α > 1 and lim←−1

ακα = 0

as required.

Lemma 6.1.5. Suppose M is a finitely generated, flat Λ-module and write Mα =

M⊗Λ O[Γα]. Then

MΠ∼−→ lim←−

α

(Mα ⊗O E).

Proof. Since Λ is Noetherian, M is finitely presented, and thus fits in an exact se-

quence of the form

(Λ)⊕m → (Λ)⊕n →M→ 0. (6.1)

Tensoring is right exact, hence we deduce presentations for MΠ and M ⊗ Πα =

98

Mα ⊗O E (Lemma 6.1.3):

Π⊕m → Π⊕n →MΠ → 0, Π⊕mα → Π⊕nα →M⊗Πα → 0.

Now it suffices to show that

Π⊕m → Π⊕n → lim←−α

(M⊗Πα)→ 0

is exact. The surjectivity of Π⊕n lim←−α (M ⊗Πα) follows from Lemma 6.1.4, so

we are left to prove exactness at the middle of the sequence. Let Q = ker(Λn →M),

since (6.1) is a presentation Q comes equipped with a surjection Λm Q. Moreover,

Q⊗Πα = ker(Π⊕nα →M⊗Πα)

because TorΛ1 (M,Πα) = 0 as M is a flat Λ-module. Therefore, we are left to show

that the induced map Πm → lim←−α (Q⊗Πα) is surjective, which follows from another

application of Lemma 6.1.4.

In particular, since IG is a finite flat algebra over Λ, Lemma 6.1.5 implies

IG ⊗Λ Π ' lim←−α

IG ⊗Λ E[Γα] ' lim←−α

(IG ⊗Λ Π)⊗Λ O[Γα]

as Λ-modules. Thus the condition of Lemma 6.1.2 is also satisfied for IG ⊗Λ Π. This

will be crucial later when we compare the automorphic and motivic p-adic L-functions

so we record it here as a corollary.

Corollary 6.1.6. Let x ∈ IG ⊗Λ Π. Then x = 0 if and only if P(x) = 0 for all

arithmetic points P ∈ A(IG ).

Proof. Since IG ⊗Λ Π ' lim←−α IG ⊗Λ Πα, an element x ∈ IG ⊗Λ Π is equal to zero if

99

and only if its image xα ∈ IG ⊗Λ Πα is equal to zero for every α ≥ 1. We have a

natural injection Πα → ⊕χQp, the sum over all the characters χ : Γα → Q×p . Hence

IG ⊗Λ Πα → ⊕χ (IG ⊗Λ,w2,χ Qp) (6.2)

because IG is Λ-flat. Since Hida families are finite etale over arithmetic points of

weight at least 2, i.e. IG ⊗ΛOχ is finite etale over Oχ for every character χ : Γα → Q×p

(here Oχ is the image of w2,χ), we see

(IG ⊗Λ Oχ)⊗Oχ Eχ ∼= IG ⊗Λ,χ Eχ

is a finite product of finite field extensions of Eχ = Frac(Oχ), the field of fractions of

Oχ, indexed by the arithmetic points P ∈ A(IG ) above w2,χ. In particular,

IG ⊗Λ,w2,χ Qp∼= ⊕PQp (6.3)

the sum over the arithmetic points P ∈ A(IG ) above w2,χ. Combining (6.2) and (6.3)

yields the desired claim immediately.

6.2 A weak Λ-adic Eichler-Shimura map

Let G ∈ Sn.o.

L (K,t(p∞); Λ) be a Λ-adic Hilbert cuspform, which corresponds to a

Λ-module homomorphism

G : hn.o.L (K,t(p

∞);O) −→ Λ.

100

The exact control for the nearly ordinary Hecke algebra implies that tensoring G with

O[Γα] over Λ produces an O[Γα]-module homomorphism

G α : hn.o.2tL,tL

(K,t(pα);O) −→ O[Γα].

Since O[Γα] = ⊕σ∈ΓαO · [σ], we may write G α = ⊕σ∈ΓαGα,σ−1 · [σ], and the O[Γα]-

linearity implies Gα,σ−1(−) = Gα,1([σ]−). In order to lighten the notation we write Gα

for the Hilbert cuspform

Gα := Gα,1 ∈ Sn.o.2tL,tL

(K,t(pα);O).

The compatibility

hn.o.2tL,tL

(K,t(pα+1);O)

Gα+1// O[Γα+1]

hn.o.2tL,tL

(K,t(pα);O)

Gα // O[Γα]

translates into ∑σ∈ker(Γα+1→Γα)

Gα+1([σ]−) = Gα(−),

or equivalently,

µ∗Gα+1 = π∗1Gα. (6.4)

Lemma 6.2.1. There is a Hecke-equivariant morphism

Ω∞ : S

n.o.

L (K,t(p∞); IG ) −→ DdR

(H2

n.o.(K,t(p∞);E)

)⊗Λ IG ,

101

where

DdR(H2n.o.(K,t(p

∞);E)) := lim←−α,$2,∗

DdR

(en.o.H

2et(S(K,t(p

α))Q, E)).

Proof. Pushing forward equation (6.4) along π2 gives $2,∗Gα+1 = UpGα. Hence, the

collection (U−αp Gα)α is compatible under projection along $2. The map

Sn.o.

L (K,t(p∞); Λ) −→ lim←−

α,$2

Sn.o.2tL,tL

(K,t(pα);O), G 7→

(U−αp Gα

)α

combined with the inclusions

Sn.o.2tL,tL

(K,t(pα);E) ' en.o.Ω

2(Stor(K,t(pα))/E) → DdR(en.o.H

2et(S

tor(K,t(pα)), E))

gives

Ω∞ : S

n.o.

L (K,t(p∞); Λ) −→ DdR

(H2

n.o.(K,t(p∞);E)

). (6.5)

Since Λ is Noetherian and hn.o.L (K,t(p

∞);O) is finite over Λ, the Hecke algebra is also

finitely presented as a Λ-module. As IG is flat over Λ it follows that

Sn.o.

L (K,t(p∞); IG ) ' S

n.o.

L (K,t(p∞); Λ)⊗Λ IG .

The claimed Hecke equivariant morphism is obtained from (6.5) by extension of

scalars.

Definition 6.2.2. For M a hn.o.L (K;O)-module we denote by

M [G ] :=m ∈M : Tm = G (T )m, ∀T ∈ hn.o.

L (K;O)

its G -isotypic submodule.

102

Consider the Hecke module

H2n.o.(K,t(p

∞);BdR) := lim←−α,$2

H2et(S(K,t(p

α))Q;E)⊗Qp BdR.

As the Hecke action commute with the Galois action, there is an induced map

DdR(H2n.o.(K,t(p

∞);O)[G ] −→ H2n.o.(K,t(p

∞);BdR)[G ]GQp . (6.6)

Definition 6.2.3. We let

ΩG : S

n.o.

L (K,t(p∞); IG )[G ] −→ H2

n.o.(K,t(p∞);BdR)[G ]GQp

be the composition of the Hecke equivariant mapΩ∞ with (6.6).

6.3 Big pairing.

To simplify notation, write H2et

(K,t(p

α);O)

= H2et(S(K,t(p

α))Q;O). Recall the group

Gα,t,L(K) = K0(pα)O×L/K,t(pα)O×L acts as diamond operators on S(K,t(p

α)). We

have an inclusion

Γα = (1 + pZp)/(1 + pαZp) → Gα,t,L(K), z 7→

∆(z)

∆(z)

,

where ∆ : 1 + pZp → 1 + pOp is the diagonal inclusion. In this way, z ∈ Γα acts as

the diamond operator 〈∆(z), 1〉. Let mα ∈ A×L,f be the image of Mpα ∈ L under the

103

diagonal embedding and consider the matrix

τα =

0 −1

mα 0

∈ G(AQ,f ).

If we denote by (−)∗ : G(AQ,f ) → G(AQ,f ) the involution g∗ = det(g)−1g then for

every α ≥ 1 there is a morphism λα : S(K,t(pα)) → S(K,t(p

α)) defined as the

composition

S(K,t(pα))

λα //

Tτα

((

S(K,t(pα))

S(ταK,t(pα)τ−1

α )

(−)∗66

.

Lemma 6.3.1. The morphism λα is defined over Q(ζMpα). Furthermore, let σ ∈ GQ

such that a ∈ A×Q,f corresponds to σ via global Artin map. Then we have

〈∆(a−1), 1〉 λα σ = σ λα

where ∆ : AQ,f → AL,f is the natural diagonal inclusion.

Proof. This is a standard computation using the reciprocity laws of Shimura varieties

at CM points. We will sketch a proof below following the notations of [Mil05], in

particular those in Chapters 12 and 13.

Let x ∈ H2 be an imaginary quadratic point defined over a CM field E. Suppose

σ ∈ Aut(C/E) and choose s ∈ A×E so that s corresponds to σ|Q under the reciprocal

of the global Artin map (this coincides with the convention [Mil05] uses). Then we

104

have the following commutative diagram

[x, h] ∈ S(ταK,t(pα)τ−1

α )

σ

(−)∗// [x, det(h)−1h] ∈ S(K,t(p

α))

σ

[x, rx(s)h] ∈ S(ταK,t(pα)τ−1

α )(−)∗〈det(rx(s)),1〉

// [x, det(h)−1rx(s)h] ∈ S(K,t(pα))

where rx(s) = NE/Q(µx(sf )) as defined by equation (52) of Chapter 12 in [Mil05], and

µx : Gm → G is the usual cocharacter of G associated to x characterized by µx(z) =

hxC(z, 1) for z ∈ C, which is defined over E. In the case of Hilbert modular varieties,

we have det(µx) is the natural embedding Gm → ResLQGm and thus det(rx(s)) =

NE/Q(sf ), by which we mean the image of sf ∈ A×E,f in A×L,f along the following

commutative diagram

A×E,fNE/Q

// (AE,f ⊗Q L)×

NE/Q

A×Q,f

iL/Q// (AQ,f ⊗Q L)× = A×L,f

where the horizontal maps are natural inclusions. Now by functoriality of class field

theory, NE/Q(sf ) ∈ A×Q,f corresponds to under the reciprocal of the global Artin map

for Q to σ|Q considered as an element in GQ

Since points of the form [x, h] with fixed x are Zariski dense in the Shimura variety

(see for example Lemma 13.5 of [Mil05]), taking into account that the map Tτα is

defined over Q and hence commutes with Galois action, we conclude that for any

σ ∈ GQ that fixes the reflex field of any CM point, if we choose a ∈ A×Q,f so that a

corresponds to σ via the global Artin map (note that here we switch back to our usual

convention of translating between Galois and adelic characters, which is the inverse

105

of Milne’s convention), we have

〈∆(a−1), 1〉 λα σ = σ λα.

Clearly such σ’s generate Aut(C/Q), and thus the above relation holds for all σ ∈

Aut(C/Q). Note that in particular if σ fixes Q(ζMpα) then σ commutes with λα,

implying λα is defined over Q(ζMpα).

A direct calculation shows that

π1,p λα+1 = λα π2,p, π2,p λα+1 = λα π1,p, (6.7)

so that

Up = λα,∗ U∗p λ∗α. (6.8)

We define a twisted group-ring-valued pairing at finite level

, α : H2et

(K,t(p

α);O(2))× H2

et

(K,t(p

α);O)−→ O[Γα]

by

xα, yαα =∑z∈Γα

⟨〈z〉∗xα, λ∗αUα

p yα⟩α[z−1], (6.9)

where the Poincare pairing 〈 , 〉α is defined modulo torsion.

Proposition 6.3.2. The pairing , α is O[Γα]-bilinear and all the Hecke operators

are self-adjoint with respect to it. In particular, , α induces a pairing on nearly

ordinary parts.

Proof. The Hecke operator Up is self-adjoint with respect to , α because U∗($p)

and Up are adjoint with respect the Poincare pairing and equation (6.8). A similar

106

argument works for all the other Hecke operators T ($q) (or U($q) if q | Mp). To

check O[Γα]-bilinearity, let b be any element in Γα, then we have 〈b〉λα = λα〈b−1〉,

which implies λ∗α〈b〉∗ = 〈b〉∗λ∗α. Therefore

xα, 〈b〉∗yαα =∑z∈Γα

⟨〈z〉∗xα, λ∗αUα

p 〈b〉∗yα⟩α[z−1]

=∑z∈Γα

⟨〈zb〉∗xα, λ∗αUα

p yα⟩α[z−1]

= [b]xα, yαα.

and similarly

〈b〉∗xα, yαα = [b]xα, yαα.

Lemma 6.3.3. The finite Galois covering µ : S(K,t(pα+1))→ S(K,t(p

α)∩U0(pα+1))

induces an isomorphism

µ∗ : H2et

(K,t(p

α) ∩ U0(pα+1);E(2))−→ H2

et

(K,t(p

α+1);E(2))Iα+1,α,t,L (K)

,

where Iα+1,α,t,L (K) = ker(Gα+1

,t,L(K)→ Gα,t,L(K)) is the Galois group.

Proof. We claim that the part of the Hochschild-Serre spectral sequence,

Ep,q2 = Hp

(Iα+1,α,t,L (K),Hq

et(K,t(pα+1);E(2))

)=⇒ Hp+q

et

(K,t(p

α) ∩ U0(pα+1);E(2)),

computing H2et

(K,t(p

α) ∩ U0(pα+1);E(2))

degenerates at the second page and the

term

E0,22 = H2

et

(K,t(p

α+1);E(2))Iα+1,α,t,L (K)

is the only non-zero subquotient. Indeed, Ep,02 = 0 for all p > 0 since Iα+1,α

,t,L (K) is

107

a finite group and H0et(K,t(p

α+1);E(2)) is uniquely divisible, while Ep,12 = 0 for all

p ≥ 0 because Hilbert modular surfaces are simply connected.

Proposition 6.3.4. Let pα+1 : O[Γα+1] → O[Γα] be the homomorphism induced by

the natural projection Γα+1 → Γα. Then the diagram

H2et

(K,t(p

α+1);O(2))× H2

et

(K,t(p

α+1);O) ,α+1

//

$2,∗×$2,∗

O[Γα+1]

pα+1

H2et

(K,t(p

α);O(2))× H2

et

(K,t(p

α);O) ,α

// O[Γα]

commutes.

Proof. We prove the proposition through a direct computation. Since the pairing is

defined modulo torsion, it suffices to prove the lemma after inverting p. We have

pα+1

(xα+1, yα+1α+1

)= pα+1

( ∑z∈Γα+1

⟨〈z〉∗xα+1, λ

∗α+1U

α+1p yα+1〉

⟩α+1

[z−1])

=∑b∈Γα

⟨ ∑z∈Γα+1, z 7→b

〈z〉∗xα+1, λ∗α+1U

α+1p yα+1

⟩α+1

[b−1].

Note that∑

z∈Γα+1, z 7→b〈z〉∗xα+1 =

∑z∈Γα+1, z 7→b〈z

−1〉∗xα+1 is invariant under the ac-

tion of Iα+1,α,t,L (K), and thus equals to µ∗ηb for some ηb ∈ H2

et

(K,t(p

α)∩U0(pα+1);E(2))

by Lemma 6.3.3. We compute that

pα+1

(xα+1, yα+1α+1

)=∑b∈Γα

⟨µ∗ηb, λ

∗α+1U

α+1p yα+1

⟩α+1

[b−1]

=∑b∈Γα

⟨ηb, µ∗λ

∗α+1U

α+1p yα+1

⟩α[b−1]

=∑b∈Γα

⟨ηb, λ

∗α+1U

αp $2,∗π

∗1yα+1

⟩α[b−1]

=∑b∈Γα

⟨π2,∗ηb, λ

∗α+1U

αp $2,∗yα+1

⟩α[b−1].

(6.10)

108

Observing that

π2,∗ηb =1

deg(µ)· π2,∗µ∗

( ∑z∈Γα+1, z 7→b

〈z−1〉∗xα+1

)= 〈b−1〉∗$2,∗xα+1

= 〈b〉∗$2,∗xα+1,

we deduce

pα+1

(xα+1, yα+1α+1

)= $2,∗xα+1, $2,∗yα+1α.

Extending scalars in the pairing , α in (6.9) and taking the projective limit yield

, ∞ : H2n.o.(K,t(p

∞);O(2))⊗ZpZurp × H2

n.o.(K,t(p∞);BdR) −→ lim←−

α

BdR[Γα]

with respect to which all Hecke operators in the anemic Hecke algebra are self-adjoint.

By restricting to the G -isotypic subspace in the second argument we obtain

, G :(VG (M)(θ−1

Q ηQ))⊗ZpZur

p ×H2n.o.(K,t(p

∞);BdR)[G ] −→(

lim←−α

BdR[Γα])⊗Λ IG .

(6.11)

Proposition 6.3.5. Let Θ = (ηQηQ)|GQp, then the pairing

, G : VG (M)(Θ−1)⊗ZpZurp ×H2

n.o.(K,t(p∞);BdR)[G ] −→

(lim←−α

BdR[Γα])⊗Λ IG (−1)

is GQp-equivariant.

Proof. Let σ ∈ GQp such that a ∈ Q×p corresponds to σ via the local Artin map.

109

Lemma 6.3.1 implies

(λα)∗ σ = 〈∆(a−1), 1〉∗ σ (λα)∗,

which, combined with (6.9) yields

σxα, σyαα =∑z∈Γα

⟨σ〈∆(z), 1〉∗xα, σ〈∆(a−1), 1〉∗ (λα)∗ Uα

p yα⟩α[z−1]

=∑z∈Γα

⟨〈∆(za−1), 1〉∗xα, (λα)∗ Uα

p yα⟩α[z−1]

=∑z∈Γα

⟨〈∆(a), 1〉∗ 〈∆(z), 1〉∗xα, (λα)∗ Uα

p yα⟩α[z−1]

where in the second equality we used the Galois equivariance of the Poincare pairing

〈 , 〉α : H2et(S(K,t(p

α))Q, O(2))× H2et(S(K,t(p

α))Q, O)→ O.

As

G ([∆(a), 1]) = χ(∆(a))[ξ−tL∆(a)

]= θ−2

Q (a)η2Q(a),

we see that

σxα, σyαα = θ−2Q (a)η2

Q(a)xα, yαα.

Therefore

, G :(VG (M)(θ−1

Q ηQ))⊗ZpZur

p × H2n.o.(K,t(p

∞);BdR)[G ]

−→(

lim←−α

BdR[Γα])⊗Λ IG (θ−2

Q η2Q)

is GQp-equivariant and the claim follows by twisting.

110

6.4 On Dieudonne modules

Given f ∈ Sord2,1 (N ;1; Q) an ordinary elliptic cuspform, one can define a linear map

ϕ : Sord2,1 (Np;1; Q) −→ Q, h 7→

⟨h, f

(p)⟩

Pet⟨f(p) , f

(p)⟩

Pet

which satisfies ϕ(T ∗` h) = ap(`, f) · ϕ(h). The natural inclusion of cuspforms in de

Rham cohomology induces an isomorphism

Sord2,1 (Np;1;E)

∼−→ DdR

(Gr0(Vf(Mp)(−1))

), h 7→ ωh,

hence we can make the following definition as in ([DR17], Section 2.3 & Equation

(118)).

Definition 6.4.1. Following (79) of [DR17], let η ∈ H1dR(Y (K ′0(p)))ord,ur[f] denote

the unique class that satisfies

Φ(η) = αf · η

and for any h ∈ Sord2,1 (Np;1; Q)

⟨ωh, (λ1)∗η

⟩dR

= ϕ(h).

Remark. For any φ ∈ Sord2,1 (Np;1; Q) we have

⟨η, ωφ

⟩dR

=⟨(λ1)∗η, (λ1)∗ωφ

⟩dR

= ϕ((λ1)∗φ

)=

⟨(λ1)∗φ, f

(p)⟩

Pet⟨f(p) , f

(p)⟩

Pet

.

111

The Hecke equivariant twist of the Poincare pairing

, Q : H1et(Y (K ′0(p), O)× H1

et(Y (K ′0(p), O(1)) −→ O,x, yQ =

⟨x, (λ1)∗y

⟩dR

(6.12)

induces a GQp-equivariant perfect pairing on f-isotypic components

, f : Vf(Mp)(−1)× Vf(Mp) −→ O.

Furthermore, by looking at the Galois action, one sees that Fil1Vf(Mp)(−1) and

Fil1Vf(Mp) are orthogonal with respect to , f . Therefore there is an induced

perfect pairing

, f : Gr0Vf(Mp)(−1)× Fil1Vf(Mp) −→ O. (6.13)

which we can use to make the identification

DdR(Fil1Vf(Mp))∼−→ HomE

(DdR

(Gr0Vf(Mp)(−1)

), E).

Definition 6.4.2. We denote by

η′ ∈ DdR(Fil1Vf(Mp))

the element corresponding to the homomorphism

DdR

(Gr0(Vf(Mp))(−1)

)−→ E, ωφ 7→

⟨ωφ, η

⟩dR.

112

It satisfies

ω, η′

f

=⟨ω, η

⟩dR

∀ ω ∈ DdR

(Gr0Vf(Mp)(−1)

).

Proposition 6.4.3. For any Λ-adic test vector G ∈ Sn.o.

(K,t; IG )[G ] there exists a

homomorphism of IG -modules

〈 , ωG ⊗ η′〉 : D

(Uf

G (M))−→ Π⊗Λ IG

whose specialization at any arithmetic point P ∈ A(IG ) of weight 2 is

P 〈−, ωG ⊗ η′〉 = 〈−, (λα)∗ωGP

⊗ η〉dR : DdR

(Uf

GP(M)

)−→ Cp.

Proof. Recall that

UfG (M) = Fil2VG (M)(Θ−1)⊗Gr0Vf(M)(−1).

By tensoring the GQp-equivariant pairing in Proposition 6.3.5

, G : VG (M)(Θ−1)⊗ZpZurp ×H2

n.o.(K,t(p∞);BdR)[G ] −→

(lim←−α

BdR[Γα])⊗Λ IG (−1)

with the GQp equivariant pairing in (6.13), we obtain

UfG (M)⊗ZpZur

p ×H2n.o.(K,t(p

∞);BdR)[G ]⊗OFil1Vf(M) −→ (lim←−α

BdR[Γα])⊗Λ IG (−1).

113

Restricting the pairing to GQp-invariants we obtain

〈 , 〉 : D(Uf

G (M))× H2

n.o.(K,t(p∞);BdR)[G ]GQp ⊗Qp DdR

(Fil1Vf(M)

)−→ Π⊗Λ IG .

(6.14)

Let ωG :=Ω

G (G ) ∈ H2n.o.(K,t(p

∞);BdR)[G ]GQp be the class represented by the

compatible collection (U−αp Gα)α of cuspforms, and let η′ ∈ DdR

(Fil1Vf(M)

)be the

class of Definition 6.4.2. Then evaluating the pairing (6.14) at ωG ⊗ η′ gives the

homomorphism ⟨, ωG ⊗ η

′⟩

: D(Uf

G (M))−→ Π⊗Λ IG . (6.15)

Now we study the specialization of the pairing at arithmetic points P ∈ A(IG ) of

weight 2 and character χ of conductor pα. Let

z =∑i

xi ⊗ yi ∈ D(Uf

G (M))

=(Fil2VG (M)(Θ−1)⊗Gr0Vf(M)(−1)

)GQp

be any element, then by construction

⟨z, ωG ⊗ η

′⟩

=∑i

xi, ωG

G·yi, η

′f.

Firstly we note that yi, η

′f

=⟨yi, η

⟩dR,

then we observe that the projection ofxi, ωG

G

to level α is

xi,α, U

−αp Gα

α

=∑z∈Γα

⟨〈z〉∗xi,α, (λα)∗Uα

p U−αp Gα

⟩[z−1]

=∑z∈Γα

⟨xi,α, (λα)∗ 〈z〉∗Gα

⟩[z−1].

114

Therefore

P xi, ωG

G

=⟨xi,α, (λα)∗

∑z

χ(z−1)〈z−1〉∗Gα

⟩=⟨xi,α, (λα)∗GP

⟩=⟨xi,P, (λα)∗GP

⟩where the last equality results from the O[Γα]-equivariance of the twisted Poincare

pairing. It follows that

P ⟨z, ωG ⊗ η

′⟩

=⟨zP, (λα)∗ωGP

⊗ η⟩

dR.

115

Chapter 7

Motivic p-adic L-functions

7.1 Perrin-Riou’s regulator

Definition 7.1.1. We consider the characters

ΨfG := Ψ−1

G ,pδp(f), Θ := (ηQηQ)|Dp ,

of a decomposition group Dp at p of GQ. We denote by UfG the representation

corresponding to the unramified character ΨfG .

Note that the arithmetic Frobenius acts on UfG as multiplication by Ψf

G (Frobp) =

αfG (T($p))−1 and Lemma 5.2.5 implies Vf

G = UfG (Θ).

Suppose P ∈ A(IG ) is an arithmetic point of weight ` and character χ, then

Θ(P) = χ−1 · (η`−1

Q )|Dp

has negative Hodge-Tate weight if ` ≥ 2. In this case the weights of the pair of

modular forms (gP, f) are balanced. For the arithmetic point P of weight 1 and

trivial character,Θ(P) = 1 has Hodge-Tare weight equal to zero and the weights

116

of the pair (g(p) , f) are Q-dominated. Therefore, if we let Vf

gP:=(Vf

G

)P

be the

specialization of VfG at P, we have isomorphisms

logBK : H1(Qp,V

fgP

) ∼−→ DdR

(Vf

gP

), if P has weight ` ≥ 2,

exp∗BK : H1(Qp,V

fgP

) ∼−→ DdR

(Vf

gP

), if P = P,

(7.1)

since VfgP

is never isomorphic to Q(1) nor the trivial representation (ΨfG (P) should

always be of infinite order because of the Weil conjectures).

Lemma 7.1.2. Let β : Z×p → E×β be a finite order character of conductor pα, where

Eβ is a finite extension of Qp, and suppose β corresponds to a Galois character

β : GQp → E×β factoring through Gal(Qp(ζpα)/Qp). Consider the GQp-representation

Eβ(β + j

), then the Eβ-vector space

DdR

(Eβ(β + j)

)= Eβ · bβ,j.

has a canonical basis bβ,j.

Proof. For any j ∈ Z, the choice of a compatible sequence of p-power roots of unity

ζ := ζpαα≥0 determines a basis ζj of the GQp-representation Qp(j) and an element

t−j ∈ BdR such that the element ζj ⊗ t−j gives a canonical basis of DdR

(Qp(j))

independent of ζ. We consider models of the GQp-representations Eβ(β), Eβ(−β)

appearing in the Galois modules Eβ ⊗Qp Qp(ζpα) where GQp acts only on the second

factor by Galois automorphisms. For a character α : Gal(Qp(ζpα)/Qp) → E×β the

element

θα =∑

τ∈Gal(Qp(ζpα )/Qp)

α−1(τ)⊗ ζτpα ∈ Eβ ⊗Qp Qp(ζpα)

satisfies (θα)σ = α(σ)θε for all σ ∈ GQp . Then Eβ(β) ∼= Eβ ·θβ and Eβ(−β) ∼= Eβ ·θβ−1 .

We choose the model Eβ · θβ ⊗ ζj of the GQp-representation Eβ(β + j) and we note

117

that

bβ,j := (θβ ⊗ ζj)⊗Eβ (θβ−1 ⊗ t−j) ∈ Eβ(β + j)⊗Qp BdR

is invariant under the GQp-action and it is independent of the choice of ζ. Therefore,

we deduce that DdR

(Eβ(β + j)

)has bβ,j as canonical Eβ-basis.

Write ΛΓ for OJZ×p K, then by ([KLZ17], Theorem 8.2.3) and ([LZ14], Theorem

4.15 – Theorem B.5) there is a(IG ⊗ΛΓ

)-linear map L : H1

(Qp,U

fG ⊗ΛΓ(−j)

)−→

D(UfG )⊗ΛΓ such that for all points P ∈ Hom(IG , Qp) and all characters of the cyclo-

tomic Galois group Gal(Q(ζp∞)/Q) of the form η · εjQ where j ∈ Z and η has finite

order, we have a commutative diagram

H1(Qp,U

fG ⊗ΛΓ(−j)

)

L // D(UfG )⊗ΛΓ

H1(Qp,U

fGP

(−j − η))

// DdR(UfGP

(−j − η))

where the rightmost vertical map is

D(UfG )⊗ΛΓ −→ DdR(Uf

GP(−j − η)), x⊗ [u] 7→ ηεjQ(u) · νP(x)⊗ bη−1,−j,

and the bottom horizontal map is given by (compare also with Theorem 8.2.8 of

[KLZ17])

(1− α1,gP

α2,gPα−1fη(Frobp)p

j) (

1− α−11,gP

α−12,gP

αfη−1(Frobp)p

−j−1)−1

cond(η) = 0

(α1,gP

α2,gPα−1fη(Frobp)p

1+j)cond(η)

G(η−1)−1 cond(η) > 0

(7.2)

118

×

(−1)−j−1

(−j−1)!· logBK j < 0

j! · exp∗BK j ≥ 0.

We pull back the map L by the automorphism of IG ⊗ΛΓ given by

1⊗ [z] 7→ Θ(z)−1 · 1⊗ [z],

by functoriality of the construction of L we obtain

H1(Qp,V

fG ⊗ΛΓ(−j)

)

L′ //(D(Uf

G )⊗ΛΓ

)⊗ΘΛΓ

id⊗Θ−1

H1(Qp,U

fG ⊗ΛΓ(−j)

) L // D(UfG )⊗ΛΓ.

(7.3)

Remark. The conventions of the local reciprocity map and Gauss sum are different

in [LZ14] and [KLZ17], which causes the character in the Gauss sum to be inverted

as compared to Theorem 8.2.8 of [KLZ17]. In fact, in [LZ14] they normalize the

reciprocity map so that geometric Frobenius elements are sent to uniformizers, which

means the identification of Gal(Q(ζp∞)/Q) with Z×p coincides with the cyclotomic

character, which is the reciprocal of our convention (see Section 2.8 of loc.cit.). Fur-

thermore, their Gauss sum is really a sum of Galois characters, whereas ours is a sum

of Hecke characters, so implicitly there is a translation via class field theory involved.

Our equation (7.2) has already taken this into account.

Proposition 7.1.3. There is an homomorphism

LfG : H1

(Qp,V

fG

)−→ D(Uf

G )

119

satisfying the following properties:

(i) For all arithmetic points P ∈ A(IG ) of weight ` ≥ 2, and character χ,

νP LfG =

(−1)`−2

(`− 2)!·Υ(P) ·

(logBK νP

)where Υ(P) =

(α1,gP

α2,gPα−1fp2−`)α ·G(χ · θ`−1

Q|Dp

)−1.

(ii) For the arithmetic point P ∈ A(IG ) of weight one,

νP LfG = Υ(P) ·

(exp∗BK νP

)where Υ(P) =

(1− α1,gP

α2,gPα−1f

)(1− α−1

1,gPα−1

2,gPαf · p−1

)−1.

Proof. First, we note that the map Θ⊗ id : ΛΓ⊗ΘΛΓ∼→ ΛΓ is an isomorphism, hence

L′ can be seen as a homomorphism L′ : H1(Qp,V

fG ⊗ΛΓ(−j)

)−→ D(Uf

G )⊗ΛΓ. For

any arithmetic point P ∈ A(IG ), there is a commutative diagram

H1(Qp,V

fG ⊗ΛΓ(−j)

)

L′ // D(UfG )⊗ΛΓ

H1(Qp,V

fgP

)// DdR

(Vf

gP

)obtain by composing (7.3) with specialization at point P and character

Θ(P)−1 = χ ·(ε1−`Q θ`−1

Q)|Dp.

Then, using (7.2), the bottom horizontal map can be computed to be

(−1)`−2

(`− 2)!·(α1,gP

α2,gPα−1fp2−`)αG(χ · θ`−1

Q|Dp

)−1 · logBK .

120

Similarly, when considering the non-arithmetic point P, the relevant character is the

trivial character Θ(P)−1 = 1, and the bottom horizontal map can be seen to be

(1− α1,gP

α2,gPα−1f

)(1− α−1

1,gPα−1

2,gPαf · p−1

)−1 · exp∗BK .

In order to define the claimed homomorphism LfG , we note that if we consider

β : IG ⊗ΛΓ −→ IG , 1⊗ [u] 7→ 〈u〉[u],

then for all P ∈ A(IG ) the following diagram commutes:

IG ⊗ΛΓ

P⊗Θ(P)−1

''

β// IG

P

O.

Therefore, the composition

H1(Qp,V

fG

)H1(Qp,id⊗1

)

LfG // D(Uf

G )

H1(Qp,V

fG ⊗ΛΓ(−j)

) L′ // D(UfG )⊗ΛΓ

β

OO

satisfies the claimed properties.

7.2 The motivic p-adic L-function.

We are finally ready to define our motivic p-adic L-function. Because of the lack

of a satisfactory generalization of the Λ-adic Eichler-Shimura isomorphism as in the

modular curve case given by [Oht95], a priori it lacks certain integrality properties

(in particular, it lies in a finite algebra over Π instead of Λ). However, later on we

121

will still be able to compare it with our previously constructed automorphic p-adic

L-function and show they are essentially the same.

Recall we have the class κfp (G ) ∈ H1

(Qp,V f

G (K)). Also recall the IG -modules

homomorphism

〈 , ωG ⊗ η′〉 : D

(Uf

G (M))−→ Π⊗Λ IG

associated to any Λ-adic test vector G ∈ Sn.o.

(K,t; IG )[G ] given by Proposition 6.4.3.

Definition 7.2.1. Let G ∈ Sn.o.

(K,t; KG )[G ] be any Λ-adic test vector as above.

The motivic p-adic L-function is defined as

L motp (G , f) :=

⟨Lf

G

(κfp (G )

), ωG ⊗ η

′⟩∈ Π⊗Λ IG .

122

Chapter 8

p-adic Gross-Zagier formulas

The main goal of this section is to give a formula for certain values of the syntomic

Abel-Jacobi map in terms of p-adic modular forms. Given the definition of the motivic

p-adic L-function in terms of the pairing introduced in Chapter 6, we will be interested

in the values

AJsyn

(∆α)(

(λα)∗ωGP⊗ η

)∈ Cp.

The basic tool is the so-called finite polynomial cohomology, as first introduced in

[Bes00] and later generalized in [BLZ16]. Below we will first review the basic theory

following [BLZ16], and then carry out the explicit calculations in our situation.

8.1 P -syntomic cohomology

Let K/Qp be a finite extension, we denote by K0 the maximal unramified subfield of

K and by q be the cardinality of the residue field.

Definition 8.1.1. A filtered (ϕ,N,GK)-module over K is a finite dimensional Qurp -

vector space D endowed with a Qurp -semilinear bijective Frobenius endomorphism ϕ

and a Qurp -linear monodromy operator N satisfying Nϕ = pϕN . The absolute Galois

123

group GK acts Qurp -semilinearly on D and there is a decreasing, separated, exhaustive

filtration of the K-vector space

DK :=(D ⊗Qur

pQp

)GKby K-vector subspaces FiliDK .

One writes Dst for the K0-vector space DGK of GK-invariant elements. A filtered

(ϕ,N,GK)-module D such that the GK-action is unramified and N = 0 is said to be

crystalline. In this case one can show that

D = Dst ⊗K0 Qurp and DK = Dst ⊗K0 K.

Definition 8.1.2. A crystalline filtered (ϕ,N,GK)-module over K is said to be con-

venient for a choice of polynomial P (T ) ∈ 1+TK[T ] if P (Φ) and P (qΦ) are bijective

endomorphisms of DK , where Φ denotes the extension of scalars of the K0-linear

operator ϕ[K0:Qp] on Dst.

For a variety X/K , we let H•HK(Xh) and H•dR(Xh) be the extensions of Hyodo-Kato

and de-Rham cohomology groups defined by Beilinson [Bei13] and by

ιBdR : H•HK(Xh)⊗K0 K −→ H•dR(Xh)

the comparison morphism (which is an isomorphism if X has a semistable model over

OK). For the filtered (ϕ,N,GK)-modules D•(Xh) = Dpst

(H•et(XK ,Qp)

)it was shown

by Beilinson [Bei13] that

D•(Xh)st = H•HK(Xh) and D•(Xh)K = H•dR(Xh).

124

Cohomology of filtered (ϕ,N,GK)-modules.

For a polynomial P (T ) ∈ 1 + TK[T ] one can define the complex

C•st,P (D) : Dst,K ⊕ Fil0DK −→ Dst,K ⊕Dst,K ⊕DK −→ Dst,K

where the first map is (u, v) 7→ (P (Φ)u,Nu, u − v), and the second is (w, x, y) 7→

Nw − P (qΦ)x. The cohomology of this complex is denoted by

H•st,P (D) := H•(C•st,P (D)

).

Theorem 8.1.3. ([BLZ16] Theorem 2.1.2) There is a P -syntomic descent spectral

sequence

Ei,j2 = Hi

st,P

(Dj(Xh)(r)

)=⇒ Hi+j

syn,P (Xh, r)

compatible with cup products.

8.2 Syntomic Abel-Jacobi map

Let X/K be a proper and smooth d-dimensional variety. The commutativity of the

following diagram

CHi(X)clsyn

vv

clet

((

H2isyn(XK,h, i)

ρsyn//

H2iet(XK ,Qp(i))

Gr0syn _

// Gr0et

H0st,1−T (D2i(Xh)(i))

∼ // H2iet(XK ,Qp(i))

GK

(8.1)

125

where ρsyn is Nekovar-Niziol period morphism, follows from the compatibility of the

syntomic descent spectral sequence for syntomic cohomology and the Hochshield-Serre

spectral sequence for etale cohomology ([BLZ16] Theorem 2.1.2).

Remark. The bottom horizontal map of diagram (8.1) is an isomorphism by ([BLZ16]

Theorem 1.1.4), hence the middle horizontal map is injective.

Therefore, if we let

CHi(X)0 := ker(clet : CHi(X) −→ H2i

et(XK ,Qp(i))GK)

denote the subgroup of null-homologous cycles, then the syntomic and the p-adic etale

Abel-Jacobi maps can be compared

CHi(X)0

AJsyn

vv

AJetp

))

H1st,1−T (D2i−1(Xh)(i))

expst // H1(K,H2i−1et (XK ,Qp(i)))

through the generalized Bloch-Kato exponential map.

Furthermore, if V is a quotient of H2i−1et (XK ,Qp(i)) such that D = Dpst(V ) is a

convenient quotient of D2i−1(Xh)(i) with respect to the polynomial 1 − T , then the

natural inclusion DK → C1st,1−T (D) induces an isomorphism

DK

Fil0DK

∼= H1st,1−T (D),

126

and one can refine the comparison as follows

CHi(X)0

AJsyn,D

xx

AJetp,V

&&

DK/Fil0expBK // H1

e(K,V ).

Remark. According to Definition 3.10 of [BK90], the Bloch-Kato exponential surjects

onto H1e(K,V ) and its kernel is given by Dcris(V )ϕ=1/H0(K,V ). In particular, when

Dcris(V )ϕ=1 = 0 we can write

AJsyn,D = logBK AJetp,V ,

where logBK = exp−1BK.

If we letD∗(1) denote the Tate dual ofD, which is a submodule ofD2(d−i)+1(Xh)(d+

1− i), then DK/Fil0 =(Fil0D∗(1)K

)∨and we can write

AJsyn,D : CHi(X)0 −→(Fil0D∗(1)K

)∨. (8.2)

Evaluation using P -syntomic cohomology.

Let ∆ ∈ CHi(X)0 be a null-homologous cycle. For any class

η ∈ Fil0D∗(1)K ⊂ Fild+1−iH2(d−i)+1dR (Xh),

choose a polynomial P (T ) ∈ 1 + TK[T ] such that P (1) 6= 0, P (q−1) 6= 0 and η ∈

H0st,P (D∗(1)) ([BLZ16] Proposition 1.4.3). Suppose that η is in the kernel of the

“knight’s move” map

H0st,P

(D2(d−i)+1(Xh)(d+ 1− i)

)−→ H2

st,P

(D2(d−i)(Xh)(d+ 1− i)

),

127

so that η can be lifted to syntomic cohomology. Then for any choice of lift η ∈

H2(d−i)+1syn,P

(Xh, d+ 1− i

)we can write

AJsyn,D(∆)(η) = trX,syn,P

(clsyn(∆) ∪ η

)(8.3)

thanks to the compatibility of the P -syntomic descent spectral sequence with cup

products

Fil1H2isyn,1−T

(Xh, i

)

× H2(d−i)+1syn,P

(Xh, d+ 1− i

)

// H2d+1syn,P

(Xh, d+ 1

)/Fil2 ∼= K

∼

H1st,1−T

(D2i−1(Xh)(i)

)

× H0st,P

(D2(d−i)+1(Xh)(d+ 1− i)

)// H1

st,P

(Dpst(Q(1))

) ∼= K

H1st,1−T (D) × H0

st,P (D∗(1))?

OO

// H1st,P

(Dpst(Q(1))

) ∼= K.

8.3 Abel-Jacobi map of Hirzebruch-Zagier cycles

For every arithmetic point P ∈ A(IG ) of weight 2 and level pα, the Galois represen-

tation VgPis crystalline as a GQp(ζpα )-representation. Throughout this subsection,

we will consider all our geometric structures, including the moduli schemes and the

cycles, to be defined over Fα := Qp(ζpα). Similarly, we will regard all Galois repre-

sentations, as well as Deudonnee functors, to be defined with respect to the absolute

Galois group GQp(ζpα ).

Let P ∈ A(IG ) be an arithmetic point of weight 2 and character χ of conductor

pα, and write χ = χ NL/Q , then the specialization of VG ,f(M) at P is a quotient

of H3et(Zα(K)Fα ,Qp(2)) such that DgP,f := Dpst(VGP,f(M)) is a convenient quotient

of D3(Zα(K))(2). Furthermore, DgP,f∼= (DgP,f)

∗(1) since VGP,f(M) is Kummer

self-dual.

Definition 8.3.1. Let ωP ∈ en.o.Fil2H2dR(S(K,t(p

α))/Fα) be the de Rham cohomol-

128

ogy class associated with the specialization gP ∈ Sn.o.2tL,tL

(K,t(p

α);χθ−1L χ−1,1;O

)of

the KG -adic cuspform G .

Recall the class η of Definition 6.4.1. The tensor product (λα)∗ωP ⊗ η belongs

to the convenient (ϕ,N,ΓFα)-module (DgP,f)∗(1)Fα

∼= (DgP,f)Fα and we will evaluate

AJsyn

(∆α)(

(λα)∗ωP ⊗ η)

= AJsyn

((λα, id)∗∆

α

)(ωP ⊗ η

).


λα wpα2= 〈$α

p2, 1〉 wpα2

λα

when restricted to the first connected component of S(K(pα)).

Proof. Recall that by definition wpα2= Tτp2

να and λα = (−)∗ Tτα . A direct

calculation on the complex uniformization of Shimura varieties shows that

λα Tτp2= 〈$α

p2, 1〉 Tτp2

λα.

On the other hand, identifying both S(K(pα))(C) and S(KX(pα))(C) with

∐hi=1 Γi\H2,

we see the first connected component Γ1\H2 is identified via

(z1, z2) ∈ H2 7→ [(z1, z2), I2] ∈ S(K(pα))(C),

so λα restricted to Γ1\H2 is precisely the map (z1, z2) 7→ (−1/(Mpαz1),−1/(Mpαz2),

which clearly commutes with να, the identity map on H2.

129

The diagonal embedding ζ : Y (K ′0(pα))→ S(K(pα)) naturally factors

Y (K ′0(pα)) ι //

ζ''

S∗(K∗(pα))

ξ

S(K(pα))

through the natural diagonal embedding map to a Hilbert-Blumenthal variety

ι : Y (K ′0(pα))→ S∗(K∗(pα)).

Lemma 8.3.3. Let Yα := Y (K ′0(pα)) and Z∗(pα) := S∗(K∗(p

α)) × Y (K ′0(p)). Con-

sider the null-homologous cycle

Ξα := (id, εf∗ )∗(ι λα, π1,α)∗[Yα] ∈ CH2(Z∗(pα))0(Fα)⊗Z Zp.

Then

(λα, id)∗∆α = (wpα2

ξ, id)∗Ξα.

Proof. Since the image of the diagonal embedding only involves the first connected

component of S(K(pα)), Lemma 8.3.2 allows us to “push” (λα, id)∗ all the way back:

(λα, id)∗∆α = (λα, id)∗(id, εf)∗(〈$α

p2, 1〉 wpα2

ζ, π1,α)∗[Yα]

= (id, εf∗ )∗(〈$−αp2, 1〉 λα wpα2

ζ, π1,α)∗[Yα]

= (id, εf∗ )∗(〈$−αp2, 1〉〈$α

p2, 1〉 wpα2

λα ζ, π1,α)∗[Yα]

= (id, εf∗ )∗(wpα2 ζ λα, π1,α)∗[Yα]

= (wpα2 ξ, id)∗Ξ

α.

130

The functoriality of the construction of the syntomic Abel-Jacobi map gives

AJsyn

((λα, id)∗∆

α

)(ωP ⊗ η

)= AJsyn

(Ξα)(ωP ⊗ η

)(8.4)

where the de Rham cohomology class

ωP := (wpα2 ξ)∗ωP ∈ Fil2H2

dR(S∗(K∗(pα))/Fα)

is associated with the cuspform

gP := (να ξ)∗(gP|τ−1pα2

) ∈ S2tL,tL

(K∗(p

α);O).

If R(T ), Q(T ) ∈ 1 + T · Fα[T ] are polynomials such that R(p−2Φ)ωP and Q(Φ)η

are zero, then

ωP ∈ H0st,R

(D2(S∗(K∗(p

α)))(2)), η ∈ H0

st,Q

(D1(Y1)

).

Lemma 8.3.4. Suppose that Q(p) 6= 0, then there exists lifts

ωP ∈ H2syn,R(S∗(K∗(p

α)), 2), η ∈ H1syn,Q(Y1, 0)

of ωP and η to syntomic cohomology.

Proof. As the Hilbert-Blumenthal surface S∗(K∗(pα)) is simply connected, we have

H2st,R

(D1(S∗(K∗(p

α)))(2))

= 0, thus the descent spectral sequence induces a surjec-

tion H2syn,R(S∗(K∗(p

α)), 2) H0st,R

(D2(S∗(K∗(p

α)))(2)). In the modular curve case

we can compute that

H2st,Q

(D0(Y1)

)= H2

st,Q

(Dpst(Qp)

)= Fα/Q(p)Fα

131

is zero as long as Q(p) 6= 0, in which case we also obtain a surjection H1syn,Q(Y1, 0)

H0st,Q

(D1(Y1)

).

It follows by equation (8.3) that if (R ? Q)(1) 6= 0 and (R ? Q)(p−1) 6= 0 we can

compute values of the syntomic Abel-Jacobi map as

AJsyn

((λα, id)∗∆

α

)(ωP ⊗ η

)= trZ∗ (pα),R?Q

(clsyn

(Ξα)∪(ωP ⊗ η

)).

By the projection formula ([BLZ16], Theorem 2.5.3) and noting that (εf)∗η = η,

we can compute the syntomic trace for Z(pα) as a syntomic trace for the curve

Yα = Y (K ′(pα)):

AJsyn

((λα, id)∗∆

α

)(ωP ⊗ η

)=⟨(ι λα)∗ωP, (π1,α)∗η

⟩Yα,R?Q

. (8.5)

Explicit cup product formulas.

In order to continue the computation of syntomic regulators it is convenient to make

the choice of lifts more explicit. Any lift ωP ∈ H2syn,R(S∗(K∗(p

α)), 2) of ωP can be

explicitly described as in Section 2.4 of [BLZ16]:

ωP =[u, ωP;w

]where u ∈ RΓB,2HK(S∗(K∗(p

α))), w ∈ RΓB,1HK(S∗(K∗(pα)))

satisfy du = 0, dw = R(p−2Φ)u, and ιBdR(u) = ωP. Similarly, a lift η ∈ H1syn,Q(Y1, 0)

of η can be described as

η = [u′, η;w′] where u′ ∈ RΓB,1HK(Y1), w′ ∈ RΓB,0HK(Y1)

satisfy du′ = 0, dw′ = Q(Φ)u′ and ιBdR(u′) = η. Now we consider any two polynomials

a(T1, T2) and b(T1, T2) satisfying (R ? Q)(T1T2) = a(T1, T2)R(T1) + b(T1, T2)Q(T2).

132

Then by Proposition 2.4.1 of [BLZ16], the class

(ι λα)∗ωP ∪ (π1,α)∗η ∈ H3syn,R?Q(Yα, 2)

can be described by [u′′, v′′;w′′] = [0, 0;w′′] where,

u′′ = (ι λα)∗u ∪ (π1,α)∗u′ = 0, v′′ = (ι λα)∗ωP ∪ (π1,α)∗η = 0

for dimension reasons, and

w′′ = a(p−2Φ,Φ)((ι λα)∗w ∪ (π1,α)∗u′) + b(p−2Φ,Φ)((ι λα)∗u ∪ (π1,α)∗w′)

= a(p−2Φ,Φ)((ι λα)∗w ∪ (π1,α)∗u′).

Combining (8.5) with the definition of the syntomic trace map ([BLZ16], Definition

3.1.2 & Equation (2)) we obtain

AJsyn

((λα, id)∗∆

α

)(ωP ⊗ η

)= trYα,R?Q

((ι λα)∗ωP ∪ (π1,α)∗η

)= −ιBdR

((R ? Q)(Φ)−1w′′

)=−a(p−2Φ, αf)

(R ? Q)(p−1)·[(λα)∗ι∗[ιBdR(w)] ∪dR (π1,α)∗η

](8.6)

where the last equality follows from the facts that η is a eigenvector for Φ of eigenvalue

αf and the Frobenius endomorphism of Dpst(Qp(1))st,Fα is multiplication by p−1.

8.4 Overconvergent Hilbert Modular Forms

In the next section we will present the syntomic lift in terms of overconvergent Hilbert

modular forms. In this section we briefly review the the theory of overconvergent

133

Hilbert modular forms following [KL05] (see also [TX16], which also has an excellent

account of the material and we will also follow it closely), and slightly adapt their

constructions to our situation.

Roughly speaking, Kisin and Lai [KL05] defined overconvergent Hilbert modular

forms of level K∗1(pα) as overconvergent sections of automorphic vector bundles over

a certain rigid analytic subspace of S∗(K∗1(pα)). To more precisely introduce the

overconvergent Hilbert modular forms, we need to first summarize the geometric

definitions and constructions made in op.cit. Let c ⊂ L be a fractional ideal of L and

M(c, µQpα) be the Hilbert-Blumenthal moduli scheme classifying pairs (A = (A, λ), i)

over Zp with λ a c-polarization and i : d−1 ⊗ µQpα → A a µQpα-level structure.

Following section 2.3 of [TX16], for any open compact subgroup U0(Q) ⊂ K ⊂

Γ(Q), we may replace the tame µQ-level structure with the more general K-level struc-

ture. Let M(c, K, µpα) be the moduli scheme classifying triples (A = (A, λ), iQ, ip)

where A is above, iQ is a K-level structure, and ip is a µpα-level structure.

The Qp-varietyM(c, K, µpα)Qp is canonically identified with a Shimura variety for

G∗. Choose an adele t ∈ AL,f generating the ideal cd. Let

Kc,?(pα) =

t−1 0

0 1

K?(pα)

t 0

0 1

and

K∗c,?(pα) = Kc,?(p

α) ∩G∗(Af )

for ? ∈ 0, 1, , so that the lower left corner lies in cdOL and the upper right in

(cd)−1OL. Note that clearly Kc,1(pα) does not depend on the choice of t. Then

M(c, K, µpα)Qp is the canonical Qp-model for the G∗-Shimura variety S∗(K∗c,1(pα)).

Denote by Mur(c, K, µpα) the Zp-scheme obtained by gluing unramified cusps to

134

M(c, K, µpα) (c.f. 3.2.1 of op.cit.). Let Mur(c, K, µpα) be the formal completion of

Mur(c, K, µpα) along its special fiber and

Mur(c, K, µpα) := Mur(c, K, µpα) ∩ M(c, K, µpα)

the open formal subscheme of Mur(c, K, µpα). Furthermore, let Mur(c, K, µpα)ord be

the rigid analytic generic fiber of Mur(c, K, µpα), and

Mur(c, K, µpα)ord := Mur(c, K, µpα)ord ∩M(c, K, µpα)rigQp ,

whereM(c, K, µpα)rigQp denotes the rigid analytic variety associated with the Qp-variety

M(c, K, µpα)Qp . Kisin and Lai defined a system of admissible open strict neighbor-

hoods Mur(c, K, µpα)ord(r) given by pulling back the corresponding open neighbor-

hoods on the tame level Hilbert modular varieties Mur(c, K)ord given by the Hasse

invariant (3.2.2 of op.cit.), and set

Mur(c, K, µpα)ord(r) = Mur(c, K, µpα)ord(r) ∩M(c, K, µpα)rigQp .

Note that here the convention on the indexing parameter r is such that as r → 1− the

open neighborhood Mur(c, K, µpα)ord(r) shrinks towards Mur(c, K, µpα)ord. Finally,

let A →M(c, K, µpα) be the universal abelian scheme, which extends to a universal

semi-abelian scheme A over Mur(c, K, µpα).

When c = d−1, we will omit it from the notation and simply write M(K,µpα),

etc. Then M(K,µpα)Qp is the canonical Qp-model for S∗(K∗1(pα)).

Now we are ready to define the overconvergent Hilbert modular forms on Qp-

model for S∗(K∗1(pα)). Here we use the automorphic bundles introduced in [TX16],

which are slightly more general than the ones used in [KL05], as the former adopts the

135

more flexible double digit weights which makes it easier to later descend to Hilbert

modular surfaces for G. Let ωk,w be the automorphic line bundle on M(K,µpα) of

weight (k, w) defined in 2.12 of [TX16]. Here we are using the weight convention

consistent with the current paper, and the k in op.cit. is precisely our k, whereas

their w, which is just an integer, is our m + 2. The line bundle ωk,w extends to ωk,w

over Mur(K,µpα). Slightly abusing the notations, we will also use the same symbols

to denote their appropriate rigid-analytificatons.

Let S∗(K∗1(pα))rig be the rigid analytic variety associated to S∗(K∗1(pα)), and let

j :Mur(K,µpα)ord → S∗(K∗1(pα))rig be the natural inclusion of rigid analytic varieties.

Definition 8.4.1. (c.f. [KL05], Definition 4.2.1 and Section 3.3, [TX16]) The space

of overconvergent Hilbert modular forms of weight (k, w) and level K∗1(pα) with co-

efficients in an extension E of Qp is

M †k,w(K∗1(pα), E) := H0(S∗(K∗1(pα))rig

E , j†ωk,w).

Note that each section of ωk,w on the (rigid-analytification of the) whole generic

fiber S∗(K∗1(pα)) clearly restricts to an overconvergent section on the strict neighbor-

hoods of Mur(c, K, µpα)ord, and thus the space of classical Hilbert modular forms of

weight (k, w) and level K∗1(pα) canonically embeds into the space of overconvergent

forms M †k,w(K∗1(pα)).

Kisin and Lai showed that the usual Hecke correspondences defined on the Shimura

variety S∗(K∗1(pα)) restrict to correspondences on the strict neighborhoods (c.f. 1.9-

1.11 and 4.1 of [KL05], in particular Lemma 4.1.10), and they induce well-defined

Hecke-operators (4.2.3 of op.cit.). In particular, there is the U(p) operator.

For completeness we now recall the definition of U(p) on forms of weight (2tL, tL)

(i.e. working with trivial coefficient sheaves). Let Mp(K,µpα) denote the moduli

136

scheme classifying triples (A, iQ, ip, C) where A is a d−1-polarized HBAV, iQ a K-

level structure, ip : d−1 ⊗ µpα → A[pα] a µpα-level structure, and C ⊂ A a closed

subscheme stable under the action of OL etale locally isomorphic to the constant

group scheme OL/p and meets ip(µpα) only at the identity section. We have the

natural projection

pr1 :Mp(K,µpα) M(K,µpα) : (A, iQ, ip, C) 7→ (A, iQ, ip).

There is another projection

pr2 :Mp(K,µpα) M(K,µpα)

sending (A, iQ, ip, C) to (A/C, iQ/C, ip/C), where A/C is the HBAV A/C together

with the d−1-polarization as specified in 1.9 of op.cit., and iQ/C (respectively, ip/C)

is the composition of iQ (respectively, ip) with the canonical projection A A/C.

The U(p) operator on cohomology groups is induced by pr1,∗pr∗2.

Remark.

1. Here we are slightly simplifying the construction in 1.9 of op.cit., as we are

looking at a = pOL in their notation, which clearly represents the trivial in the

narrow class group. Thus implicitly we are omitting the data ε : pc ' c, which

is understood to be multiplication by p−1.

2. It can be shown the U(p) correspondence and operator here is equivalent to

the double coset operator associated to

p 0

0 1

, where p is considered as an

element in AL,f via diagonal embedding, and thus the notation U(p) is justified.

Note thatMp(K,µpα)Q is precisely the G∗-Shimura variety S∗(K∗1(pα)∩U0(p))),

137

and pr1 corresponds to the natural projection

S∗(K∗1(pα) ∩ U0(p))) S∗(K∗1(pα)),

whereas pr2 is the composition

S∗(K∗1(pα) ∩ U0(p)))γ' S∗(K∗1(pα) ∩ U0(pα+1)))→ S∗(K∗1(pα))

where γ = Tg with g =

p−1 0

0 1

∈ G∗(AQ,f ), and the second map is the natu-

ral projection. On the other hand, the double coset operator U(p) corresponds

to (pr1 γ−1)∗(pr2 γ−1)∗, which equals to pr1,∗(γ−1)∗(γ

−1)∗pr∗2 = pr1,∗pr∗2.

Following 3.2.5 of op.cit, there is also a Frobenius morphism on the formal scheme

Frobp : M(K,µpα) → M(K,µpα) defined as follows. Let (A, iQ, ip) ∈ M(K,µpα).

Denote by Cm the multiplicative part of A[pm] for each m > 0. Then ip defines

an isomorphism between d−1 ⊗ µpm and Cm for each m ≤ α. Let A′ be the HBAV

A/C1 with the induced d−1-polarization, and i′p : d−1 ⊗ µpα → Kα+1 any morphism

satisfying p · i′p = ip. Then i′p is independent of the choice modulo C1, and thus gives

a well-defined embedding i′p : d−1 ⊗ µpα → A′. Furthermore, let i′Q = iQ/C1 be the

composition with the canonical projection as above. Then Frobp is the map sending

(A, ip, iQ) to (A′, i′p, i′Q). Furthermore, thanks to Proposition 3.2.6 of op.cit., Frobp

induces morphisms on the ordinary locus as well as the strict neighborhoods. We

then define the Frobenius operator Φ on the space of overconvergent modular forms

as Frob∗p.

Furthermore, we have the diamond operators at p. The group K∗0(pα)/K∗1(pα)

acts on S∗(K∗1(pα)) via the morphism Tg for g ∈ K∗0(pα). It is naturally identified

with the Galois group of the covering S∗(K∗1(pα)) S∗(K∗0(pα)). Since the ordinary

138

locus Mur(K,µpα)ord and strict neighborhoods Mur(K,µpα)ord(r) are all pullbacks

of the respective rigid spaces along the natural projection S∗(K∗1(pα)) S∗(K∗),

which factors through S∗(K∗0(pα)), we see K∗0(pα)/K∗1(pα) acts on the ordinary locus

and strict neighborhoods. In particular, K∗0(pα)/K∗1(pα) acts on M †k,w(K∗1(pα), E),

extending the usual diamond operator action on the classical Hilbert modular forms.

Finally, we introduce the Sp operator defined in 3.10 of [TX16], which corre-

sponds to the Hecke action of

$−1p 0

0 $−1p

∈ G∗(AQ,f ), where we recall that

$p ∈ AL,f is the element which is p at places above p and 1 everywhere else. Let

πSp : M(K,µpα) → M(K,µpα) be the map sending (A, iQ, ip) to (A/A[p], i′′Q, i′′p),

where i′′p is any embedding of d−1 ⊗ µpα into A such that p · i′′p = ip, inducing a well-

defined embedding i′′ : d−1 ⊗ µpα → A/A[p], and i′′Q = iQ/A[p] as before. Then Sp

operator is given by π∗Sp .


πSp pr1 = Frobp pr2

as morphisms Mp(K,µpα)→M(K,µpα).

Proof. Follow immediately from the definitions.

Corollary 8.4.3. We have

U(p)Φ = p2Sp

as operators acting on M †2tl,tL

(K∗1(pα)).

Proof. This is essentially Lemma 3.20 of [TX16], and follows from a direct calculation:

U(p)Φ = pr1,∗pr∗2Frob∗p = pr1,∗(Frobp pr2)∗ = pr1,∗(πSp pr1)∗ = pr1,∗pr∗1π∗Sp = p2Sp.

139

Now we would like to descend to our Hilbert modular surface S∗(K∗(pα)) in ques-

tion. This is possible because the condition on determinant of G∗ implies that the

congruence condition on the upper left corner of K∗1(pα) = K1(pα) ∩G∗(Af ) requires

its p-component to come from Z×p modulo pα, so that K∗1(pα) ⊂ K∗(pα).

Recall we have K∗0(pα)/K∗1(pα) acting as diamond operators. Consider the sub-

group K∗(pα)/K∗1(pα). Note that

1 0

0 ∆(a)

: a ∈ Z×p /(1 + pαZp) is a set of rep-

resentatives for K∗(pα)/K∗1(pα), where ∆ : Z×p /(1 + pαZp) → O×L,p/1 + pαOL,p is the

diagonal embedding. We denote by D∗α this group of representatives. Then D∗α is nat-

urally identified with the Galois group of the covering map S∗(K∗1(pα))→ S∗(K∗(pα)).

We have seen that D∗α act on the ordinary locus and the strict neighborhoods. In

particular, by quotienting out the action of the Galois group, we obtain the ordinary

locus Sur,ord of S∗(K∗(p

α))rig, as well as a system of strict neighborhoods of Sur,ord .

Therefore, still using j : Sur,ord → S∗(K∗(p

α))rig to denote the natural inclusion, we

may define

M †k,w(K∗(p

α), E) := H0(S∗(K∗(pα))rig

E , j†ωk,w).

As noted in Remark 2.17 in [TX16], one has the Θ operators on the overconver-

gent modular forms. In our particular case of interest of forms of ((2, 0), (1, 0)) and

((0, 2), (0, 1)), the Θ operators become

d1 : M †(0,2),(0,1)(K

∗(p

α), E)→M †2L,tL

(K∗(pα), E)

and

d2 : M †(2,0),(1,0)(K

∗(p

α), E)→M †2L,tL

(K∗(pα), E),

where for an overconvergent form f with q-expansion f =∑

λ aλqλ on one component

one has d1f =∑

λ aλqλ and d2f =

∑λ λaλq

λ. Here implicitly we are using our fixed

140

embedding Q → Qp corresponding to a place of Q above the place p of L, and λ ∈ L

is the algebraic conjugate. Translating to the p-adic Fourier coefficients, one sees that

ap(y, d1f) = yp1ap(y, d1f) and ap(y, d2f) = yp2ap(y, d1f).

Replacing the real quadratic field L with Q, one obtains the usual elliptic over-

convergent modular forms of Coleman as in [Col97]. More precisely, writing Yα :=

Y (K ′0(pα)), one has the space of overconvergent modular forms of weight k and level

K ′(pα) given by

M †k(K ′(p

α), E) := H0(Y rigα , j†ωk),

where j : Y ordα → Y rig

α is again the natural inclusion of rigid spaces and j† is with

respect to a system of strict neighborhoods of Y ordα .

Lemma 8.4.4. The diagonal embedding ι : Yα → S∗(K∗(pα)) maps ordinary locus

to ordinary locus, restricting to a map Y ordα → S(K(p

α)),ord.

Proof. Note that ζ sits in the following commutative diagram

Y (′K1(pα)) ι1 //

S∗(K∗1(pα))

Yα ι // S∗(K∗(p

α)).

By the moduli description of the diagonal embedding ι1 on the top, it extends to

an embedding of the incomplete integral models, and thus ordinary loci. Moreover,

the two vertical projections clearly map the respective ordinary locus of the modular

variety on the top to the ordinary locus of the bottom, and since ordinary loci of

both Y (K ′1(pα)) and Yα are the pullback of the ordinary locus of Y (K ′), we see the

vertical map on the left induces a surjection on the ordinary loci. The claim of the

lemma thus follows from the commutativity of the diagram.

Thus a system of strict neighborhoods of S∗(K∗(pα))ord also restricts to a system

141

of strict neighborhoods of Y ordα . Furthermore, by the theory of dual BGG complexes

(c.f. Section 2 of [TX16]), we have a quasi-isomorphic embedding of the dual BGG

complex on S(K(pα))

ω0,0 → ω(0,2),(0,1) ⊕ ω(0,2),(0,1) → ω2tL,tL

into the de Rham complex

O(S∗(K∗(pα)))→ Ω1(S∗(K∗(p

α)))→ Ω2(S∗(K∗(pα))),

which in particular gives a restriction map

ι∗(ω(0,2),(0,1) ⊕ ω(0,2),(0,1)) → ι∗Ω1(S(K(pα))) = Ω1(Yα) = ω2.

Therefore we have a natural diagonal restriction map

ι∗ : M †(0,2),(0,1)(K(p

α), E)⊕M †(2,0),(1,0)(K(p

α), E)→M †2(K ′(p

α), E).

8.5 Relation to p-adic modular forms.

We are now in a situation very similar to that of [DR17]: both are trying to compute

the pairing on a modular curve of a unit root lifting of an ordinary elliptic modular

form with some de Rham cohomology class obtained from a syntomic cohomology

class on a surface, only that in our case it comes from a Hilbert modular form on a

Hilbert modular surface, instead of a product of elliptic modular forms on the product

of two modular curves. We first briefly summarize the set-up of at the beginning of

Chapter 4 of [DR17].

142

The compactified modular curve X0(p) := X(K ′0(p)) admits a proper regular

model X0(p)/Zp whose special fiber is the union of two curves, each isomorphic to the

special fiber of the smooth integral model X of X(K ′). One writes W∞,W0 for

the standard admissible covering of X0(p)an by wide open neighborhoods obtained as

the inverse image under the specialization map of the two distinguished curves in the

special fiber. The two wide open neighborhoods are interchanged by the involution

λ1 : X0(p)an → X0(p)an defined over Qp(ζp). The de Rham cohomology group

H1dR(X0(p)/Qp) is endowed with an action of a Frobenius map Φ commuting with the

Up Hecke operator. The ordinary unit root subspace

H1dR(X0(p)/Qp)

ord,ur ⊆ eordH1dR(X0(p)/Qp)

is spanned by the eigenvectors of Φ whose eigenvalue is a p-adic unit. The natural

map

resW∞ : H1dR(X0(p)/Qp)

ord,ur 0−→ H1rig(W∞)

induced by restriction is the zero map, while

resW0 : H1dR(X0(p)/Qp)

ord,ur[φ]∼−→ H1

rig(W0)[φ]

is an isomorphism for any eigenform φ ∈ S2,1(K ′; Q) ([DR17], Lemma 4.2).

Lemma 8.5.1. For any ω ∈ H1dR(X(K ′0(pα))/Qp) the de Rham pairing

⟨(λα)∗ω, (π1,α)∗η

⟩dR

=⟨eordω, (π2,α)∗(λ1)∗η

⟩dR

depends only on the p-adic cuspform associated to eordω.

143

Proof. Since the class η ordinary and eord (π1,α)∗ = (π1,α)∗ eord we can compute

⟨(λα)∗ω, (π1,α)∗η

⟩dR

=⟨e∗ord(λα)∗ω, (π1,α)∗η

⟩dR

=⟨(λα)∗eordω, (π1,α)∗η

⟩dR

=⟨eordω, (λα)∗(π1,α)∗η

⟩dR

=⟨eordω, (λ

−1α )∗(π1,α)∗η

⟩dR

=⟨eordω, (π2,α)∗(λ−1

1 )∗η⟩

dR

=⟨eordω, (π2,α)∗(λ1)∗η

⟩dR,

where we have used the fact λα is an isomorphism and it intertwines π1 and π2 (see

(6.7)). Finally, note the class η is supported on the wide open W0. Hence, from the

explicit description of the Poincare pairing in Equation (109) of [DR17] and the fact

that the involution λ1 : X0(p)an → X0(p)an interchanges W0 with W∞, we see that

the pairing depends only on the restriction resW∞(pα)

(eordω

).

We now describe eordζ∗[ιBdR(w)] in terms of p-adic cuspforms. Recall L/Q is a

real quadratic field and pOL = p1p2 is a split prime, and that the cuspform gP ∈

S2tL,tL(K,t(pα);O) is an eigenform for the Hecke operators Up1 , U

∗p2

with eigenvalues

Up1 gP = α1,gP· gP and U∗p2

gP = α2,gP· gP = α−1

2,gPp · gP.

Thus, the cuspform (wpα2)∗gP ∈ S2tL,tL(K(p

α);O) is an eigenform for the Up-operator

and Corollary 4.3.8 allows us to compute the Hecke eigenvalue by

Up · (wpα2)∗gP = (wpα2

)∗Up1U∗p2〈$p2 , 1〉gP

=(χθ

−1L χ−1($p2) · α1,gP

α−12,gP

p)· (wpα2

)∗gP

= χθ−1L χ−1($p2) · α1,gP

α−12,gP

p · (wpα2)∗gP.

(8.7)

144

Working over Hilbert-Blumenthal varieties.

To further simplify notation, in the rest of this chapter we will write S = S∗(K∗(pα)).

We also fix a toroidal compactification of S over Qp, which we denote by Stor.

We now need to slightly modify the operators Up1 and Up2 to overcome the minor

technical issues that they do not commute with ν∗α and they do not descent to the

Hilbert-Blumenthal variety S = S∗(K∗(pα)).

Suppose p1 is narrowly principal in OL, p1 = p1OL for a totally positive generator

p1 ∈ OL and set p2 := p/p1 ∈ OL, where p is considered as an element in A×L,f via

diagonal embedding. Arguing as in Lemma 4.3.4, the Hecke operator U(p1), which is

by definition the double coset operator associated with

p1 0

0 1

, commutes with ν∗α.

Moreover, as p1$−1p1∈ O×L , we see U(p1) and Up1 only differs by a diamond operator.

More precisely,

U(p) = 〈1, p−1$p〉Up, U(pi) = 〈1, p−1i $pi〉Upi for i = 1, 2.

We also note that as p1, p2 ∈ G = G(Q)+G∗(AQ,f ), the operators U(p1) and U(p2) are

well-defined on S and commute with the natural map ξ : S → S(K(pα)).

Lemma 8.5.2. The cuspform gP := ξ∗(wpα2)∗gP is an eigenform for the Hecke opera-

tors U(p1) and U(p2)

U(p1) · gP = α1 · gP, U(p2) · gP = α2 · gP,

where

α1 := χθ−1L χ−1((p1)−1

p2) · α1,gP

and α2 := χθ−1L χ−1((p2)−1

p2$p2) · α−1

2,gPp.

145

Proof. We observe that

U(p1) · gP = ξ∗ U(p1) (wpα2)∗gP

= ξ∗ (να)∗ U(p1) (Tτp2)∗gP.

As

U(p1) (Tτp2)∗ = Up1 〈1, p−1

1 $p1〉 (Tτp2)∗

= Up1 (Tτp2)∗ 〈(p1)−1

p2, p−1

1 $p1 · (p1)2p2〉

= (Tτp2)∗ Up1 〈(p1)−1

p2, p−1

1 $p1 · (p1)2p2〉,

where in the second quality we used (4.2), we obtain

U(p1) · gP = ξ∗ (να)∗ (Tτp2)∗ Up1 〈(p1)−1

p2, $p1 · (p1)−2

p2〉gP

= χθ−1L χ−1((p1)−1

p2) · α1,gP

· gP.

Similarly,

U(p2) · gP = χθ−1L χ−1((p2)−1

p2$p2) · α−1

2,gPp · gP.

Remark. Note that

α1α2 = α1,gP

α−12,gP

p,

which is the eigenvalue of U(p) for gP.

Similarly, if we define

V (p1) := 〈1, p1$−1p1〉Vp1 , V (p2) := 〈1, p2$

−1p2〉Vp2 and V (p) := V (p1)V (p2),

146

then the action of pV (pi) is precisely right multiplication by

p−1i 0

0 1

∈ G for

i = 1, 2, which is again well-defined on S. Note that

U(pi)V (pi) = UpiVpi = 1,

whereas V (pi)U(pi) is the operator of taking pi-depletion, i = 1, 2.

Finally, we look at the action of the operator Sp on gP.

Lemma 8.5.3. The operator Sp acts trivially on gP.

Proof. By definition, Sp acts as 〈$p, 1〉, which, by the same argument as in Lemma

4.3.4, commutes with να. On the other hand, 〈$p, 1〉 = 〈$pp−1, 1〉 where p is regarded

as an element of AL,f via diagonal embedding, and as $pp−1 has trivial components

at p, it commutes with Tτpα2. Therefore Sp commutes with wpα2

, and we have

Sp · gP = ξ∗(wpα2)∗〈$pp

−1, 1〉gP = ξ∗(wpα2)∗(χθ

−1L χ−1($pp

−1)gP

)= gP,

where in the last equality we used the facts that $pp−1 has trivial p-component and

χ|Q is trivial.

Choice of the polynomial.

Consider

Ri(Ti) = (1− αiTi) for i = 1, 2, R(T ) = 1− α1α2T.

Setting T = T1T2, one can write

R(T ) = R1(T1)R2(T2) + α2T2R1(T1) + α1T1R2(T2).

147

Thus by plugging in T1 = V (p1), T2 = V (p2), so that T = V (p1)V (p2) = V (p), we

see the cuspform R(V (p))gP can be expressed as

R(V (p))gP = (gP)[P] + α2V (p2)(gP)[p1] + α1V (p1)(gP)[p2].

Definition 8.5.4. We choose R(T ) ∈ 1+T ·Fα[T ] to be the characteristic polynomial

of the Frobenius endomorphism p−2Φ acting on H2dR(S)[gP].

By construction, all roots of R(T ) have complex absolute value p−1, and we have

R(p−2Φ)ωP = 0.

Moreover, R(T ) is divisible by R(T ), i.e.,

R(T ) = R(T )R(T )

for some polynomial R(T ) ∈ 1 + T · Fα[T ]

Let ιrig : RΓ•dR(Stor)→ RΓ•rig(Sord) be the natural map induced by restriction from

Stor to the ordinary locus Sord, which is understood to be in the derived category. We

may choose RΓ•dR(Stor) to be an injective resolution of

0→ S0,0(K∗ (p

α))(d1,d2)−→ S(2,0),(1,0)(K

∗ (p

α))⊕S(0,2),(0,1)(K∗ (p

α))d2−d1−→ S2tL,tL(K

∗ (p

α))→ 0,

and RΓ•rig(Sord) to be

0→ S†0,0(K∗ (p

α))(d1,d2)−→ S†(2,0),(1,0)(K

∗ (p

α))⊕S†(0,2),(0,1)(K∗ (p

α))d2−d1−→ S†2tL,tL(K

∗ (p

α))→ 0,

so that ιrig respects the natural inclusion of classical modular forms into overconver-

gent forms.

148

Lemma 8.5.5. Let p be an OL-prime ideal above p. For any overconvergent cuspform

g ∈ S†2tL,tL(K∗(pα)) there exist p-depleted overconvergent forms g1 ∈ S†(2,0),(1,0)(K

∗(p

α))

and g2 ∈ S†(0,2),(0,1)(K∗(p

α)) such that

d2g1 − d1g2 = g[p].

In particular, g1 and g2 both lie in the kernel of the U(p)-operator.

Proof. Since U(p)g[p] = 0 and U(p) acts invertibly on H2rig(Sord), the cuspform g[p]

represents the trivial class in the rigid cohomology. Thus there exist cuspforms g1 ∈

S†(2,0),(1,0)(K∗(p

α)) and g2 ∈ S†(0,2),(0,1)(K∗(p

α)) satisfying d2g1 − d1g2 = g[p]. Since

p-depletion is an idempotent operation commuting with the differential operators d1

and d2, then g[p]1 and g

[p]2 satisfy all the conditions of the lemma.

Hence, there are cuspforms

h1, b1, c1 ∈ S†(2,0),(1,0)(K∗(p

α)) and h2, b2, c2 ∈ S†(0,2),(0,1)(K∗(p

α))

such that

• h1, h2 are P-depleted, and R(V (p))(gP)[P] = d2h1 − d1h2,

• b1, b2 are p1-depleted, and R(V (p))V (p2)(gP)[p1] = d2b1 − d1b2,

• c1, c2 are p2-depleted, and R(V (p))V (p1)(gP)[p2] = d2c1 − d1c2.

Lemma 8.5.6. Set

H1 =(h1 + α2b1 + α1c1

)and

H2 =(h2 + α2b2 + α1c2

),

149

then there is an overconvergent function f ∈ S†0,0(K∗(pα)) such that

ιBdR(w) =(H1 + d1f, H2 + d2f

).

Proof. By the construction of H1 and H2 we see

d2H1 − d1H2 = R(V (p)) ·((gP)[P] + α2V (p2)(gP)[p1] + α1V (p1)(gP)[p2]

)= R(V (p))gP.

As p−2Φ acts as V (p)S−1p = V (p) on gP, we see d(H1, H2) = d · ιBdR(w), which means

ιBdR(w) − (H1, H2) defines a class in H1rig(Stor). Since (H1, H2) consists of p1 and p2

depleted cuspforms, U(p)(H1, H2) = 0, and we can compute that

U(p)[ιBdR(w)− (H1, H2)] = [U(p)ιBdR(w)] = 0 in H1rig(Stor)

since [U(p)ιBdR(w)] is in the image of H1dR(Stor), which is trivial because the geo-

metrically connected components of Stor are simply connected. Therefore the class

[ιBdR(w)− (H1, H2)] is trivial because the U(p)-operator is invertible on H1rig(Stor), and

the existence of such an f follows immediately.

8.6 The formula

Now we are ready to carry out the explicit calculation of the syntomic regulator (8.6).

To avoid overly cumbersome notations, we will use the same notation for an elliptic

modular form of weight 2 and the corresponding differential on the modular curve,

and it shall be clear from the context which one we are referring to.

150

We choose Q(T ) = 1− α−1fT , so that

(R ? Q)(T ) = R(α−1fT)

= R(α−1fT)· (1− α1,gP

α−12,gP

α−1fpT ).

Since αf has complex absolute value√p, while all roots of R(T ) has complex absolute

value p−1, we see Q(p) 6= 0 and R ? Q(1) 6= 0, R ? Q(p−1) 6= 0. If we write

(R ? Q)(T1T2) = a(T1, T2)R(T1) + b(T1, T2)Q(T2),

as in Section 8.3, then a(T1, αf) = 1 because plugging in T2 = αf yields

R(T1) = (R ? Q)(T1 · αf) = a(T1, αf)R(T1).

Combining with Lemma 8.5.1, we see that equation (8.6) simplifies to the following

expression

AJsyn

((λα, id)∗∆

α

)(ωP ⊗ η

)=

−1

(R ? Q)(p−1)

⟨eordι

∗[ιBdR(w)], (π2,α)∗(λ1)∗η⟩

dR,Yα.

(8.8)

Combining the straightforward computation eordι∗(d1f, d2f) = eorddι

∗f = 0 with

Lemma 8.5.6, we have

eordι∗[ιBdR(w)] = eordι

∗(H1 + d1f,H2 + d2f)

= eordι∗(H1, H2

). (8.9)

An important observation is that this last classical elliptic cuspform of level K ′(pα)

is independent of the choice of p-adic Hilbert modular forms h1, h2, b1, b2, c1, c2.

Lemma 8.6.1. Suppose (g1, g2) ∈ Mp-adic(2,0),(1,0)(K

∗(p

α)) ⊕Mp-adic(0,2),(0,1)K

∗(p

α)) be a pair

151

of p-adic Hilbert modular forms (not necessarily overconvergent) such that

d2g1 − d1g2 = 0,

and that either g1 is p1-depleted or g2 is p2-depleted. Then

eordι∗(g1, g2) = 0.

Proof. Since this is a statement involving only sections on the identity component of

the Hilbert-Blumenthal surface, we may carry out the computation using the usual

Fourier expansion there.

First, suppose g1 is p1-depleted. For λ ∈ L, write qλ = exp(2πi(λz1 + λz2)), where

λ ∈ L is the algebraic conjugate. Suppose g1 =∑

λ aλqλ and g2 =

∑λ bλq

λ, where λ

runs through the totally positive elements in some lattice in L. Then d2g1− d1g2 = 0

implies

λbλ − λaλ = 0, or equivalently bλ =λ

λ· aλ.

Here aλ and bλ are understood to be in Qp, and so are λ and λ via our fixed embedding

Q → Qp which corresponds to a place of Qp above p1.

On the other hand, as g1 is p1-depleted, d−11 g1 :=

∑λaλλqλ = limi→∞ d

pi−11 g1 is a

well-defined p-adic modular form, and so is ι∗(d−11 g1). Therefore

ι∗(g1, g2)(z) =∑n>0

qn∑λ+λ=n

(aλ + bλ) =∑n>0

nqn∑λ+λ=n

aλλ

= dι∗(d−1

1 g1

)

is killed by eord. The case that g2 is p2-depleted is completely analogous.

152

Corollary 8.6.2. We have the following identity

eordι∗[ιBdR(w)] = R(p−1U−1

p ) · eordζ∗(d−1

2 (ν∗α(gP|τ−1pα2

))[P]).

Proof. Lemma 8.6.1 gives us the freedom to choose any representatives h1, h2, b1,

b2, c1, c2, even if they are not overconvergent. In particular, we may choose h2 =

b1 = c2 = 0, and h1 = d−12

(R(V (p))(gP)[P]

), b2 = −d−1

1

(R(V (p))V (p2)(gP)[p1]

),

c1 = d−12

(R(V (p))V (p1)(gP)[p2]

).

Then equation (8.9) tells that


∗(h1 + α1c1, α2b2

).

Note that for j ≥ 0, the y-th Fourier coefficient of d−11

(V (p)jV (p2)(gP)[p1]

)is zero

unless vp1(y) = j and vp2(y) ≥ j + 1, so that we may apply Lemma 5.3.6 of [LSZ16],

which implies eordι∗(d−1

1

(V (p)jV (p2)(gP)[p1]

))= 0, and hence eordι

∗b2 = 0. Similarly,

eordι∗c1 = 0. Therefore


∗h1.

Now by Proposition 2.11 of [BCF], which tells ι∗V (p) = p−1V (p)ι∗, we see

ι∗h1 = R(p−1V (p)) · ι∗d−12 ((gP)[P]) = R(p−1V (p)) · ζ∗

(d−1

2 (w∗pα2 (gP)[P]))

where we recall the definition (gP)[P] = (να ξ)∗(gP|τ−1pα2

) = ξ∗(w∗pα2 (gP)[P]). The

claimed identity now follows immediately from the fact that eordV (p) = U(p)−1eord.

Finally we are ready to calculate the syntomic regulator in terms of p-adic cusp-

forms.

153

Theorem 8.6.3. The following formula holds

AJsyn

((λα, id)∗∆

α

)(ωP ⊗ η

)=

−αα−1f

1− α1,gPα−1

2,gPα−1f

·⟨eordζ

∗(d−12 (w∗pα2 (gP)[P]), f

(p)⟩⟨

f(p) , f

(p)⟩ .

Proof. Combining (8.8) and Corollary 8.6.2 we have

AJsyn

((λα, id)∗∆

α

)(ωP ⊗ η

)=

−1

R(α−1fp−1)

·⟨eordζ

∗(d−12 (w∗pα2 (gP)[P]), (π2,α)∗(λ1)∗R(p−1U−1

p )η⟩

dR,Yα

=−R(α−1

fp−1)

R(α−1fp−1)

·⟨eordζ

∗(d−12 (w∗pα2 (gP)[P]), (π2,α)∗(λ1)∗η

⟩dR,Yα

=−1

1− α1,gPα−1

2,gPα−1f

·⟨eordζ

∗(d−12 (w∗pα2 (gP)[P]), (π2,α)∗(λ1)∗η

⟩dR,Yα

.

Now Proposition 4.6.1 and Corollary 3.1.4 together imply that eordζ∗(d−1

2 (w∗pα2 (gP)[P])

is of level K ′0(p), so we are really pairing with (π1,α)∗ζ∗(d−1

2 (w∗pα2 (gP)[P]) on Yα. Thus

⟨eordζ

∗(d−12 (w∗pα2 (gP)[P]), (π2,α)∗(λ1)∗η

⟩dR,Yα

=⟨eordζ

∗(d−12 (w∗pα2 (gP)[P]), (π1,α)∗(π2,α)∗(λ1)∗η

⟩dR,Y1

=⟨eordζ

∗(d−12 (w∗pα2 (gP)[P]), (λ1)∗Uα−1

p η⟩

dR,Y1

=αα−1f·⟨eordζ

∗(d−12 (w∗pα2 (gP)[P]), (λ1)∗η

⟩dR,Y1

=αα−1f

⟨eordζ

∗(d−12 (w∗pα2 (gP)[P]), f

(p)⟩⟨

f(p) , f

(p)⟩ ,

where in the second equality we used the fact that λ1 intertwines Up and U∗p , and in

the third one, Uα−1p η = αα−1

f· η. This concludes the proof.

154

8.7 An explicit reciprocity law

The main technical result of our work is the following theorem relating the automor-

phic and motivic p-adic L-functions constructed, which can be seen as an instance of

the reciprocity laws:

Theorem 8.7.1. For any G ∈ Sn.o.

(K,t; KG )[G ], let

ζG ,f := −αf ·(1− G (T($p1)T($−1

p2))α−1

f

)∈ IG

then

ζG ,f ·Lmotp (G , f) = L an

p (G , f) in Π⊗Λ IG .

In particular, L motp (G , f) belongs to IG [ζ−1

G ,f].

Proof. By Corollary 6.1.6 it suffices to show that both elements have the same spe-

cializations at all arithmetic points of IG of weight 2. Let P ∈ A(IG ) be an arithmetic

point of weight 2 and character χ of conductor pα. By Proposition 7.1.3,

P LfG = Υ(P) ·

(logBK P

)for Υ(P) =

(α1,gP

α2,gPα−1f

)α ·G(χ−1 · θ−1

Q|Dp

)−1.

By Propositions 6.4.3

L motp (G , f)(P) =

⟨P Lf

G

(κfp (G )

), (λα)∗ωGP

⊗ η⟩

dR

= Υ(P) ·⟨

logBK

(κfp (G )(P)

), (λα)∗ωGP

⊗ η⟩

dR.

155

From the definition of κn.o.α in (4.18) and Theorem 8.6.3 we continue with

L motp (G , f)(P) = α−α1,gP

·Υ(P) ·⟨AJsyn(∆α), (λα)∗ωGP

⊗ η⟩

dR

= α−α1,gP·Υ(P) · AJsyn

((λα, id)∗∆

α

)(ωGP⊗ η

)=−α−α2,gP

α−1f·G(χ−1 · θ−1

Q|Dp

)−1

1− α1,gPα−1

2,gPα−1f

·⟨eordζ

∗(d−12 (w∗pα2 (gP)[P]), f

(p)⟩⟨

f(p) , f

(p)⟩ .

(8.10)

Comparing with Corollary 4.6.2, we obtain

L autp (G , f)(P) = −αf(1− α1,gP

α−12,gP

α−1f

) ·L motp (G , f)(P)

= ζG ,f(P) ·L motp (G , f)(P), ∀ P ∈ A(IG ),

as desired.

Corollary 8.7.2. For any arithmetic point P ∈ A(IG ), the automorphic p-adic L-

function is non-vanishing at P if and only if the motivic p-adic L-function is non-

vanishing at P:

L autp (G , f)(P) 6= 0 ⇐⇒ L mot

p (G , f)(P) 6= 0.

Proof. The claim follows from Theorem 8.7.1 and the fact that ζG ,f(P) 6= 0 for any

arithmetic point P ∈ A(IG ).

156

Chapter 9

On the equivariant BSD-conjecture

9.1 Artin representations and Selmer groups

Let (W, %) be a d-dimensional self-dual Artin representation with coefficients in a

number field D. Suppose % factors through the Galois group Gal(H/Q) of a number

field H

GQ%

//

%% %%

GLd(D)

Gal(H/Q)+

88

.

Let E/Q be a rational elliptic curve, then its algebraic rank with respect to the Artin

representation % is defined as

ralg(E, %) = dimD E(H)%D,

the dimension of the %-isotypic component E(H)%D = HomGal(H/Q)(%, E(H) ⊗ D) of

the Mordell-Weil group. For p a rational prime and ℘ | p an OD-prime ideal, we can

157

consider the p-adic Galois representations

V℘(E) = H1et(EQ, D℘(1)), W℘ = W ⊗D D℘.

As the Artin representation % is self-dual, the Kummer map allows us to identify the

group E(H)%D with a subgroup of the Bloch-Kato Selmer group H1f (Q,W℘ ⊗ V℘(E)).

Then, local Tate duality together with the global Poitou-Tate exact sequence can be

used to show that global cohomology classes not crystalline at p bound the size of the

%-isotypic component of the Mordell-Weil group of E/Q ([DR17], Section 6.1).

Lemma 9.1.1. Let κ1, . . . , κd ∈ H1(Q,W℘ ⊗ V℘(E)) be global cohomology classes

with linearly independent images in the singular quotient H1sing(Qp,W℘ ⊗ V℘(E)) at

p. Then the %-isotypic part of E(H) is trivial:

ralg(E, %) = 0.

Proof. By Lemma 6.1 of [DR17], the local cohomology group H1(Qp,W℘ ⊗ V℘(E)) is

a 2d-dimensional D℘-vector space and the local Tate pairing induces a perfect duality

of d-dimensional spaces

〈 , 〉 : H1f (Qp,W℘ ⊗ V℘(E))× H1

sing(Qp,W℘ ⊗ V℘(E)) −→ D℘.

The global Poitou-Tate exact sequence implies that the image of the localization at p

locp : H1(Q,W℘ ⊗ V℘(E)) −→ H1(Qp,W℘ ⊗ V℘(E))

is d-dimensional ([DR17], Lemma 6.2). Therefore, the existence of global cohomology

classes κ1, . . . , κd whose localizations generate the singular quotient at p implies that

158

the restriction of locp to the Bloch-Kato Selmer group H1f (Q, V℘(E)⊗W℘) is the zero

map. The commutativity of the diagram

E(H)%D//

_

⊕p|pHomG(Hp/Qp)(W℘, E(Hp)⊗D℘) _

H1f (Q,W℘ ⊗ V℘(E)) 0 // H1(Qp,W℘ ⊗ V℘(E))

and the injectivity of the vertical Kummer maps imply the triviality of the top hori-

zontal morphism. Since E(H)⊗D → E(Hp)⊗D℘ is injective for all OH-prime ideal

p | p, we deduce that dimD E(H)%D = 0.

9.2 Existence of prime p

The goal of this last section is to apply the idea of p-adic deformation to the setting

where the self-dual Artin representation is 4-dimensional and arises as the tensor

induction

As(%) = ⊗-IndQL(%)

of a totally odd, irreducible two-dimensional Artin representation % : GL → GL2(D)

of the absolute Galois group of a real quadratic field L. We suppose that % has

conductor Q and that the tensor induction of the determinant det(%) is the trivial

character. Let E/Q be a rational elliptic curve of conductor N , and for any rational

prime p, we consider the Kummer self-dual p-adic Galois representation of GQ

V%,E = As(%)⊗ Vp(E).

By modularity ([Wil95],[TW95], [PS16]) there is a primitive Hilbert cuspform g% ∈

StL,tL(Q;D) of parallel weight one, and a primitive elliptic cuspform fE ∈ S2,1(N ;Q)

159

of weight 2 associated to % and E respectively. The twisted L-function L(E,As(%), s

)has meromorphic continuation to C and a functional equation centered at s = 1, at

which the L-function is holomorphic.

Before we delve into the arithmetic results, let us first show that we can pick a

prime p satisfying all assumptions. Recall that up till now we have the following

assumptions on p in proving the explicit reciprocity law:

Assumption 9.2.1. Suppose p is a rational prime satisfying the following:

• f is ordinary and non-Eisenstein at p;

• pOL = p1p2 splits into the product of two narrowly principal ideals, and

• there are no totally positive units of L congruent to -1 modulo p.

We first show that (for any given real quadratic field L) the set of prime p satisfying

the last two conditions has positive density.

First of all, we record the following lemma regarding narrow class fields of real

quadratic fields, whose proof follows immediately from some elementary and explicit

calculations and is thus omitted:

Lemma 9.2.2. Let L = Q(√d) be a real quadratic field, and let H+

L denote the

narrow class field of L. Then

• if d ≡ 1 (mod 4) then H+L ∩Q(ζ8) = Q;

• if d ≡ 3 (mod 4) then Q(i) ⊆ H+L and H+

L ∩Q(√±2) = Q;

• if d ≡ 6 (mod 8) then H+L ∩Q(i) = Q, H+

L ∩Q(√

2) = Q and Q(√−2) ⊆ H+

L ;

• if d ≡ 2 (mod 8) then H+L ∩Q(i) = Q, H+

L ∩Q(√−2) = Q and Q(

√2) ⊆ H+

L .

160

Recall we have the lattice of subfields of Q(ζ16)

Q(ζ16)

Q(ζ16)+ F Q(ζ8)

Q(√

2) Q(i) Q(√−2)

Q

(9.1)

where F/Q is the splitting field of the polynomial X4 + 4X2 + 2.

Proposition 9.2.3. Let L = Q(√d) be a real quadratic field, then the primes p ≡ 9

(mod 16) that also split in L into narrowly principal factors have positive density.

Proof. It follows directly from Lemma 9.2.2 that:

• if d ≡ 1 (mod 4) then Q(ζ16) ∩H+L = Q;

• if d ≡ 3 (mod 4) then Q(ζ16) ∩H+L = Q(i);

• if d ≡ 6 (mod 8) then Q(ζ16) ∩H+L = Q(

√−2).

When d ≡ 2 (mod 8) we claim that Q(ζ16) ∩H+L = Q(

√2). Indeed, in this case the

intersection could be either Q(√

2), Q(ζ16)+ or F and we show that the latter two

options cannot occur. Let A denote either Q(ζ16)+ or F . When d = 2 then Q(ζ16) ∩

H+L = Q(

√2) because A/Q(

√2) is ramified at 2. When d 6= 2, then L(

√2)/Q(

√2) is

a proper extension unramified at 2. It follows that L(√

2) ·A/L(√

2) is ramified at 2

and cannot be contained in H+L .

Since the rational primes p ≡ 9 (mod 16) are precisely those totally split in Q(ζ8)

and inert in the extension Q(ζ8) ⊆ Q(ζ16), the analysis above of the intersection

Q(ζ16) ∩H+L together with Chebotarev’s density theorem finishes the proof.

161

Let ε ∈ O×L,+ be a generator of the group of totally positive units and p a rational

prime split in L. Then the requirement that there is no totally positive unit congruent

to −1 modulo p is satisfied when the subgroup 〈ε〉 of (OL/p)×, generated by the

reduction of ε, has odd order.

Lemma 9.2.4. Let ε and L be as above, then the totally real number field L(√ε) is

either equal to L or it is biquadratic over Q. In the latter case, write ε = a+ b√d for

a, b ∈ Z+, then the subfields of L(√ε) are given by

L(√ε)

Q(√

2(a+ 1)) Q(√d) Q(

√2(a− 1))

Q .

Proof. If the fundamental unit of L is not totally positive, then ε is a square in L and

L(√ε) = L. If ε is the fundamental unit, then the number field L(

√ε) is the splitting

field of the polynomial

X4 − TrL/Q(ε)X2 + 1 = (X2 − ε)(X2 − 1/ε),

hence it is biquadratic over Q and totally real. Using the relation NL/Q(ε) = 1 one

sees that (√a+ 1

2+

√a− 1

2

)2

= ε

and the claim follows.

Remark. The number field L( 8√ε) is not Galois over Q. Its Galois closure is obtained

by adding an 8-th root of unity. Indeed J = L( 8√ε, ζ8) is the splitting field of the

162

polynomial

X16 − TrL/Q(ε)X8 + 1 = (X8 − ε)(X8 − 1/ε).

It is clear from this description that J/Q is a solvable extension.

Lemma 9.2.5. Let ε, L and J be as above. Then for all but finitely many primes

p ≡ 9 (mod 16) that are also totally split in J , there is no totally positive unit in L

congruent to −1 modulo p.

Proof. Suppose that p ≡ 9 (mod 16) and totally split in J . If p is an OL-prime

ideal above p then (OL/p)× ∼= (Z/pZ)× and for all but finitely many such primes the

reduction ε of ε modulo p is an 8-th power. It follows that ε generates a subgroup of

order dividing (p − 1)/8. Since p ≡ 9 (mod 16) this order is odd and the subgroup

cannot contain −1.

Corollary 9.2.6. Let L be a real quadratic field, then the set of primes p split

in L with narrowly principal factors and such that there is no totally positive unit

congruent to −1 mod p has positive density.

Proof. By Proposition 9.2.3 and Lemma 9.2.5, all the primes which are totally split

in J , H+L , Q(ζ8) and inert in the extension Q(ζ8) ⊆ Q(ζ16) satisfy the requirements.

Clearly the splitting conditions for J and H+L are compatible, and from Proposition

9.2.3 we know that also the splitting conditions for H+L and Q(ζ16) are compatible too.

We are left to understand J ∩ Q(ζ16). Clearly Q(ζ8) is contained in the intersection

because J = L( 8√ε, ζ8). One can check that

[J : Q] =

16 · 2 if Q(

√2) ⊆ L(

√ε)

16 · 4 if Q(√

2) 6⊆ L(√ε),

163

[L( 4√ε, i) : Q] = 16, and

L( 4√ε, i) ∩Q(ζ16) =

Q(ζ8) if Q(

√2) ⊆ L(

√ε)

Q(i) if Q(√

2) 6⊆ L(√ε).

Suppose by contradiction that Q(ζ16) ⊆ J . Then J = Q(ζ16) · L( 4√ε, i) because

Q(ζ16) ·L( 4√ε, i) is a subfield of the same degree as J . Therefore the natural injection

Gal(J/Q) → Gal(Q(ζ16)/Q)×Gal(L( 4√ε, i)/Q)

produces a contradiction because Gal(J/Q) contains an element of order 8 while the

other two Galois groups have exponent 4. In summary, we showed that J ∩Q(ζ16) =

Q(ζ8) so that all the required splitting conditions are compatible. Chebotarev’s den-

sity theorem finishes the proof.

Proposition 9.2.7. Let K/Q be an S5-quintic extension whose Galois closure K/Q

contains a real quadratic field L. Suppose E/Q is a rational elliptic curve, then there

are infinitely many ordinary primes p for E/Q such that

• p splits in L into narrowly principal factors;

• there is no totally positive unit in L congruent to −1 modulo p;

• the conjugacy class of Frp in G(K/Q) ∼= S5 is that of 5-cycles.

Proof. Since K/L is a non-solvable extension, we deduce that K ∩ J = L and K ∩

H+L = L. Moreover, K ∩ Q(ζ16) is either Q(

√2) or Q according to whether L =

Q(√

2) or not. Given that 5-cycles are in the kernel of the surjection Gal(K/Q)

Gal(L/Q), one can prove the existence of a set of positive density consisting of rational

primes satisfying the listed conditions as in Corollary 9.2.6. It then remains to show

164

that infinitely many of such primes are of good ordinary reduction for the given

elliptic curve. When E/Q does not have complex multiplication, the ordinary primes

have density one so there are infinitely many ordinary primes that satisfy the listed

conditions. When the elliptic curves E/Q has complex multiplication by a quadratic

imaginary field B, a prime is ordinary for E/Q if it splits in B. As this new splitting

requirement is compatible with those coming from the conditions above, Chebotarev’s

density theorem gives the claim.

9.3 Proof of main theorems

Theorem 9.3.1. Let p be a rational prime satisfying Assumption 9.2.1 such that

the eigenvalues of Frobp on As(%) are all distinct modulo p. If g(p) is any ordinary

p-stabilization of g, then the special L-value L(E,As(%), 1

)does not vanish if and

only if the global cohomology class κ(g(p) , f) is not crystalline at p.

Proof. Corollary 3.3.2 shows that the non-vanishing of L(E,As(%), 1

)is equivalent

to the non-vanishing of the automorphic p-adic L-function L autp (G , f) at the weight

1 arithmetic point P ∈ A(IG ) corresponding to g(p) , which in turn is equivalent to

L motp (G , f)(P) 6= 0 by Corollary 8.7.2. Unwinding the definitions in Proposition

7.1.3, this last statement holds if and only if κfp (G )(P) 6= 0, which, in light of

Proposition 5.4.7, is the same as ∂p(κp(g

(p) , f)

)6= 0.

Corollary 9.3.2. Let p be as above. If g(p)% is any ordinary p-stabilization of g%, then

L(E,As(%), 1

)6= 0 ⇐⇒ ∂p

(κ(g(p)

% , fE))6= 0 ∈ H1

sing

(Qp,V%,E

)Proof. This is a special case of Theorem 9.3.1.

165

Theorem 9.3.3. Suppose that N is coprime to Q and split in L, and p be as above.

Then

ran

(E,As(%)

)= 0 =⇒ ralg

(E,As(%)

)= 0.

Proof. As the eigenvalues of Frobp on As(%) are all distinct modulo p, the cuspform

g% has 4 distinct p-stabilizations

g(α1α2)% , g(α1β2)

% , g(β1α2)% and g(β1β2)

% .

Therefore if ran

(E,As(%)

)= 0, by applying Corollary 9.3.2 to each of these four

p-stabilization, we obtain four global cohomology classes

κ(g(α1α2)% , fE

), κ

(g(α1β2)% , fE

), κ

(g(β1α2)% , fE

), κ

(g(β1β2)% , fE

)∈ H1

(Q,V%,E

)which are not crystalline at p. Given the assumption on the Frobenius eigenvalues

of As(%), Proposition 5.4.7 implies that the images of the global classes are linearly

independent in the singular quotient H1sing

(Q,V%,E

). The result follows by invoking

Lemma 9.1.1.

Corollary 9.3.4. Let K/Q be a non-totally real S5-quintic extension of positive

discriminant, such that it contains a real quadratic field L whose fundamental unit

has norm -1. Suppose N is odd, unramified in K/Q and split in L, then

ran(E/K) = ran(E/Q) =⇒ ralg(E/K) = ralg(E/Q).

Proof. By Corollary 4.2 of [For18] there exists a parallel weight one Hilbert eigenform

166

gK over L of level Q prime to N such that As(%gK ) ∼= IndQK1− 1. Then

ran

(E,As(%gK )

)= ran(E/K)− ran(E/Q)

ralg

(E,As(%gK )

)= ralg(E/K)− ralg(E/Q).

Let K/Q denote the Galois closure of K satisfying G(K/Q) ∼= S5, and write L for

the discriminant field of K. Thanks to Proposition 9.2.7 we can choose a rational

prime p - 5NQ split in L such that its Frobenius conjugacy class in G(K/Q) is that

of 5-cycles and satisfies Assumption 9.2.1. Then the eigenvalues of Frobp on As(%gK )

are all distinct modulo p by Proposition 5.3.9 and the corollary follows by applying

Theorem 9.3.3.

167

Part II

Local computations for a

GSp4 × GL2 Euler system

168

Chapter 10

Introduction

10.1 Recent progress in Euler systems

The theory of Euler systems is one of the most powerful tools available for studying

the arithmetic of Galois representations, and especially for controlling Selmer groups.

It was first introduced by Kolyvagin [Kol90], inspired by works of Thaine [Tha88] and

his own [Kol88], and later formalized by Rubin [Rub00]. However, the construction

of Euler systems is a difficult problem, and there are relatively few known examples

of Euler systems.

A common strategy for constructing Euler systems is pushing forward cohomology

classes on certain special cycles of Shimura varieties of varying levels. The prototypical

example is Kolyvagin’s Euler system of Heegner points [Kol88], which uses codimen-

sion one cycles (Heegner points) in modular curves, although one should note that

this is not exactly an Euler system in the sense of [Rub00], but rather a so-called

“anticyclotomic” Euler system. Another related construction is Kato’s Euler system

[Kat04], which utilizes the cup product of Siegel units on modular curves.

Recently a series of examples have been constructed using this technique, includ-

169

ing the work of [LLZ14] (embedding of Shimura varieties corresponding to GL2 →

GL2 ×GL1 GL2), [LLZ18] (GL2 → ResF/QGL2 with F a real quadratic field), and

[LSZ17] (GL2 ×GL1 GL2 → GSp4). The common starting point for all these re-

cent constructions is Siegel units, realized in the appropriate motivic cohomology of

(products of) modular curves. Pushing forward these cohomology classes via the

corresponding closed embedding of Shimura varieties one gets motivic cohomology

classes in the target Shimura variety. The embeddings are suitably “twisted” in order

to get the desired levels in the Shimura variety, which can be then realized as one

base Shimura variety base-changed to various cyclotomic extensions of Q. One then

considers the etale realization of such motivic cohomology classes to find elements in

etale cohomology. Finally, an application of the Hochschild-Serre spectral sequence

yields Galois cohomology classes satisfying the desired norm relations.

In [LLZ14] and [LLZ18], the proof of the norm relations involves explicit yet labori-

ous double coset computations. In [LSZ17], a novel approach using a multiplicity-one

argument in local representation theory is adopted, bypassing the complicated cal-

culations on GSp4 that would have been practically intractable. More specifically,

Loeffler, Skinner and Zerbes use methods of smooth representation theory to reduce

the norm-compatibility statement to a far easier, purely local statement involving

Bessel models of unramified representations of GSp4(Ql). This reduction is possible

thanks to a case of the local Gan-Gross-Prasad conjecture due to Kato, Murase and

Sugano [KMS03], showing that the space of SO4(Ql)-invariant linear functionals on

an irreducible spherical representation of SO4(Ql)× SO5(Ql) has dimension as most

one. This technique promises to be applicable in many other settings where local

multiplicity one results of this type are known. In [Gro18], the norm relations for the

Euler system of Hilbert modular surfaces as first constructed in [LLZ18] is improved

and reproved using this type of multiplicity one arguments.

170

Exactly the same multiplicity one result can also be applied in the setting of the

natural embedding of GL2×GL1 GL2 into GSp4×GL1 GL2, as suggested in [LSZ17] and

[LZ18]. In this part of the thesis, we set up the analogous local representation tools

and establish the key technical results that will lead to norm compatibility relations

for an Euler system for GSp4×GL1 GL2, which is a joint work in progress with Chi-Yun

Hsu and Ryotaro Sakamoto.

10.2 Outline of the proof and main results

We now give a brief overview of this part of the thesis. The main goal is to establish

results in the setting of GL2×GL1 GL2 → GSp4×GL1 GL2 that are analogous to those

in Section 3 of [LSZ17]. In particular, we would like analogues of Proposition 3.10.4

of op.cit., which will lead to norm relations in the vertical (i.e., p-) direction, and

of Corollary 3.10.5, which will lead to tame norm relations. The latter one is the

most crucial, as the vertical norm relations can still be proved relatively easily by

explicit geometric arguments, but a similar argument for tame norm relations would

be extremely tedious.

Construction of Euler system classes

To provide a little bit more of context, we first give a very brief description of the

construction of the proposed Euler system for GSp4 ×GL1 GL2, which follows very

closely the construction in [LSZ17], and therefore our account will also closely follow

the introduction there.

Let H = GL2 ×GL1 GL2 and G = GSp4 ×GL1 GL2. We will construct a G(AQ,f )-

171

equivariant map, the Lemma-Eisenstein map,

LE : I ⊗H(H(AQ,f )) H(G(AQ,f )) −→ lim−→U

H5mot(YG(U),D)

where I is a certain explicit representation of GL2(AQ,f ) regarded as an H(AQ,f )-

representation via the projection to its first factor H GL2, H(−) denotes the

Hecke algebra, U runs over all open compact subgroups of G(AQ,f ), YG(U) is the

Shimura variety for G of level U , D is some relative Chow motive over YG arising

from algebraic representations of G. Let K be a level subgroup of G(AQ,f ), unramified

outside p and a finite set S of primes and with a certain special form at p. For any

integer n coprime to S, the base-extension variety YG(K)×SpecQ SpecQ(µn) is again

a Shimura variety for G of some level Kn ⊂ K. Combining this identification with

some explicit choices of test data as input to LE , we can define the motivic Euler

system classes

zM,m ∈ H5mot(YG(K)×SpecQ SpecQ(µMpm),D),

for m ≥ 0 and M coprime to pS.

To get an Euler system associated to a Galois representation in the usual sense, we

choose Π = σ ⊗ τ an automorphic representation of G satisfying certain conditions,

where σ (respectively, τ) is an automorphic representation of GSp4 (respectively,

GL2). Let WΠ be the p-adic Galois representation associated to Π, and W ∗Π be its

dual representation. Then Π∗f ⊗W ∗Π appears with multiplicity 1 as a direct summand

of lim−→UH4

et(YG(U)Q,D). Choosing a vector ϕ ∈ Πf thus results in a homomorphism

of Galois representations Π∗f ⊗W ∗Π → W ∗

Π, which factors through (Π∗f )K ⊗W ∗

Π if ϕ

is K-invariant. This, combined with the Hochschild-Serre spectral sequence gives a

map

H5et(YG(K)×SpecQ SpecQ(µMpm),D) −→ H1(Q(µMpm),W ∗

Π).

172

The images of the etale realizations of zM,m give a collection of cohomology classes

zΠM,m ∈ H1(Q(µMpm),W ∗

Π), and we will prove that they satisfy the norm-compatibility

relations of an Euler system.

Norm relations

We will focus on the tame norm relations. The vertical norm relations can be dealt

with in a similar fashion and are easier. Proving the norm relation amounts to com-

paring zΠM,m and norm(zΠ

lM,m), where l is a prime away from Mp. We have constructed

a G(AQ,f )-equivariant bilinear pairing

(I ⊗H(H(AQ,f )) H(G(AQ,f ))

)⊗ Πf −→ H1(Q(µMpm),W ∗

Π),

which, via Frobenius reciprocity (for more details see Section 13.2 below), corresponds

to an H(AQ,f )-equivariant pairing

z : I ⊗ Πf −→ H1(Q(µMpm),W ∗Π).

Now the classes zΠM,m and norm(zΠ

lM,m) are given by the values of Z with different

choices of test data vM , vlM ∈ I ⊗ Πf . In our case the representation I will be a

direct sum of principal series τ for GL2, and our choice of the test data is such that

vlM and vM coincide at all places except for l. Moreover, the target of z has trivial

H(AQ,f )-action, it suffices to replace it with the 1 dimensional trivial representation

C. Therefore we are in the situation of comparing the values of different local test

data under the H(Ql)-equivariant pairing

zl : τl ⊗ Πl −→ C. (10.1)

173

By a known case of the Gan-Gross-Prasad conjecture due to [KMS03], the space of

such H(Ql)-equivariant maps is 1-dimensional (the “multiplicity one” input), and we

may construct an explicit basis element zχ,ψ using zeta integrals (Proposition 13.1.3).

Thus the desired norm relations are reduced to local statements involving our explicit

pairing.

Discussion of main results and outline

In the main results of this part of the thesis (Corollary 13.3.3 and Theorem 13.4.1), we

find the appropriate local test data that lead to norm-compatibility and establish the

corresponding local relations. These will constitute the foundation and technical core

of the anticipated Euler system: not only does the proof of the norm compatibility

relations rely heavily on these local results, the local data we constructed so that

we have the correct form of relations also essentially dictates how the Euler system

should be constructed.

The strategy for constructing the local input data to the pairing (10.1) is as follows.

The test vector in τl will be some form of Siegel sections, which is fairly standard. For

the test vector in Πl, we will look at a linear combination of the images of the normal-

ized spherical vector ϕ0 in Πl under various Hecke operators. The intuition behind

looking at such linear combinations is that the normalized zeta integral (12.2) of ϕ0 at

s is, up to a constant factor of L(s, σ⊗τ)−1, a double sum∑

i≥0

∑j≥0 of the Whittaker

function of ϕ0 at certain values (for a precise statement see (12.3)), and (in an over-

simplified situation) hitting ϕ0 with the Klingen, Siegel, or both, Ul-operator will re-

sult in double sums∑

i≥0

∑j≥1,

∑i≥1

∑j≥1, or

∑i≥1

∑j≥1, of the same summands as

before. Thus a linear combination∑

i≥0

∑j≥0−

∑i≥0

∑j≥1−

∑i≥1

∑j≥0 +

∑i≥1

∑j≥1

will leave us with precisely the single term corresponding to i = j = 0, which in our

case is 1, which, after taking the normalizing factor L(s, σ ⊗ τ)−1 into consideration,

174

gives us exactly what we want. In reality, because of the way our explicit bilinear

form zχ,ψ is constructed, we can only use Hecke operators in the Klingen bi-invariant

Hecke algebra to simplify the calculation (this is one of the main technical differences

compared to the situation of [LSZ17]; see the end of the proof of Theorem 13.1.5),

and therefore there will be extra terms showing up, forcing us to use more Hecke

operators to compensate for them.

Finally we give a brief overview of all the chapters. Chapter 2 is preliminary in

nature, where we collect the standard tools in the theory of local representations of

GL2 and GSp4. Chapter 3 is a preparation for constructing the desired local data

and proving the main results. We give explicit presentations of the Hecke operators

we need, and compute their actions on the local zeta integral (Proposition 12.3.2),

thus coming up with the desired linear combination (Corollary 12.3.3), which is the

technical heart of this part of the thesis. Finally in Chapter 4, we put everything

together to deduce the main results.

10.3 Notations and conventions

Throughout this part of the thesis, we will use the following notations.

• Let J be the skew-symmetric 4× 4 matrix over Z given by

1

1

−1

−1

.

We let GSp4 be the group scheme over Z defined by

GSp4(R) := (g, µ) ∈ GL4(R)×GL1(R) | gT · J · g = µJ

for any commutative ring R. We write µ : GSp4 → GL1 for the symplectic

175

multiplier map.

• Put G := GSp4×GL1 GL2 and H := GL2×GL1 GL2, with GSp4 → GL1 the

symplectic multiplier map µ and GL2 → GL1 the determinant map. We also

denote by µ the obvious induced maps G→ GL1 and H → GL1.

• Let ι denote the embedding H → G given by

(

a b

c d

,

a′ b′

c′ d′

) 7→ (

a b

a′ b′

c′ d′

c d

,

a′ b′

c′ d′

).

• For any prime number l, the Klingen congruence subgroup of level ln is the

subgroup of GSp4(Zl) defined by

Kl(ln) = g ∈ GSp4(Zl) : g ≡

∗ ∗ ∗ ∗

∗ ∗ ∗

∗ ∗ ∗

∗

mod ln,

and the Borel congruence subgroup of level ln is the subgroup B(ln) of GSp4(Zl)

consisting of matrices which are upper-triangular modulo ln.

176

• For integers m,n ≥ 0, we define the following compact open subgroups of G(Ql):

Km,n := g ∈ G(Q`) : g ≡ (

∗ ∗ ∗ ∗

∗ ∗ ∗

∗ ∗ ∗

∗

,

∗ ∗∗ ∗

) mod `n, µ(g) ≡ 1 mod `m

Bm,n := g ∈ G(Q`) : g ≡ (

∗ ∗ ∗ ∗

∗ ∗ ∗

∗ ∗

1

,

∗ ∗∗

) mod `n, µ(g) ≡ 1 mod lm.

177

Chapter 11

Local representation theory

We first review the basic theory and tools of local representations of GL2. The basic

reference is Chapter 4 of [Bum97]. Chapter 3 of [LSZ17] and Chapter 1 of [Gro18]

also have excellent summaries of the materials, and we follow their accounts closely.

11.1 Principal series of GL2(Ql)

Let |.| denote the standard l-adic norm on Ql normalized such that |l| = 1/l. For χ

a smooth character of Q×l , we write L(χ, s) for the local L-factor defined as

L(χ, s) = L(χ|.|, 0) =

(1− χ(l)l−s)−1 if χ|Z×l = 1,

1 otherwise.

Definition 11.1.1. Given two smooth characters χ and ψ of Q×l , let I(χ, ψ) be the

space of smooth functions f : GL2(Ql)→ C such that

f(

a b

d

g) = χ(a)ψ(d)|a/d|1/2f(g),

178

equipped with a GL2(Ql)-action of right translation.

This is the normalized induction from the standard Borel group. It is well-known

that if χ/ψ 6= |.|±1, then I(χ, ψ) is an irreducible representation. If χ/ψ = |.|, then

I(χ, ψ) has an irreducible codimension one invariant subspace. If χ/ψ = |.|−1, then

I(χ, ψ) has a one-dimensional invariant subspace and the quotient representation is

irreducible.

Write dx (respectively, d×x, dg) for the Haar measure on Ql (respectively, Q×l ,

GL2(Ql)) normalized so that Zl (respectively, Z×l , GL2(Zl)) has unit volume. There

is a natural pairing I(χ, ψ)× I(χ−1, ψ−1)→ C defined by

〈f1, f2〉 =

∫GL2(Zl)

f1(g)f2(g)dg,

under which I(χ−1, ψ−1) is identified with the dual of I(χ, ψ).

Definition 11.1.2. Let χ and ψ be smooth characters of Ql. A flat section of the

family of representations I(χ|.|s1 , ψ|.|s2) indexed by s1, s2 ∈ C is a function GL2(Ql)×

C→ C, (g, s) 7→ fs1,s2(g) such that for all fixed s1, s2, the function g 7→ fs1,s2(g) is in

I(χ|.|s1 , ψ|.|s2)), and the restriction of fs1,s2 to GL2(Zl) is independent of s1 and s2.

Remark. From the Iwasawa decomposition (Proposition 4.5.2 of [Bum97]), one sees

that every f ∈ I(χ, ψ) extends to a unique flat section.

Next we recall the definition of the intertwining operator. Fix characters χ, ψ of

Q×l and write them as

χ = ξ1|.|s1 , ψ = ξ2|.|s2 ,

where ξ1, ξ2 are unitary characters of s1, s2 ∈ C. Let f ∈ I(χ, ψ). For g ∈ GL2(Ql),

179

consider

Mf(g) =

∫Qlf(w

1 x

1

g)dx,

where w =

−1

1

. This integral is well-defined if Re(s1 − s2) > 0:

Proposition 11.1.3. ([Bum97], Proposition 4.5.6) If Re(s1 − s2) > 0, then the

integral defining Mf(g) is absolutely convergent and defines a nonzero intertwining

map

M : I(χ, ψ)→ I(ψ, χ), f 7→Mf.

The next proposition allows us to extend the intertwining operator to all s1, s2 ∈ C

such that χ 6= ψ.

Proposition 11.1.4. ([Bum97], Proposition 4.5.7) Fix f ∈ I(χ, ψ), and let fs1,s2

denote the flat section passing through f . Then for fixed g ∈ GL2(Ql), the integral

Mfs1,s2(g), originally defined for Re(s1 − s2) > 0, has analytic continuation to all

s1, s2 ∈ C where χ 6= ψ, and defines a nonzero intertwining operator

M : I(χ, ψ)→ I(ψ, χ).

In particular, by Schur’s lemma, when I(χ, ψ) is irreducible (and hence so is

I(ψ, χ)), the intertwining operator M is an isomorphism. On the other hand, when

χ/ψ = |.| (respectively, χ/ψ = |.|−1), the kernel of M is precisely the unique nontrivial

irreducible sub-representation of I(χ, ψ), and M maps the quotient isomorphically

onto the irreducible sub-representation of I(ψ, χ).

180

11.2 Siegel sections

We follow the account of [LSZ17], Section 3.2. See also Section 1.5 of [Gro18].

Definition 11.2.1. Let S(Q2l ,C) be the space of Schwartz functions on Q2

l . For

φ ∈ S(Q2l ,C), we use φ to denote its Fourier transform

φ(x, y) =

∫ ∫el(xv − yu)φ(u, v)dudv,

where el(x) is the standard additive character of Ql taking 1/ln to exp(2πi/ln).

Proposition 11.2.2. ([LSZ17], Proposition 3.2.2) Let φ ∈ S(Q2l ,C) and χ, ψ be

characters of Q×l . There is a well-defined element fφ,χ,ψ ∈ I(χ, ψ) defined by integrals

satisfying

fg·φ,χ,ψ(h) = χ(det g)−1| det g|−1/2fφ,χ,ψ(hg),

fg·φ,χ,ψ(h) = ψ(det g)−1| det g|−1/2fφ,χ,ψ(hg).

In particular, if ψ = |.|−1/2, the map

S(Q2l ,C)→ I(χ, ψ), φ 7→ Fφ,χ,ψ := fφ,χ,ψ

is GL2(Ql)-equivariant.

Proposition 11.2.3. ([LSZ17], Proposition 3.2.3) We have

M(fφ,χ,ψ) =ε(ψ/χ)

L(χ/ψ, 1)fφ,χ,ψ,

where ε(ψ/χ) is the local ε-factor (a non-zero scalar, equal to 1 if χ/ψ is unramified).

Definition 11.2.4. For integers t ≥ 0, we define functions φt ∈ S(Q2l ,C) as follows:

181

• for t = 0, φ0 := ch(Zl)ch(Zl);

• for t > 0, φt := ch(ltZl)ch(Z×l ).

Note that φt is preserved by the action of the group K0(lt) :=

a b

c d

∈

GL2(Zl) : c ≡ 0 mod lt.

Lemma 11.2.5. Let χ, ψ be unramified characters. Then the function fφt,χ,ψ is

supported on B(Ql)K0(lt), where B ⊂ GL2 is the standard Borel subgroup, and

fφt,χ,ψ(1) =

1 if t = 0;

L(χ/ψ, 1)−1 if t ≥ 1.

11.3 Principal series of GSp4(Ql)

Next we collect some basic results on principal series representations of GSp4(Ql). We

follow closely the account of [LSZ17], Section 3.5. The standard reference is [RS07].

Definition 11.3.1. Let χ1, χ2, ρ be smooth characters of Q×l such that

|.|±1 /∈ χ1, χ2, χ1χ2, χ1/χ2. (11.1)

We define χ1 × χ2 o ρ to be the representation of GSp4(Ql) given by the space of

smooth functions f : GSp4(Ql)→ C satisfying

f(

a ∗ ∗ ∗

b ∗ ∗

cb−1 ∗

ca−1

g) =

a2b||c|3/2

χ1(a)χ2(b)ρ(c)f(g),

182

with GSp4(Ql) acting by right translation. The representation χ1 × χ2 o ρ is called

an irreducible principal series.

Remark. This representation has central character χ1χ2ρ2. The condition on χ1, χ2

implies it is irreducible and generic. In fact, this is the only type among the 11 groups

of irreducible, admissible, non-supercuspidal representations of GSp4(Ql) which is

both generic and spherical (see table A.1 of [RS07] and table 3 of [Sch05]).

If η is a smooth character of Q×l , we may regard it as a character of GSp4(Ql)

via the multiplier map. Then twisting χ1 × χ2 o ρ by η results in the representation

χ1 × χ2 o ρη.

Lemma 11.3.2. Let σ = χ1×χ2 o ρ be an irreducible principal series as above, and

η a smooth character of Q×l . Then the twist σ ⊗ η of σ by η is equivalent to η if and

only if at least one of the following conditions is satisfied:

• η = 1;

• η = χ1 and χ21 = 1;

• η = χ2 and χ22 = 1;

• η = χ1χ2 and χ21 = χ2

2 = 1.

Proof. By Theorem 4.2 of [ST93], σ⊗ η = χ1×χ2 o ρη is isomorphic to σ if and only

if (χ1, χ2, ρ) and (χ1, χ2, ρη) are in the same orbit of the Weyl group acting on char-

acters of the diagonal torus. Using the explicit representatives of the 8-element Weyl

group as given in Section 2.1 of [RS07], we see (χ1, χ2, ρη) must be one of the follow-

ing: (χ1, χ2, ρ), (χ2, χ1, ρ), (χ−11 , χ2, ρχ1), (χ2, χ

−11 , ρχ1), (χ1, χ

−12 , ρχ2), (χ−1

2 , χ1, ρχ2),

(χ−11 , χ−1

2 , ρχ1χ2), (χ−12 , χ−1

1 , ρχ1χ2). The desired claim then follows immediately.

183

Definition 11.3.3. Let σ = χ1 × χ2 o ρ be an irreducible principal series as defined

above. The local spin L-factor of σ is defined as

L(σ, s) = L(σ ⊗ |.|s, 0) = L(ρ, s)L(ρχ1, s)L(ρχ2, s)L(ρχ1χ2, s).

We record the following characterization of irreducible generic unramified repre-

sentations.

Proposition 11.3.4. ([LSZ17], Proposition 3.5.3; see also [RS07], Section 2.2) Let

σ = χ1 ⊗ χ2 o ρ be an irreducible principal series. Then σ is unramified if and

only if all three characters χ1.χ2, ρ are all unramified. Moreover, every irreducible,

generic, unramified representation of GSp4(Ql) is isomorphic to χ1 ⊗ χ2 o ρ for a

unique Weyl-group orbit of unramified characters (χ1, χ2, ρ) satisfying (11.1).

Proposition 11.3.5. Let σ = χ1 × χ2 o ρ and τ = I(χ, ψ) be irreducible principal

series representations, of GSp4(Ql) and GL2(Ql), respectively. Suppose neither of the

following cases happens:

• χi = χ/ψ is quadratic for either i = 1 or i = 2;

• χ1χ2 = χ/ψ and both χ1, χ2 are quadratic.

Then σ ⊗ τ remains irreducible as a representation of G(Ql) ⊂ GSp4(Ql)×GL2(Ql).

Proof. Let Z = (aI4, bI2) ∈ GSp4(Ql) × GL2(Ql) : a, b ∈ Q×l be the center of

GSp4(Ql)×GL2(Ql). As σ⊗τ is irreducible as a GSp4(Ql)×GL2(Ql)-representation,

Z acts as scalars. Thus σ ⊗ τ is irreducible as a G(Ql)-representation if and only if

it is also irreducible as an Z ·G(Ql)-representation.

Now we are in the situation of [GK82], Lemma 2.1. By part (d) of the cited

lemma, it suffices to prove there is no character ν = ν1ν2 of GSp4(Ql) × GL2(Ql)

184

trivial on Z · G(Ql), such that σ ⊗ τ ' (σ ⊗ ν1) ⊗ (τ ⊗ ν2). Here ν1 (respectively,

ν2) is a character of Q×l , regarded as a character of GSp4(Ql) via the multiplier map

(respectively, a character of GL2(Ql) via determinant).

The condition that ν is trivial on G(Ql) implies that ν1ν2 = 1. Thus neither of ν1

and nu2 can be trivial. Moreover, τ ' τ ⊗ ν2 implies ν2 = χ/ψ and is quadratic. On

the other hand, Lemma 11.3.2 implies either ν1 is trivial, or equals to one of χ1 and

χ2 and is quadratic, or equals to χ1χ2 and both χ1 and χ2 is quadratic. It is clear

none of the cases is possible given our assumption.

From now on, we will assume σ⊗ τ satisfies the assumption of Proposition 11.3.5.

185

Chapter 12

Hecke operators and zeta integrals

12.1 Local zeta integrals

Let σ (respectively, τ) be a cuspidal automorphic representation of GSp4(Ql) (re-

spectively, GL2(Ql)) with central character ωσ (respectively, ωτ ), and suppose σ is

generic. We fix a character Ψ of Ql, a Whittaker models W of σ with respect to Ψ,

and a Whittaker model W of τ with respect to Ψ−1.

For ϕ1 ∈ σ and ϕ2 ∈ τ , let Wϕ1 and Wϕ2 be their respective Whittaker functions.

For any unramified character η of Q×l , consider the following simplified zeta inte-

gral

`(ϕ1⊗ϕ2, η, s) :=

∫Q×l ×Q

×l

Wϕ1(

xy

x

y

1

)Wϕ2(

xy

)η(xy)|x|s−2|y|sd×xd×y.

(12.1)

More generally, for any ϕ ∈ σ ⊗ τ , we may write it as a finite sum of simple tensors

and define `(ϕ, η, s) by linearity.

186

Definition 12.1.1. Let σ, τ and η be as above. For every ϕ ∈ σ ⊗ τ , define

Z(ϕ, η, s) = L(s, σ ⊗ τ ⊗ η)−1`(ϕ, η, s). (12.2)

Suppose ϕ1 and ϕ2 are the normalized spherical vectors in the respective auto-

morphic representations. To simplify notations, for i, j ∈ Z put

W1(i, j) = Wϕ1(

li+2j

li+j

lj

1

), W2(i, j) = Wϕ2(

li+jlj

),

and X = η(l)l−s. Note that W1(i, j) 6= 0 if and only if i, j ≥ 0, while W2(i, j) 6= 0 if

and only if i ≥ 0. Then we have

`(ϕ1 ⊗ ϕ2, η, s) =∑i≥0

∑j≥0

W1(i, j)W2(i, j)l2(i+j)X i+2j. (12.3)

Proposition 12.1.2. The integral `(ϕ, η, s) is absolutely convergent for Re(s) 0,

and it has analytic continuation to all s ∈ C which lies in C[ls, l−s]. Furthermore, if

ϕ = ϕ1 ⊗ ϕ2 is the normalized spherical vector, we have

Z(ϕ, η, s) = L(2s, η2ω)−1,

where ω = ωσωτ is the central character of σ ⊗ τ .

Proof. Suppose τ = I(χ2, ψ2), and write α = χ2(l), β = ψ2(l). Then ωτ (l) = αβ. We

have

li/2W2(i, j) = αjβjαi+1 − βi+1

α− β= tr(Symi ⊗ detj)(tτ ),

187

the trace of the irreducible algebraic representation of GL2 with highest weight (i, j),

where tτ =

αβ

is the semisimple conjugacy class of the L-group assciated to τ .

Similarly, for the GSp4(Ql) representation σ, let tσ =

ξ1ξ2

ξ1ξ3

ξ2ξ4

ξ3ξ4

be the semisimple conjugacy class in the L-group. By equation (3.2.4) in [Bum89],

l3i/2+2jW1(i, j) = (ξ1ξ2ξ3ξ4)jTi,j(ξ1, ξ2, ξ3, ξ4) = tr(ρi,j ⊗ µj)(tσ),

where Ti,j is the polynomial as defined in equation (3.2.2) of [Bum89], ρi,j is the

finite dimensional irreducible algebraic representation of GSp4 as defined on p.91 of

[Bum89], and µ is the usual similitude character of GSp4. Here we remark that

ρi,j ⊗ µj is precisely V j,i in the notation of [LSZ17]. Therefore

`(ϕ1 ⊗ ϕ2, η, s) =∑i≥0

∑j≥0

tr(Symi ⊗ detj)(tτ )tr(ρi,j ⊗ µj)(tσ)X i+2j.

This last sum is precisely ζ(s,W, f) as defined in the equation above (A.1.3) in the

appendix of [GPSR87]. By Proposition A.1 there, this last quantity equals to

L(2s,∧2(τ ⊗ η)⊗ ωσ)−1L(s, σ ⊗ τ ⊗ η) = L(2s, η2ω)−1L(s, σ ⊗ τ ⊗ η)

as desired.

188

12.2 Double coset computations

In this section we calculate decomposition of various double cosets into right cosets, in

preparation of the computation of the actions of associated Hecke operators. We will

be mostly interested in double cosets of Km,n defined in the introduction. However,

calculations involving the Klingen congruence subgroups Kl(ln) are generally easier.

We first prove a technical lemma allowing us to reduce the decomposition of Km,n-

double cosets into that of Kln-double cosets.

Lemma 12.2.1. Let G be a group, H ⊂ K ⊂ G two subgroups, and g ∈ G any

element. Then the natural map HgH/H → KgK/K is injective if and only if gHg−1∩

H = gKg−1 ∩H. It is surjective if and only if HgK = KgK.

Proof. We have a natural bijection of right coset spaces

K/(gKg−1 ∩K) ∼= KgK/K, γ 7→ γg,

and similarly a bijection γ 7→ γg for K replaced by H, which implies we may look

at the natural map H/(gHg−1 ∩ H) → K/(gKg−1 ∩ K) instead. This last map

is injective if and only if gHg−1 ∩ H = gKg−1 ∩ H. The claim on surjectivity is

straightforward.

Taking G = GSp4(Ql)×GL2(Ql), K = Kl(ln)×GL2(Zl) and H = Km,n, which is

precisely the subgroup of K consisting of elements of the form (k1, k2) with µ(k1) =

det(k2) ∈ 1 + lmZl. Then the condition gHg−1 ∩H = gKg−1 ∩H is satisfied for all g

as conjugation does not change the value of µ on GSp4 or the determinant on GL2.

On the other hand, if g is diagonal, then HgK = KgK, as we can always write any

element of K as the product of an element in H and a diagonal element, and the latter

189

commutes with g. Thus by the above lemma the natural map HgH/H → KgK/K

is a bijection. In particular, we have proven the following:

Corollary 12.2.2. Suppose g = (g1, g2) ∈ GSp4(Ql) × GL2(Zl) and we have the

following decompositions of double cosets:

Kl(ln)g1Kl(ln) =∐i

γiKl(ln), GL2(Zl)g2GL2(Zl) =∐j

γ′jGL2(Zl).

If (γi, γ′j) ∈ Km,ngKm,n for all pairs (i, j), then

Km,ngKm,n =∐i,j

(γi, γ′j)Km,n.

Therefore under mild hypotheses we can build up decomposition of Km,n-double

cosets from the respective coset decomposition for Kl(ln) and GL2(Zl).

First we record deal with the case of GL2(Zl).

Lemma 12.2.3. We have the following decomposition of GL2(Zl) double cosets:

GL2(Zl)

l1

GL2(Zl) =∐

0≤u<l

l u

1

GL2(Zl) t

1

l

GL2(Zl),

GL2(Zl)

l21

GL2(Zl) =∐

0≤u<l2

l2 u

1

GL2(Zl) t∐

0<t<l

l t

l

GL2(Zl)

t

1

l2

GL2(Zl).

Proposition 12.2.4. Assume n ≥ 4. We have the following decompositions of double

190

cosets of the Klingen parabolic Kl(ln) into disjoint union of left cosets:

Kl(ln)

l

l

1

1

Kl(ln) =

∐0≤u,v,w<l

l u w

l v u

1

1

Kl(ln)

t∐

0≤λ,κ<l

l λ κ

1

l −λ

1

Kl(ln),

Kl(ln)

l2

l

l

1

Kl(ln) =

∐0≤λ,µ<l,0≤κ<l2

l2 lλ lµ κ

l µ

l −λ

1

Kl(ln),

Kl(ln)

l3

l2

l

1

Kl(ln) =

∐0≤t,λ<l,0≤µ<l2,0≤κ<l3

l3 l2λ lµ κ

l2 lt µ− tλ

l −λ

1

Kl(ln)

t∐

0≤µ<l,0≤λ<l2,0≤κ<l3

l3 lλ l2µ κ

l µ

l2 −λ

1

Kl(ln),

191

Kl(ln)

l4

l2

l2

1

Kl(ln) =

∐0≤λ,µ<l2,0≤κ<l4

l4 l2λ l2µ κ

l2 µ

l2 −λ

1

Kl(ln),

Kl(ln)

l2

l2

1

1

Kl(ln) =

∐0≤u,v,w<l2

l2 u w

l2 v u

1

1

Kl(ln)

t∐

0≤λ,κ<l2

l2 λ κ

1

l2 −λ

1

Kl(ln)

t∐

0<t<l,0≤λ,µ<l,0≤κ<l2

l2 lλ lµ+ tλ κ

l t µ

l −λ

1

Kl(ln).

Proof. The decompositions follow the same line of argument as in the proof of part

(i) of Lemma 6.1.1 of [RS07], except that instead of the paramodular subgroup K(ln)

the stricter condition on the upper right corner of the Klingen congruence subgroup

means we cannot get rid of the upper right corner in the decomposition.

We sketch the proof of the last decomposition. By Iwahori factorization (c.f.

192

equation (2.7) of [RS07] we have

Kl(ln)

l2

l2

1

1

Kl(ln)

=

1 Zl Zl Zl

1 Zl

1 Zl

1

Z×l

Zl Zl

Zl Zl

Z×l

1

lnZl 1

lnZl 1

lnZl lnZl lnZl 1

l2

l2

1

1

Kl(ln)

=

1 Zl Zl Zl

1 Zl

1 Zl

1

Z×l

Zl Zl

Zl Zl

Z×l

l2

l2

1

1

Kl(ln)

=

1 Zl Zl Zl

1 Zl

1 Zl

1

1

1 Zl

1

1

l2

l2

1

1

Kl(ln)

∪

1 Zl Zl Zl

1 Zl

1 Zl

1

l2

1

l2

1

Kl(ln)

∪

1 Zl Zl Zl

1 Zl

1 Zl

1

1

1 l−1Z×l

1

1

l2

l

l

1

Kl(ln).

193

Here at each step we try to push

l2

l2

1

1

to the left. The second equality

follows from the fact that

l2

l2

1

1

−1

1

lnZl 1

lnZl 1

lnZl lnZl lnZl 1

l2

l2

1

1

=

1

lnZl 1

ln+2Zl 1

ln+2Zl ln+2Zl lnZl 1

is a subgroup of Kl(ln), while the last one follows from the decomposition

GL2(Zl)

l21

GL2(Zl) =

1 Zl

1

l2

1

GL2(Zl)

t

1 l−1Z×l

1

l

l

GL2(Zl) t

1

l2

GL2(Zl).

194

We now analyze each of the three parts in the union. For the first one, since

1 Zl Zl Zl

1 Zl Zl

1 Zl

1

=

1 Zl Zl Zl

1 Zl

1 Zl

1

1

1 Zl

1

1

=

1 Zl Zl

1 Zl Zl

1

1

1 Zl

1

1 Zl

1

and

1 Zl

1

1 Zl

1

commutes with

l2

l2

1

1

, we see the first part reduces to

1 Zl Zl

1 Zl Zl

1

1

l2

l2

1

1

Kl(ln).

For the second one, note

1 Zl Zl Zl

1 Zl

1 Zl

1

=

1 Zl Zl

1

1 Zl

1

1 Zl

1 Zl

1

1

195

and

1 Zl

1 Zl

1

1

commutes with

l2

1

l2

1

, so we are left with

1 Zl Zl

1

1 Zl

1

l2

1

l2

1

Kl(ln).

For the last one, it is a union of left cosets of the form

1 λ µ κ

1 µ

1 −λ

1

1

1 t/l

1

1

l2

l

l

1

Kl(ln)

with λ, µ, κ ∈ Zl and t ∈ Z×l . By multiplying the coset representatives by elements

in Kl(ln) it suffices to use coset representatives with 0 ≤ λ < l and 0 < t < l, and

furthermore we may restrict to those with 0 ≤ µ < l and 0 ≤ κ < l2.

196

Putting them together, we see

Kl(ln)

l2

l2

1

1

Kl(ln) =

⋃0≤u,v,w<l2

l2 u w

l2 v u

1

1

Kl(ln)

∪⋃

0≤λ,κ<l2

l2 λ κ

1

l2 −λ

1

Kl(ln)

∪⋃

0<t<l,0≤λ,µ<l,0≤κ<l2

l2 lλ lµ+ tλ κ

l t µ

l −λ

1

Kl(ln).

It is clear that all of these left cosets are mutually disjoint.

Remark. 1. Let Kl(ln)g1Kl(ln) be any of the five double cosets considered in Propo-

sition 12.2.4 and γKl(ln) be any of the left cosets in the coset decomposition,

then γ can be written as γ = k1g1k2 with k1, k2 ∈ Kl(ln) and µ(k1) = µ(k2) = 1.

2. Similarly, let GL2(Zl)g2GL2(Zl) be either of the two double cosets considered in

Lemma 12.2.3 and γ′GL2(Zl) any of the left cosets in the coset decomposition,

then γ′ can be written as γ′ = k′1g2k′2 with k′1, k

′2 ∈ GL2(Zl) and det(k′1) =

det(k′2) = 1.

In particular, as elements of Kl(ln)×GL2(Zl), (k1, k′1) and (k2, k

′2) both belong to

the subgroup Km,n, so (γ, γ′) ∈ Km,n(g1, g2)Km,n. Therefore the assumption of Corol-

lary 12.2.2 is satisfied and we may compute decomposition of Km,n(g1, g2)Km,n into

197

left Km,n-cosets by combining the respective decompositions for Kl(ln) and GL2(Zl).

12.3 Hecke operators and local zeta integrals

Definition 12.3.1. We define Hecke operators at l given by the following double

coset opertors:

• TKl1 (l) = 1

Vol(Km,n)ch(Km,n(

l

l

1

1

,

l1

)Km,n);

• TKl2 (l) = 1

Vol(Km,n)ch(Km,n(

l2

l

l

1

,

ll

)Km,n);

• TKl3 (l) = 1

Vol(Km,n)ch(Km,n(

l2

l

l

1

,

l21

)Km,n);

• TKl4 (l) = 1

Vol(Km,n)ch(Km,n(

l2

l2

1

1

,

ll

)Km,n);

198

• TKl5 (l) = 1

Vol(Km,n)ch(Km,n(

l3

l2

l

1

,

l2l

)Km,n);

• TKl6 (l) = 1

Vol(Km,n)ch(Km,n(

l4

l2

l2

1

,

l2l2

)Km,n).

• U1(l) = 1Vol(Bm,n)

ch(Bm,n(

l

l

1

1

,

l1

)Bm,n);

• U2(l) = 1Vol(Bm,n)

ch(Bm,n(

l2

l

l

1

,

ll

)Bm,n).

We will compute `(TKli (l)(ϕ1 ⊗ ϕ2), η, s) for i = 1, .., 6 using the double coset

decompositions given in last section.

199

Proposition 12.3.2. We have the following identities:

`(TKl1 (l)(ϕ1 ⊗ ϕ2), η, s) = l2X−1

(∑i≥0

∑j≥1

+∑i≥1

∑j≥0

)W1(i, j)W2(i, j)l2(i+j)X i+2j

+ lX−1∑i≥2

∑j≥0

W1(i, j)W2(i− 2, j + 1)l2(i+j)X i+2j

+ l3X−1∑i≥0

∑j≥1

W1(i, j)W2(i+ 2, j − 1)l2(i+j)X i+2j,

`(TKl2 (l)(ϕ1 ⊗ ϕ2), η, s) = l2X−2

∑i≥0

∑j≥1

W1(i, j)W2(i, j)l2(i+j)X i+2j,

`(TKl3 (l)(ϕ1 ⊗ ϕ2), η, s) = l2X−2

((l − 1)

∑i≥1

∑j≥1

−∑

i=0,j≥1


+ l2X−2∑i≥2

∑j≥1

W1(i, j)W2(i− 2, j + 1)l2(i+j)X i+2j

+ l4X−2∑i≥0

∑j≥1

W1(i, j)W2(i+ 2, j − 1)l2(i+j)X i+2j,

`(TKl4 (l)(ϕ1 ⊗ ϕ2), η, s) = l2X−2

((l − 1)

∑i≥1

∑j≥1

−∑

i=0,j≥1


+ l2X−2∑i≥2

∑j≥0

W1(i, j)W2(i− 2, j + 1)l2(i+j)X i+2j

+ l4X−2∑i≥0

∑j≥2

W1(i, j)W2(i+ 2, j − 1)l2(i+j)X i+2j,

`(TKl5 (l)(ϕ1 ⊗ ϕ2), η, s) = l4X−3

(∑i≥1

∑j≥1

+∑i≥0

∑j≥2


+ l3X−3∑i≥2

∑j≥1

W1(i, j)W2(i− 2, j + 1)l2(i+j)X i+2j

+ l5X−3∑i≥0

∑j≥2

W1(i, j)W2(i+ 2, j − 1)l2(i+j)X i+2j,

`(TKl6 (l)(ϕ1 ⊗ ϕ2), η, s) = l4X−4

∑i≥0

∑j≥2

W1(i, j)W2(i, j)l2(i+j)X i+2j.

Proof. We will prove the identity for TKl4 (l). The others are completely analogous.

200

By Proposition 12.2.4 together with the remark following Lemma 12.2.3, we see

TKl4 (l)(ϕ1 ⊗ ϕ2) = (

∑γ∈C

γ · ϕ1)⊗ (

ll

· ϕ2),

where γ runs through the disjoint union C = C1 t C2 t C3 with

C1 =

l2 u w

l2 v u

1

1

| 0 ≤ u, v, w < l2,

C2 =

l2 λ κ

1

l2 −λ

1

| 0 ≤ λ, κ < l2,

C3 =

l2 lλ lµ+ tλ κ

l t µ

l −λ

1

| 0 < t < l, 0 ≤ λ, µ < l, 0 ≤ κ < l2,

and “·” denotes the action of right multiplication. Thus the zeta integral `(TKl4 (l)(ϕ1⊗

ϕ2), η, s) becomes the following double sum

∑γ∈C

∑i,j∈Z

Wϕ1(

li+2j

li+j

lj

1

γ)Wϕ2(

li+j+1

lj+1

)l2(i+j)X i+2j. (12.4)

201

We will break up the above sum into the contributions of γ in each of C1, C2 and

C3 separately.

• For γ ∈ C1, we have

li+2j

li+j

lj

1

l2 u w

l2 v u

1

1

=

1 li+ju li+2jw

1 liv li+ju

1

1

li+2+2j

li+2+j

lj

1

so by definition of Whittaker functions

Wϕ1(

li+2j

li+j

lj

1

γ) = Ψ(liv)W1(i+ 2, j).

Note that ∑0≤v<l2

Ψ(liv) =

l2 if i ≥ 0;

0 if i = −1 or − 2.

Thus the contribution to equation (12.4) is

l6∑i≥0

∑j≥0

W1(i+ 2, j)W2(i, j + 1)l2(i+j)X i+2j

=l2X−2∑i≥2

∑j≥0

W1(i, j)W2(i− 2, j + 1)l2(i+j)X i+2j.

(12.5)

Here note that the terms with i < −2 in the sum on the left vanishes because

W1(i+ 2, j) = 0, and similarly for terms with j < 0.

202


li+2j

li+j

lj

1

l2 λ κ

1

l2 −λ

1

=

1 ljλ li+2jκ

1

1 −ljλ

1

li+2j+2

li+j

lj+2

1

so we are summing

Ψ(ljλ)W1(i− 2, j + 2)W2(i, j + 1)l2(i+j)X i+2j.

Since ∑0≤λ<l2

Ψ(ljλ) =

l2 if j ≥ 0;

0 if j = −1 or − 2;

the contribution to equation (12.4) is

l4∑i≥2

∑j≥0

W1(i− 2, j + 2)W2(i, j + 1)l2(i+j)X i+2j

=l4X−2∑i≥0

∑j≥2

W1(i, j)W2(i+ 2, j − 1)l2(i+j)X i+2j.

(12.6)


li+2j

li+j

lj

1

l2 lλ lµ + tλ κ

l t µ

l −λ

1

=

1 ljλ li+j(µ + tλ

l) li+2jκ

1 li−1t li+jµ

1 −ljλ

1

li+2j+2

li+j+1

lj+1

1

so we are summing

Ψ(li−1t+ ljλ)W1(i, j + 1)W2(i, j + 1)l2(i+j)X i+2j.

203

We have

∑0≤λ<l,0<t<l

Ψ(li−1t+ ljλ) =∑

0<t<l

Ψ(li−1t)∑

0≤λ<l

Ψ(ljλ)

=

l2 − l if i ≥ 1 and j ≥ 0;

−l if i = 0 and j ≥ 0;

0 if j = −1.

Thus the contribution to equation (12.4) is

l4(l − 1)∑i≥1

∑j≥0

W1(i, j + 1)W2(i, j + 1)l2(i+j)X i+2j

− l4∑j≥0

W1(0, j + 1)W2(0, j + 1)l2jX2j

=l2(l − 1)X−2∑i≥1

∑j≥1

W1(i, j)W2(i, j)l2(i+j)X i+2j

− l2X−2∑j≥1

W1(0, j)W2(0, j)l2jX2j.

(12.7)

Combining equations (12.5), (12.6) and (12.7) in the above three cases, we obtain

the identity as desired.

Corollary 12.3.3. Let Ts be the operator

Ts =1− l−s−2η(l)TKl1 (l) + 2l−2s−3η(l)2TKl

2 (l) + l−2s−3η(l)2TKl3 (l)

+ l−2s−3η(l)2TKl4 (l)− l−3s−4η(l)3TKl

5 (l) + l−4s−4η(l)4TKl6 (l).

Then

`(Ts(ϕ1 ⊗ ϕ2), η, s) = 1.

204

Proof. A direct calculation using Proposition 12.3.2 shows

`(ϕ1 ⊗ ϕ2, η, s)− l−2X`(TKl1 (l)(ϕ1 ⊗ ϕ2), η, s) + 2l−3X2`(TKl

2 (l)(ϕ1 ⊗ ϕ2), η, s)

+ l−3X2`(TKl3 (l)(ϕ1 ⊗ ϕ2), η, s) + l−3X2`(TKl

4 (l)(ϕ1 ⊗ ϕ2), η, s)

− l−4X3`(TKl5 (l)(ϕ1 ⊗ ϕ2), η, s) + l−4X4`(TKl

6 (l)(ϕ1 ⊗ ϕ2), η, s) = 1,

which gives the desired result.

Later on we will be interested in the case where η is the trivial character, so we

will make the following definition

Definition 12.3.4. Let T = T0 denote the Hecke operator

1− l−2TKl1 (l) + 2l−3TKl

2 (l) + l−3TKl3 (l) + l−3TKl

4 (l)− l−4TKl5 (l) + l−4TKl

6 (l).

205

Chapter 13

Towards norm relations

13.1 A local bilinear form

We keep the notations of the last section. Let χ, ψ be two characters of Q×l such that

χ/ψ 6= |.|±1 and χψ = ω−1. Take η = ψ|.|1/2.

Lemma 13.1.1. For ϕ ∈ τ ⊗ σ, let zs,ϕ : H(Ql)→ C be the function defined by

zs,ϕ(h) = Z(h · ϕ, η, s). (13.1)

Then zs,ϕ ∈ I(|.|−sψ−1, |.|sχ−1)⊗ I(|.| 12 , |.|− 12 ).

Proof. This follows from a straightforward calculation.

In particular, we have an H(Ql)-equivariant map

zs : τ ⊗ σ → I(|.|−sψ−1, |.|sχ−1)⊗ I(|.|12 , |.|−

12 ), ϕ 7→ zs,ϕ.

206

Proposition 13.1.2. Let ϕ0 ∈ τ ⊗ σ denote the normalized spherical vector. Then

zs,ϕ0(1) = L(ψ/χ, 2s+ 1)−1

zs,Ts·ϕ0(1) = L(s, σ ⊗ τ ⊗ η)−1

Proof. The first part follows directly from Proposition 12.1.2 and the fact η2ω =

|.|ψ/χ. The second part is a direct consequence of Corollary 12.3.3.

Proposition 13.1.3. Let

〈, 〉 : I(ψ|.|, χ|.|)⊗ I(|.|−1/2|, |.|1/2)× I(ψ−1|.|, χ−1|.|)⊗ I(|.|1/2|, |.|−1/2) −→ C

be the canonical duality pairing. Then the bilinear form zχ,ψ ∈ HomH(I(χ, ψ)⊗ τ ⊗

σ,C) defined by

zχ,ψ(f ⊗ φ) = 〈M(f)⊗ 1, z0,ϕ〉,

is nonzero.

Proof. As χ/ψ 6= |.|±1, according to equation (13.2) below, zχ,ψ(Fφ0 , ϕ0) 6= 0, which

in particular implies zχ,ψ is nonzero.

Remark. The assumption χ/ψ 6= |.| can be lessened just as in [LSZ17]. In fact, as we

are only using the constant function 1 ∈ I(|.|−1/2|, |.|1/2) in the pairing defining zχ,ψ,

it factors through I(ψ|.|, χ|.|)⊗ C× I(ψ−1|.|, χ−1|.|)⊗ C −→ C, where the first C is

trivial sub-representation of I(|.|−1/2|, |.|1/2), and the second C is the trivial quotient

representation of I(|.|1/2|, |.|−1/2). Now write τ = I(χ′, ψ′), and assume the functions

L(σ⊗ψψ′, s) and L(ψ/χ, s+1/2)L(ψ′/χ′, s+1/2) do not both have a pole at s = 1/2.

Then the H(Ql)-equivariant map

σ ⊗ τ z0−→ I(ψ−1, χ−1)⊗ I(|.|12 , |.|−

12 ) I(ψ−1, χ−1)⊗ C

207

factors through the unique irreducible quotient of σ⊗ τ , and has its image landing in

the unique irreducible sub-representation of I(ψ−1, χ−1). This follows directly from

Lemma 3.7.2 of [LSZ17], which in turn builds on an explicit computation using Mackey

theory. Then we may define zχ,ψ(f ⊗ φ) = lims→0 L(ψ/χ, 2s + 1)〈M(fs) ⊗ 1, zs,ϕ〉.

However, for ease of exposition, we will keep assuming χ/ψ 6= |.| and τ is irreducible.

Exactly the same as in [LSZ17], we have the following multiplicity one result:

Theorem 13.1.4. (Kato-Murase-Sugano) Write τ = I(χ′, ψ′). Suppose neither χ/ψ

nor χ′/ψ′ is quadratic or |.|±1. Then

dim HomH(Ql)(I(χ, σ)⊗ τ ⊗ σ,C) ≤ 1.

Proof. This is exactly Theorem 3.7.5 of [LSZ17].

To further simplify notation, fixing the characters χ, ψ, for every φ ∈ S(Q2l ,C),

we write

Fφ := Fφ,χ,ψ = fφ,χ,ψ ∈ I(χ, ψ).

We choose ψ = |.|−1/2 and χ = |.|1/2+kτ , where k ≥ 1 is an integer and τ is a

finite-order unramified character. Then η = 1, and by Proposition 11.2.2, the map

φ 7→ Fφ,χ,ψ is GL2(Ql)-equivariant. Moreover, the assumptions

Theorem 13.1.5. Let z ∈ HomH(Ql)(I(χ, σ)⊗ τ ⊗ σ,C). Then for any t ≥ 1,

z(Fφt , ϕ0) =1

lt−1(l + 1)(1− lk

τ(l))z(Fφ0 , ϕ0),

z(Fφt , U · ϕ0) =1

lt−1(l + 1)L(τ ⊗ σ, 0)−1z(Fφ0 , ϕ0).

Proof. We know HomH(Ql)(I(χ, ψ) ⊗ τ ⊗ σ,C) is 1-dimensional and zχ,ψ spans it, so

it suffices to prove the claims for z = zχ,ψ

208

By definition, Fφt is the value at s = 0 of the Siegel section fφt,χ|.|−s,ψ|.|s , and by

Proposition 11.2.3, we have

M(fφt,χ|.|−s,ψ|.|s) = L(χ/ψ, 1− 2s)−1fφt,ψ|.|s,χ|.|−s .

On the other hand, by Lemma 11.2.5, fφt,ψ|.|s,χ|.|−s restricted to GL2(Zl) is supported

on K0(lt). Since φt is invariant under K0(lt), it follows that fφt,ψ|.|s,χ|.|−s⊗1 restricted

to H(Zl) is a scalar multiple of ch(K0,H(lt)), where K0.H(lt) = g ∈ H(Zl) : g ≡

(

∗ ∗∗

,

∗ ∗∗ ∗

) mod lt. Moreover, since ϕ0 is the spherical vector, it is fixed

by H(Zl), which means zs,ϕ0 is constant on H(Zl). Therefore

〈M(fφt,χ|.|−s,ψ|.|s), zs,ϕ0〉 = L(χ/ψ, 1− 2s)−1

∫K0,H(lt)

fφt,ψ|.|s,χ|.|−s(h)zs,ϕ0(h)dh

= L(χ/ψ, 1− 2s)−1fφt,ψ|.|s,χ|.|−s(1)zs,ϕ0(1)Vol(K0,H(lt))

=

L(χ/ψ, 1− 2s)−1L(ψ/χ, 1 + 2s)−1Vol(H(Zl)) if t = 0;

L(χ/ψ, 1− 2s)−1L(ψ/χ, 1 + 2s)−2Vol(K0,H(lt)) if t > 0.

Here in the last equality we have used Lemma 11.2.5 and Proposition 13.1.2. It follows

that

zχ,ψ(Fφt , ϕ0) = lims→0

L(ψ/χ, 2s+ 1)〈M(fφt,χ|.|−s,ψ|.|s), zs,ϕ0〉

=

L(χ/ψ, 1)−1Vol(H(Zl)) if t = 0;

L(χ/ψ, 1)−1L(ψ/χ, 1)−1Vol(K0,H(lt)) if t > 0.

(13.2)

AsVol(K0,H(lt))

Vol(H(Zl))= 1

[H(Zl):K0,H(lt)]= 1

lt−1(l+1)and L(ψ/χ, 1)−1 = 1−l−1ψ(l)/χ(l) = 1− lk

τ(l),

this proves the first part of the claim.

209

For the second part, note that each TKli (l) is given by double coset operators which

are invariant under K0,H(lt), which means zs,T0·ϕ0 is constant on K0,H(lt). Then a

similar argument as above shows

zχ,ψ(Fφt , T0 · ϕ0) = lims→0

L(χ/ψ, 1− 2s)−1zs,T0·ϕ0(1)Vol(K0,H(lt))

= L(χ/ψ, 1)−1L(0, σ ⊗ τ)−1Vol(K0,H(lt)),

and the desired conclusion follows immediately.

13.2 An application of Frobenius reciprocity

Let H(G(Ql)) denote the Hecke algebra of locally constant compactly supported C-

valued functions on G(Ql), with the algebra defined by convolution:

(ξ1 ∗ ξ2)(g) =

∫G(Ql)

ξ1(gh−1)ξ2(h)dh

for ξ1, ξ2 ∈ H(Ql). We regard the G(Ql)-representation σ⊗ τ as a left H(Ql)-module

via

ξ · ϕ =

∫G(Ql)

ξ(g)(g · ϕ)dg

for any ξ ∈ H(Ql) and ϕ ∈ σ ⊗ τ , so that g1 · (ξ · (g2 · ϕ)) = ξ(g−11 (−)g−1

2 ) · ϕ.

Definition 13.2.1. For smooth representations τ1, τ2 of GL2(Ql) and σ of GSp4(Ql),

we define

X(τ1, (σ ⊗ τ2)∨) := HomH(G(Ql))(τ1 ⊗H(H(Ql)) H(G(Ql)), (σ ⊗ τ2)∨),

where τ1 is regarded as an H(Ql)-representation via projection of H to its first GL2-

factor.

210

Proposition 13.2.2. With notations as above, there is a canonical bijection

X(τ1, (σ ⊗ τ2)∨) ∼= HomH(Ql)(σ ⊗ τ1 ⊗ τ2,C).

Proof. This is analogous to Proposition 3.9.1 of [LSZ17]. Note that τ1 ⊗H(H(Ql))

H(G(Ql)) is the Hecke module corresponding to the smooth G(Ql)-representation

c-IndGHτ1 ([Ren10],III.2.6), where c-Ind denotes compact induction. Thus

X(τ1, (σ ⊗ τ2)∨) = HomG(Ql)(c-IndGHτ1, (σ ⊗ τ2)∨)

∼= HomG(Ql)(σ ⊗ τ2, (c-IndGHτ1)∨).

According to [Ren10], III.2.7, there is a canonical isomorphism

(c-IndGHτ1)∨ ∼= IndGH(τ∨1 ).

Finally, by Frobenius reciprocity for non-compact induction ([Ren10], III.2.5), we see

HomG(Ql)(σ ⊗ τ2, IndGH(τ∨1 )) ∼= HomH(Ql)((σ ⊗ τ2)|H , τ∨1 )

∼= HomH(Ql)(σ ⊗ τ1 ⊗ τ2,C)

as desired.

Unwinding the definitions of all the isomorphisms used above, the bijection in

Proposition 13.2.2 is given explicitly as follows: given Z ∈ X(τ1, (σ⊗ τ2)∨), the corre-

sponding z ∈ HomH(Ql)(σ ⊗ τ1 ⊗ τ2,C) is characterized by

Z(f ⊗ ξ)(ϕ⊗ g) = z(f ⊗ ξ · (ϕ⊗ g)). (13.3)

Lemma 13.2.3. Let σ⊗τ be a smooth representation of G(Ql), where σ (respectively,

211

τ) is a smooth representation of GSp4(Ql) (respectively, GL2(Ql)). For every R ∈

H(G(Ql)), let R′ ∈ H(G(Ql)) be the function defined by R′(g) = R(g−1). Then for

any Φ ∈ (σ ⊗ τ)∨ and ϕ ∈ σ ⊗ τ , we have

Φ(R · ϕ) = R′ · Φ(ϕ).

Proof. By linearity of Φ we have

Φ(R · ϕ) = Φ(

∫G

(Ql)R(g)g · ϕdg) =

∫G(Ql)

R(g)Φ(g · ϕ)dg.

On the other hand,

R′ · ϕ =

∫G(Ql)

R(g−1)g · Φ(ϕ)dg =

∫G(Ql)

R(g−1)Φ(g−1 · ϕ)dg

which equals to Φ(R · ϕ) as required.

In light of the above lemma, we let U ′i(l) denote the Hecke operator associated to

Ui(l) as in the lemma, i = 1, 2. In particular, if we write Ui(l) as

1

Vol(Bm,n)ch(Bm,nuiBm,n)

where ui ∈ G(Ql) is given as in Definition 12.3.1, then U ′i(l) is given by

1

Vol(Bm,n)ch(Bm,nu

−1i Bm,n).

We slightly generalize the notation in Definition 11.2.4:

Definition 13.2.4. For integers s, t ≥ 0, let φ1,s,t ∈ S(Q2l ,C) denote the function

212

ch(lsZl × (1 + ltZl)), and let

K1(ls, lt) =

a b

c d

∈ GL2(Zl) : c ≡ 0 mod ls, d ≡ 1 mod lt.

Remark. Note that when s ≥ t, K1(ls, lt) is a subgroup of GL2(Zl) preserving φ1,s,t.

Now we take τ1 to be S(Q2l ,C), where GL2(Ql) acts via (g · φ)(x, y) = φ((x, y)g).

Let W be an arbitrary smooth complex representation of G(Ql). We denote by X(W )

the space of homomorphisms S(Q2l ,C) ⊗H(H(Ql)) H(G(Ql)) → W which are G(Ql)-

equivariant. In other words, in the definition of X(τ1, (σ⊗τ2)∨) we take τ1 = S(Q2l ,C)

and replace (σ ⊗ τ2)∨ with W .

Lemma 13.2.5. Let ξ ∈ H(G(Ql)) be invariant under left-translation by the princi-

pal congruence subgroup of level lT in H(Zl) for some integer T ≥ 0. Then for any

Z ∈ X(W ) and integers s ≥ t ≥ T , the expression

1

Vol(K1(ls, lt))Z(φ1,s,t ⊗ ξ)

is independent of s and t.

Proof. This is analogous to Lemma 3.10.2 of [LSZ17]. Note that lsZl × (1 + ltZl) is

precisely the orbit of (0, 1) ∈ Q2l under the right action of K1(ls, lt), which implies

that

φ1,T,T =∑γ∈J

γ · φ1,s,t,

where J is a set of representatives for the quotient K1(lT )/K1(ls, lt). We may take

J to be a subset of the principal congruence subgroup of GL2(Zl) modulo lT , so by

assumption for all γ ∈ J , the element (γ, γ) ∈ H(Zl) will act trivially on ξ. By

213

H(Ql)-equivariance of Z, we see

Z(φ1,T,T ⊗ ξ) =∑γ∈J

Z(γ · φ1,s,t ⊗ ξ) =∑γ∈J

Z(φ1,s,t ⊗ (γ, γ)−1 · ξ) = (#J) · Z(φ1,s,t ⊗ ξ)

=Vol(K1(lT ))

Vol(K1(ls, lt))Z(φ1,s,t ⊗ ξ)

which concludes the proof.

In light of Lemma 13.2.5, we will write Z(φ1,∞ ⊗ ξ) for this limiting value.

13.3 Main results towards vertical norm relations

The goal of this section is to prove Corollary 13.3.3, which is an analogue of Proposi-

tion 3.10.4 of [LSZ17] and serves as the building block for the vertical norm relation

of our anticipated Euler system. Results obtained in this part do not reply on the

multiplicity one argument and are proved directly.

Before stating the results, it is a good time to review all the assumptions we have

made on the representations. Except explicitly stated otherwise, the same assump-

tions will also be effective in the next section.

• σ = χ1 × χ2 o ρ is an irreducible principal series of GSp4(Ql), where χ1, χ2, ρ

are smooth characters of Q×l such that |.|±1 /∈ χ1, χ2, χ1χ,χ1/χ2;

• τ = I(χ′, ψ′) is an irreducible principal series of GL2(Ql), where χ′, ψ′ are

smooth characters of Q×l satisfying χ′/ψ′ is neither |.|±1 nor quadratic, so that

Proposition 11.3.5 applies.

• χ = |.|1/2+kτ and ψ = |.|−1/2 are smooth characters of Q×l where k ≥ 1

is an integer, and τ is an unramified character of finite order. Moreover,

χψχ′ψ′χ1χ2ρ2 = 1.

214

Again we remark here that as discussed before, some of the above assumptions can

be lessened, especially the requirements χ′/ψ′ 6= |.| and k ≥ 1.

For a, b, c ∈ Zl and integer m ≥ 1, let

ηa,b,cm = (

1 al−m bl−m

1 bl−m

1 −al−m

1

,

1 cl−m

1

) ∈ G(Ql).

Proposition 13.3.1. Let k ≥ maxm,n be an integer, and c ∈ Zl such that c2 ∈

lkZl. For any Z ∈ X(W ), we have

Z(φ1,∞ ⊗ ch(η0,1,lck+1 Bm,n)) =

U ′2(l)

l2· Z(φ1,∞ ⊗ ch(η0,1,c

k Bm,n)) if k ≥ 1;

U ′2(l)−1

l2−1· Z(φ1,∞ ⊗ ch(η0,1,c

k Bm,n)) if k = 0.

Proof. A direct computation shows that when k ≥ maxm,n, the conjugate of the

principal congruent subgroup of level l2k in H(Zl) ⊂ G(Zl) by η0,1,ck is a subgroup

of Bm,n. This implies both ch(η0,1,ck+1Bm,n) and ch(η0,1,c

k Bm,n) are invariant under the

left action of the principal congruent subgroup of level l2k+2 in H(Zl). By Lemma

13.2.5, we may choose any s ≥ t ≥ T := 2k + 2 and replace φ1,∞ with φ1,s,t in the

computations. We will make an additional assumption s ≥ t + maxk, 1 to further

simplify the calculations later on.

The double coset decomposition in Proposition 12.2.4 together with Lemma 13.2.3

215

implies that

U ′2(l) · Z(φ1,s,t ⊗ ch(η0,1,ck Bm,n))

=∑

0≤λ,µ<l,0≤κ<l2Z(φ1,s,t ⊗ ch(η0,1,c

k (

l2 lλ lµ κ

l µ

l −λ

1

,

ll

)Bm,n)).

Since

η0,1,ck (

l2 lλ lµ κ

l µ

l −λ

1

,

ll

) = (

l2 κ− λl−k

1

,

ll

)ηlkλ,1+lkµ,lck+1 ,

where on the right hand side we are implicitly using the natural inclusion H → G,

by H(Ql)-equivariance we see that each term in the last summation equals

Z((

l2 κ− λl−k

1

,

ll

)−1 · φ1,s,t ⊗ ch(η0,1,ck Bm,n)).

Since s ≥ k + t, we have

l2 κ− λl−k

1

−1

· φ1,s,t = φ1,s+2,t. Thus

U ′2(l) · Z(φ1,s,t ⊗ ch(η0,1,ck Bm,n)) = l2

∑0≤λ,µ<l

Z(φ1,s+2,t ⊗ ch(ηlkλ,1+lkµ,lck+1 Bm,n)). (13.4)

216

First we deal with the case k ≥ 1. Write

A = (

1 + lkµ

1

,

1 + lkµ

−lkλ 1

) ∈ H(Zl),

which fixes φ1,s+2,t, and

B = (

1 + lkµ

1 + lkµ

−lkλ 1

1

,

1 + lkµ+ cλ cµ+ vc2l−k

−lkλ 1− cλ

) ∈ Bm,n

which lies in Bm,n as k ≥ maxm,n and c2 ∈ lkZl. We have

A−1ηlkλ,1+lkµ,lck+1 B = η0,1,lc

k+1 .

Therefore (13.4) becomes

U ′2(l) · Z(φ1,s,t ⊗ ch(η0,1,ck Bm,n)) = l4Z(φ1,s+2,t ⊗ ch(η0,1,lc

k+1 Bm,n)),

which implies

U ′2(l) · Z(φ1,∞ ⊗ ch(η0,1,ck Bm,n)) = l4

Vol(K1(lT ))

Vol(K1(ls, lt))Z(φ1,s+2,t ⊗ ch(η0,1,lc

k+1 Bm,n))

= l2Vol(K1(lT ))

Vol(K1(ls+2, lt))Z(φ1,s,t ⊗ ch(η0,1,lc

k+1 Bm,n))

= l2Z(φ1,∞ ⊗ ch(η0,1,lck+1 Bm,n))

as desired.

Next assume k = 0, which forces m = n = 0, i.e. Bm,n = G(Zl). The same

217

argument works as in the previous case except when µ = l − 1, in which case A no

longer lies in H(Zl). Since

ηλ,l,lc1 = (

1 −λ/l

1

,

1

1

)ηλ,0,01 η0,l,lc1 ,

where η0,l,lc1 ∈ B0,0 = G(Zl) and (

1 −λ/l

1

,

1

1

) fixes φ1,s+2,t as s+ 2 > t+ 1,

we see that each summand with µ = l − 1 in the right hand side of (13.4) equals to

Z(φ1,s+2,t ⊗ ch(ηλ,0,01 B0,0)).

For λ 6= 0, writting B = (

λ1

,

1

−λ

) we have

ηλ,0,01 = Bη0,1,01 B−1

and B ∈ H(Zl) ⊂ B0,0 fixes φ1,s+2,t, so

Z(φ1,s+2,t ⊗ ch(ηλ,0,01 B0,0)) = Z(φ1,s+2,t ⊗ ch(η0,1,01 B0,0))

just as in the cases where µ 6= l − 1. Finally for µ = 0, we clearly have η0,0,01 B0,0 =

η0,1,c0 B0,0 = B0,0. Combining above we see (13.4) becomes

U ′2(l) · Z(φ1,s,t ⊗ η0,1,c0 ch(B0,0))

=l2(l2 − 1)Z(φ1,s+2,t ⊗ ch(η0,1,lc1 B0,0)) + l2Z(φ1,s+2,t ⊗ ch(B0,0,0))

which immediately implies the desired claim by passing to φ1,∞ as in the previous

case.

218

Proposition 13.3.2. Let k ≥ maxm, 1 be an integers, and 0 6= c ∈ Zl such that

lkc−1 ∈ lZl. For any Z ∈ X(W ), we have

Z(φ1,∞ ⊗ ch(η0,1,ck+1Bm,n)) =

U ′1(l)

l3· Z(φ1,∞ ⊗ ch(η0,1,c

k Bm,n)).

Moreover, in the case c = lk, we have

Z(φ1,∞⊗ch(η0,1,lk

k+1 Bm,n)) =U ′1(l) · Z(φ1,∞ ⊗ ch(η0,1,0

k Bm,n))− l2Z(φ1,∞ ⊗ ch(η0,1,0k+1 Bm,n))

l2(l − 1).

Proof. The proof is very similar to that of Proposition 13.3.1. We sketch it below.

Similar to above we can choose s ≥ t ≥ T := 2k + 2 + n and replace φ1,∞ with

φ1,s,t in the computation.

By the double coset decomposition for U1(l) we see

U ′1(l) · Z(φ1,s,t ⊗ ch(η0,1,ck Bm,n))

=∑

0≤u,v,w,x<l

Z(φ1,s,t ⊗ ch(η0,1,ck (

l u v

l w u

l

1

,

l x

1

)Bm,n)).

As

η0,1,ck (

l u v

l w u

l

1

,

l x

1

) = (

l v

1

,

l w

1

)η0,1+lku,c+lk(x−w)k+1 ,

219

the sum become

l∑

0≤u,w,x<l

Z(φ1,s+1,t ⊗ ch(η0,1+lku,c+lk(x−w)k+1 Bm,n)). (13.5)

Since k ≥ 1, η0,1+lku,c+lk(x−w)k+1 is conjugate to η0,1,c

k+1 via the element

(

(1 + lku)2

1 + lkc−1(x− w)

,

(1 + lku)(1 + lkc−1(x− w))

1 + lku

)

in H(Ql) ∩Bm,n, which also fixes φ1,s+1,t. Thus we obtain

U ′1(l) · Z(φ1,s,t ⊗ ch(η0,1,ck Bm,n)) = l4Z(φ1,s+1,t ⊗ ch(η0,1,c

k+1Bm,n))

which gives the desired result as in the previous proof.

Finally, we quickly dispose of the case when c = lk. The above argument still

holds except when l divides 1 + x − w, so that the conjugating matrix is no longer

invertible over Zl. In this case, η0,1+lku,c+lk(x−w)k+1 and η0,1+lku,0

k+1 only differ at the GL2-

component by an element in GL2(Ql), so the two define the same left coset of Bm,n.

As η0,1+lku,0k+1 is conjugate to η0,1,0

k+1 via (

1 + lku

1

,

1 + lku

1

) ∈ H(Ql)∩Bm,n,

we see the contribution of the terms with l dividing 1 + x − w to the sum (13.5) is

l3Z(φ1,s+1,t ⊗ ch(η0,1,0k+1 Bm,n)), and in total we have

U ′1(l) · Z(φ1,s,t ⊗ ch(η0,1,lk

k Bm,n)) =l3(l − 1)Z(φ1,s+1,t ⊗ ch(η0,1,lk

k+1 Bm,n))

+ l3Z(φ1,s+1,t ⊗ ch(η0,1,0k+1 Bm,n)),

which completes the proof.

220

Now choose k = 2m, c = lm in Proposition 13.3.1 and k = 2m + 1, c = lm+1

in Proposition 13.3.2. To further simplify notation let U(l) = U1(l)U2(l) and ηm :=

η0,1,lm

2m . We immediately obtain the following

Corollary 13.3.3. Assume n ≤ 2m. Then for any Z ∈ X(W ), we have

Z(φ1,∞ ⊗ ch(ηm+1Bm,n))

=

U ′(l)l5· Z(φ1,∞ ⊗ ch(ηmBm,n)) if m ≥ 1;

(U ′(l)−1)·Z(φ1,∞⊗ch(Bm,n))−l2(l2−1)Z(φ1,∞⊗ch(η0,1,02 Bm,n))−l2(l2−1)

l2(l−1)(l2−1)if m = 0.

Remark. This is the analogue of Proposition 3.10.4 of [LSZ17], and will be the main

technical tool to prove the norm relations in the vertical (i.e. p-) direction. The

reason we are interested in the U(l) operator instead of only U1(l) or U2(l) as in the

previous two propositions is as follows. Consider the case where we take a cuspidal

Siegel modular eigenform of genus 2 and weight 3 which corresponds to our σ, and

an elliptic eigenform of weight 2 which corresponds to our τ . The degree 8 Galois

representation V = V ∗σ ⊗ V ∗τ we will be interested in, which is the dual of the Galois

representation associated to σ⊗τ , has Hodge-Tate weights 0, 1, 1, 2, 2, 3, 3, 4 (recall

our convention is the cyclotomic character has Hodge-Tate weight 1). Conjecture

7.4 of [LZ17] predicts that to get a rank-1 Euler system we would need to look at

V (−1), which would also need to satisfy the so-called rank-1 Panchishkin condition

(see Definitions 7.1 and 7.2 of [LZ17]). In particular, we would need V (−1) to have

a GQp-stable subspace V + with all Hodge-Tate weights ≥ 1, and V (−1)/V + has all

Hodge-Tate weights ≤ 0. As V (−1) has Hodge-Tate weights −1, 0, 0, 1, 1, 2, 2, 3,

this would require V + to be the 5-dimensional subspace with Hodge-Tate weight

1, 1, 2, 2, 3. In light of [Urb05], Corollary 1, which says that Vσ has a GQp-stable line

if σ is Siegel-ordinary, and a GQp-stable plane if σ is Klingen-ordinary, the existence

221

of such a V + is guaranteed by σ ⊗ τ being both U1(l)-ordinary and U2(l)-ordinary.

13.4 Main result towards tame norm relations

We are now ready to state and prove the following analogue of Corollary 3.10.5 of

[LSZ17], from which the tame norm relations of the anticipated Euler system would

follow directly.

We now go back to the situation where W = (σ⊗τ)∨. We keep all the assumptions

made at the beginning of the last section. Recall the Hecke operator defined in

Definition 12.3.4:

T = 1− l−2TKl1 (l) + 2l−3TKl

2 (l) + l−3TKl3 (l) + l−3TKl

4 (l)− l−4TKl5 (l) + l−4TKl

6 (l).

We will also refer to T as the function on G(Ql) underlying the Hecke operator.

Theorem 13.4.1. Let K = G(Zl). There exists an element ξ ∈ H(G(Ql),Q) of the

form ξ =∑

η∈J mηch(ηK) with J s finite subset of G(Ql) and each mη ∈ Z, so that

Z(φ1,∞ ⊗ ξ) = l4L(τ ⊗ σ, 0)−1Z(φ0 ⊗ ch(K)).

Proof. Translating the second identity of Theorem 13.1.5 in terms of Z together with

the fact that the representation σ ⊗ τ is irreducible and thus generated by ϕ0 gives

Z(φt ⊗T) =1

lt−1(l + 1)Vol(Km,n)L(τ ⊗ σ, 0)−1Z(φ0 ⊗ ch(Km,n))

as elements of X((σ ⊗ τ)∨).

222

Recall φt = ch(ltZl)× Z×l and φ1,t,t = ch(ltZl)× (1 + ltZl) are related via

φt =∑

a∈(Zl/lt)×

1

a

· φ1,t,t,

hence by H(Ql)-equivariance of Z,

Z(φt⊗U)(ϕ0) =∑

a∈(Zl/lt)×Z(φ1,t,t)⊗T(

1

1

a

a

· (−))) = lt−1(l−1)Z(φ1,t,t⊗T),

where in the last identity we used the fact that T is a linear combination of double

coset operators of Km,n, and diagonal matrices are in Km,n. Therefore

Vol(Km,n)Z(φ1,∞ ⊗T) = L(τ ⊗ σ, 0)−1Z(φ0 ⊗ ch(Km,n)), (13.6)

where we have used the fact that [GL2(Zl) : K1(lt, lt)] = l2t−2(l2 − 1).

We would like to make the elements in the Hecke algebra appearing as the second

argument of Z right invariant under K = G(Zl). To achieve it we sum over the action

of representatives of K/Km,n. On the right hand side of (13.6) we obtain (omitting

the factor L(τ ⊗ σ, 0)−1 for brevity)

∑γ∈K/Km,n

γ · Z(φ0 ⊗ ch(Km,n)) = Z(φ0 ⊗∑

γ∈K/Km,n

ch(Km,n)((−) · γ)) = Z(φ0 ⊗ ch(K)).

On the left hand side, recall that l4T is a Z-linear combination of 1 (the identity opera-

tor) and TKli (l) = 1

Vol(Km,n)ch(Km,ntiKm,n) for i = 1, ..., 6 (where ti ∈ G(Ql) is as given

in Definition 12.3.1), each of which is a sum of the form∑

η∈Ji1

Vol(Km,n)ch(ηKm,n),

223

where each Ji is the set of left Km,n-coset representatives of Km,ntiKm,n as given by

Lemma 12.2.3 and Proposition 12.2.4 (see also the remark thereafter; 1 can be in-

terpreted as 1Vol(Km,n)

ch(Km,n), so we set J0 = (I4, I2) the set with only the identity

element of G(Ql)). Therefore we may write

T =1

l4Vol(Km,n)

6∑i=0

mi

∑η∈Ji

ch(ηKm,n)

with mi ∈ Z the corresponding coefficient as given in Corollary 12.3.3, and we have

∑γ∈K/Km,n

γ · Vol(Km,n)Z(φ1,∞ ⊗T) =1

l4

6∑i=0

mi

∑η∈Ji

Z(φ1,∞ ⊗∑

γ∈K/Km,n

γ · ch(ηKm,n))

=1

l4

6∑i=0

mi

∑η∈Ji

Z(φ1,∞ ⊗ ch(ηK)).

Comparing the sum of the two sides of (13.6) gives an identity of exactly the form as

required. This completes the proof.

13.5 Divisibility of mη

In this last section, we record a divisibility result of the multiplicities mη that will be

useful in the future construction of the Euler system.

In fact, once we move on to the actual construction of the Euler system, it will be

clear that in order to obtain p-adically integral cohomology classes, the Hecke element

ξ needs to satisfy the following:

Assumption 13.5.1. Let V = h = (h1, h2) ∈ H(Zl) : h1 =

∗ ∗0 1

be the stabi-

lizer of φ1,∞ in H(Zl), and put Vη = V ∩ ηK0,1η−1, where recall K0,1 = h ∈ H(Zl) :

µ(h) = 1 (mod l). Then one needs the index [V : Vη] to divide the multiplicity mη,

224

as elements of Zp.

One can easily see that the prime-to-l part of V has order (l + 1)(l − 1)2. For

the purpose of arithmetic applications (i.e. bounding Selmer groups), we will be only

interested at primes l congruent to 1 modulo p, so l + 1 will not pose a problem. On

the other hand, in general it could (and indeed often) happen that mη needs to be

divisible by (l − 1)2.

We will prove that ξ can be chosen such that the l − 1 part of the assumption

holds for every η appearing in ξ. More precisely:

Proposition 13.5.2. In the notation of Theorem 13.4.1, the element ξ can be chosen

such that each mη is divisible by l − 1. Furthermore, if the index [V : Vη] is divisible

by (l − 1)2, then so is mη.

The rest of this section is devoted to the proof of this proposition. The overall idea

is simple and has been used in the computations in the last section: many elements

in Ji, the set of coset representatives associated to the double coset operator TKli (l),

are conjugate to each other via elements in V , and thus have the same action on

Z. We will group the representatives into conjugacy classes and show the combined

multiplicities have the desired divisibility. In fact, we will show that most of those

combined multiplicities are divisible by (l − 1)2, and then examine the few rest to

show the corresponding index [V : Vη], after divided by l − 1, is coprime to l − 1 (up

to a power of 2).

Recall the notation

ηa,b,cm = (

1 al−m bl−m

1 bl−m

1 −al−m

1

,

1 cl−m

1

).

225

In addition, we will also write

θa,b,cm = (

1 al−m bl−m

1 bl−m

1 −al−m

1

,

l cl1−m

l−1

)

and

ιa,b,cm = (

1 al−m bl−m

1 bl−m

1 −al−m

1

,

l−1 cl1−m

l

).

Lemma 13.5.3. For i = 0, 1, ..., 6, let ti ∈ G(Ql) be the matrix defining the double

coset associated to TKli (l) as in Definition 12.3.1 (as before t0 is understood to be

the identity element (I4, I2)), and Ji the set of left Km,n-coset representatives of

Km,ntiKm,n as given by Lemma 12.2.3, Proposition 12.2.4 and the remark thereafter.

Then for each i = 0, 1, ..., 6, the sum

Si :=∑η∈Ji

Z(φ1,∞ ⊗ ch(ηK)) (13.7)

can be put into the form of

Si =∑η′∈J ′i

nη′,i · Z(φ1,∞ ⊗ ch(η′K)), (13.8)

where J ′i is a finite subset of G(Ql), and nη′,i ∈ (l − 1)Z, except when η is one of

η0,0,01 , θ0,0,0

1 , ι0,0,01 , in which case we only have nη′,i ∈ Z.

Proof. This lemma follows from computations similar to those in Proposition 13.3.1.

226

We will only prove it for TKl5 (l), and the rest follows analogously.

Note that J5 is the disjoint union of the following four subsets:

J15 = (Aλ,µ,t,κ1 , Bx

1 ) : 0 ≤ t, λ, x < l, 0 ≤ µ < l2, 0 ≤ κ < l3,

J25 = (Aλ,µ,t,κ1 , B2) : 0 ≤ t, λ < l, 0 ≤ µ < l2, 0 ≤ κ < l3,

J35 = (Aλ,µ,κ2 , Bx

1 ) : 0 ≤ λ, x < l, 0 ≤ µ < l2, 0 ≤ κ < l3,

J45 = (Aλ,µ,κ2 , B2) : 0 ≤ λ < l, 0 ≤ µ < l2, 0 ≤ κ < l3,

where to simplify notation we write

Aλ,µ,t,κ1 =

l3 l2λ lµ κ

l2 lt µ− tλ

l −λ

1

, Aλ,µ,κ2 =

l3 lµ l2λ κ

l λ

l2 −µ

1

,

and

Bx1 =

l2 lx

l

, B2 =

ll2

.

We also let ηa,b,cm denote (

1 al−m bl−m

1 bl−m

1 −al−m

1

,

1 cl−m

1

).

For representatives in J15 , we have

(Aλ,µ,t,κ1 , Bx1 ) = (

l3 κ

1

,

l2 lt

l

) · ηlλ,µ,l(t−x)2

227

and

l3 κ

1

−1

· φ1,s,t = φ1,s+3,t, so like in the proof of Proposition 13.3.1 the contri-

bution of J15 to (13.7) is

l ·∑

0≤λ,y<l,0≤µ<l2Z(φ1,∞ ⊗ ch(ηlλ,µ,ly2 K)).

We split the above sum into four pieces based on whether λ and y are zero or not.

When λy 6= 0, ηlλ,µ,ly2 is conjugate to ηl,µy−1λ−1,l

2 via (

λ2y

1

,

λyλ

), which

fixes φ1,s,t and belongs to K. Thus the contribution is

l∑

0<λ<l

∑0<y<l

∑0≤µ<l2

Z(φ1,∞ ⊗ ch(ηl,µy−1λ−1,l

2 K))

=l(l − 1)2∑

0≤µ<l2Z(φ1,∞ ⊗ ch(ηl,µ,l2 K)).

When λ 6= 0 but y = 0, ηlλ,µ,ly2 is conjugate to ηl,1,02 via (

λµ1

,

µλ

) if

l - µ, and conjugate to ηl,l,02 = η1,1,01 via (

λµ/l1

,

µ/lλ

) if l|µ but l2 - µ, and

conjugate to ηl,0,02 = η1,0,01 via (

λ1

,

1

λ

) if µ = 0, so by the same argument

as in the previous case the contribution is

l∑

0<λ<l

∑0≤µ<l2

Z(φ1,∞ ⊗ ch(ηl,µ,02 K))

=l2(l − 1)2Z(φ1,∞ ⊗ ch(ηl,1,02 K)) + l(l − 1)2Z(φ1,∞ ⊗ ch(η1,1,01 K))

+ l(l − 1)Z(φ1,∞ ⊗ ch(η1,0,01 K)).

228

When λ = 0 and y 6= 0, ηlλ,µ,ly2 is conjugate to η0,1,l2 via (

µyµ

,

y1

) if

l - µ, and conjugate to η0,l,l2 = η0,1,1

1 via (

µ/lyµ/l

,

y1

) if l|µ but l2 - µ,

and conjugate to η0,0,l2 = η0,0,1

1 via (

1

y

,

y1

) if µ = 0. Thus like above the

contribution in this case is

l∑

0<y<l

∑0≤µ<l2

Z(φ1,∞ ⊗ ch(η0,µ,ly2 K))

=l2(l − 1)2Z(φ1,∞ ⊗ ch(η0,1,l2 K)) + l(l − 1)2Z(φ1,∞ ⊗ ch(η0,1,1

1 K))

+ l(l − 1)Z(φ1,∞ ⊗ ch(η0,0,11 K)).

Next assume λ = y = 0. When l - µ a completely analogous argument shows we

have

l(l2 − l)Z(φ1,∞ ⊗ ch(η0,1,02 K)),

and when µ = lµ′ with 0 < µ′ < l, we have η0,µ,02 = η0,µ′,0

1 and analogously we obtain

l(l − 1)Z(φ1,∞ ⊗ ch(η0,1,01 K)).

Finally we are left with η0,0,02 , which is the identity matrix.

Combining the above four cases, we see the contribution of J15 to the sum in (13.7)

229

is given by

l(l − 1)2( ∑

0≤µ<l2Z(φ1,∞ ⊗ ch(ηl,µ,l2 K)) + lZ(φ1,∞ ⊗ ch(ηl,1,02 K)) + Z(φ1,∞ ⊗ ch(η1,1,0

1 K))

+lZ(φ1,∞ ⊗ ch(η0,1,l2 K)) + Z(φ1,∞ ⊗ ch(η0,1,1

1 K)))

+l(l − 1)(Z(φ1,∞ ⊗ ch(η1,0,0

1 K)) + Z(φ1,∞ ⊗ ch(η0,0,11 K)) + lZ(φ1,∞ ⊗ ch(η0,1,0

2 K))

+Z(φ1,∞ ⊗ ch(η0,1,01 K))

)+ lZ(φ1,∞ ⊗ ch(K))

(13.9)

In summary, the can be put into the desired form. A similar computation shows

that this is true for J25 , J3

5 and J45 as well. This completes the proof.

We will next examine, for each of i = 0, 1, ..., 6, the elements η′ ∈ J ′i appearing in

the sum Si such that nη′,i is not divisible by (l−1)2. From the calculations in the proof

of Lemma 13.5.3 (i.e. equation (13.9) and the analogues for other representatives and

other TKli (l)), the only such η′ are the following:

• For i = 0: clearly there is only η0,0,01 = (I4, I2) with multiplicity 1.

• For i = 1: η′ = η1,0,01 , η0,1,0

1 , η0,0,11 , θ1,0,0

1 , θ0,0,11 , ι0,1,01 , ι0,0,11 , with multiplicity n1,η′

equal to l − 1 up to a power of l, and η′ = η0,0,01 , θ0,0,0

1 , ι0,0,01 with n1,η′ equal to

power of l.

• For i = 2: η′ = η1,0,01 , η0,1,0

1 with n2,η′ equal to l − 1 up to a power of l, and

η′ = η0,0,01 with n2,η′ equal to a power of l.

• For i = 3: η′ = η0,0,11 , θ1,0,0

1 , θ0,1,01 , θ0,0,1

1 , θ0,0,12 , ι1,0,01 , ι0,1,01 with n3,η′ equal to l − 1

up to a power of l, and η′ = η0,0,01 , θ0,0,0

1 , ι0,0,01 with n3,η′ equal to a power of l.

• For i = 4: η′ = η0,0,11 , θ1,0,0

1 , θ1,0,02 , ι0,1,01 , ι0,0,11 , ι0,1,02 , ι0,0,12 with n4,η′ equal to l − 1

up to a power of l, and η′ = θ0,0,01 , ι0,0,01 with with n4,η′ equal to a power of l.

230

• For i = 5: η′ = η1,0,01 , η0,1,0

1 with n5,η′ equal to 2(l − 1) up to a power of

l, η′ = η0,0,11 , η1,0,0

2 , η0,1,02 , θ1,0,0

1 , θ0,1,01 , θ0,0,1

1 , θ1,0,02 , ι1,0,01 , ι0,1,01 , ι0,0,11 , ι0,1,02 with n5,η′

equal to l − 1 up to a power of l, η′ = θ0,0,01 , ι0,0,01 with n5,η′ equal to a power of

l, and η′ = η0,0,01 with with n5,η′ equal to 2 up to a power of l.

• For i = 6: η′ = η1,0,01 , η0,1,0

1 , η1,0,02 , η0,1,0

2 with n6,η′ equal to l− 1 up to a power of

l, and η′ = η0,0,01 with n6,η′ equal to a power of l.

From the proof of Theorem 13.4.1, we see that Z(φ1,∞ ⊗ ξ) is given by, up to a

l-power factor,

6∑i=0

miSi =∑η′

(6∑i=0

minη′,i) · Z(φ1,∞ ⊗ ch(η′K)),

where as before, mi ∈ Z is the coefficient of TKli (l) in the linear combination T as given

in Corollary 12.3.3. In particular, modulo l − 1 we have m0 = m3 = m4 = m6 = 1,

m1 = m5 = −1 and m2 = 2.

Lemma 13.5.4. In the above sum:

1. if η′ is one of θ0,0,11 , θ0,0,1

2 , ι0,0,11 , ι0,0,12 , η0,0,01 , θ0,0,0

1 , ι0,0,01 , then the combined multi-

plicity∑6

i=0mini,η′ is divisible by l − 1, and

2. if η′ is any other element appearing on the list above, then∑6

i=0 mini,η′ is

divisible by (l − 1)2.

Proof. Both parts follow from straightforward computations according to the list

above.

This last lemma implies that in the notation of Theorem 13.4.1, the only η ap-

pearing in ξ =∑

η∈J mηch(ηK) (after combining conjugates) with (l − 1)2 - mη are

θ0,0,11 , θ0,0,1

2 , ι0,0,11 , ι0,0,12 , η0,0,01 , θ0,0,0

1 , ι0,0,01 . It is then straightforward to check that if η is

231

any of the above seven, then l− 1 part of the index [V : Vη] is precisely l− 1: we will

only demonstrate with θ0,0,12 and the rest should be clear.

Let η = θ0,0,12 , and h = (

a b

0 1

,

a′ b′

c′ d′

) be any element in V (so that in

particular a = a′d′ − b′c′). Then

η−1hη = (

a b

0 1

,

a′ − c′ (a′ + b′ − c′ − d′)l−2

l2c′ c′ + d′

),

which lies in K0,1 if and only if b′ = c′ + d′ − a′ (mod l2). In particular,

(

1

1

,

1 b′

1

) : b′ = 0, 1, ..., l2 − 1

is a complete set of representatives for (V ∩K0,1)/(V ∩ ηK0,1η−1). On the other had,

it is clear[V : V ∩K0,1] = l − 1. Thus [V : Vη] = (l − 1)l2.

This concludes the proof of Proposition 13.5.2.

232

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