Supersingular p-adic L-functions, Maass-Shimura Operators and Waldspurger Formulas: (AMS-212)

by Daniel Kriz

Overview

A groundbreaking contribution to number theory that unifies classical and modern results

This book develops a new theory of p-adic modular forms on modular curves, extending Katz's classical theory to the supersingular locus. The main novelty is to move to infinite level and extend coefficients to period sheaves coming from relative p-adic Hodge theory. This makes it possible to trivialize the Hodge bundle on the infinite-level modular curve by a "canonical differential" that restricts to the Katz canonical differential on the ordinary Igusa tower. Daniel Kriz defines generalized p-adic modular forms as sections of relative period sheaves transforming under the Galois group of the modular curve by weight characters. He introduces the fundamental de Rham period, measuring the position of the Hodge filtration in relative de Rham cohomology. This period can be viewed as a counterpart to Scholze's Hodge-Tate period, and the two periods satisfy a Legendre-type relation. Using these periods, Kriz constructs splittings of the Hodge filtration on the infinite-level modular curve, defining p-adic Maass-Shimura operators that act on generalized p-adic modular forms as weight-raising operators. Through analysis of the p-adic properties of these Maass-Shimura operators, he constructs new p-adic L-functions interpolating central critical Rankin-Selberg L-values, giving analogues of the p-adic L-functions of Katz, Bertolini-Darmon-Prasanna, and Liu-Zhang-Zhang for imaginary quadratic fields in which p is inert or ramified. These p-adic L-functions yield new p-adic Waldspurger formulas at special values.

Product Details

ISBN-13:	9780691225739
Publisher:	Princeton University Press
Publication date:	11/09/2021
Series:	Annals of Mathematics Studies , #212
Sold by:	Barnes & Noble
Format:	eBook
Pages:	280
File size:	2 MB

About the Author

Daniel J. Kriz is an instructor in pure mathematics and a National Science Foundation postdoctoral fellow at the Massachusetts Institute of Technology.

Table of Contents

Preface xi

Acknowledgments xv

1 Introduction 1

1.1 Previous constructions and Katz's theory of p-adic modular forms on the ordinary locus 1

1.2 Outline of our theory of p-adic analysis on the supersingular locus and construction of p-adic L-functions 6

1.3 Main results 23

1.4 Some remarks on other works in supersingular Iwasawa theory 25

2 Preliminaries: Generalities 27

2.1 Grothendieck sites and topoi 27

2.2 Pro-categories 29

2.3 Adic spaces 30

2.4 (Pre-)adic spaces and (pre)perfectoid spaces 39

2.5 Some complements on inverse limits of (pre-)adic spaces 43

2.6 The proétale site of an adic space 45

2.7 Period sheaves 49

2.8 The proétale "constant sheaf" Z_p,Y 55

2.9 B+_dR,Y-local systems, OB+_dR,Y-modules with connection, and the general de Rham comparison theorem 56

3 Preliminaries: Geometry of the infinite-level modular curve 63

3.1 The infinite-level modular curve 63

3.2 Relative étale cohomology and the Weil pairing 66

3.3 The GL₂(Q_p)-action on y (and y) 67

3.4 The Hodge-Tate period and the Hodge-Tate period map 69

3.5 The Lubin-Tate period on the supersingular locus 72

3.6 The relative Hodge-Tate filtration 77

3.7 The fake Hasse invariant 78

3.8 Relative de Rham cohomology and the Hodge-de Rham filtration 79

3.9 Relative p-adic de Rham comparison theorem applied to A → Y 80

4 The fundamental de Rham periods 83

4.1 A proétale local description of OB⁽⁺⁾_dR 83

4.2 The fundamental de Rham periods 85

4.3 GL₂(Q_p)-transformation properties of the fundamental de Rham periods 86

4.4 The p-adic Legendre relation 89

4.5 Relation to Colmez's "p-adic period pairing" 94

4.6 Relation to classical (Serre-Tate) theory on the ordinary locus 97

4.7 The Kodaira-Spencer isomorphism 109

4.8 The fundamental de Rham period Z_dR 112

4.9 The canonical differential 114

5 The p-adic Maass-Shimura operator 118

5.1 The "horizontal" lifting of the Hodge-Tate filtration 118

5.2 The "horizontal" relative Hodge-Tate decomposition over O_Δ 122

5.3 Definition of the p-adic Maass-Shimura operator 126

5.4 The p-adic Maass-Shimura operator in coordinates and generalized p-adic modular forms 127

5.5 The p-adic Maass-Shimura operator with "nearly holomorphic coefficients" 131

5.6 The relative Hodge-Tate decomposition over O^†_Δ 138

5.7 The p-adic Maass-Shimura operator in coordinates and generalized p-adic nearly holomorphic modular forms 141

5.8 Relation of d^j_k and (d^†_k)j to the ordinary Atkin-Serre operator d^j_{k, AS} and Katz's p-adic modular forms 147

5.9 Comparison between the complex and p-adic Maass-Shimura operators at CM points 150

5.10 Comparison of algeraic Maass-Shimura derivatives on different levels 159

6 p-adic analysis of the p-adic Maass-Shimura operators 162

6.1 q_dR-expansions 162

6.2 Relation between q_dR-expansions and Serre-Tate expansions 172

6.3 Integrality properties of q_dR-expansions: the Dieudonné-Dwork lemma 174

6.4 Integral structures on stalks of intermediate period sheaves between O_Δ and Ô_Δ 180

6.5 The p-adic Maass-Shimura operator θ^j_k in q_dR-coordinates 182

6.6 Integrality of q_dR-expansions and the b-operator 184

6.7 p-adic analytic properties of p-adic Maass-Shimura operators 186

7 Bounding periods at supersingular CM points 197

7.1 Periods of supersingular CM points 197

7.2 Weights 205

7.3 Good CM points 208

8 Supersingular Rankin-Selberg p-adic L-functions 216

8.1 Preliminaries for the construction 216

8.2 Construction of the p-adic L-function 219

8.3 Interpolation 228

8.3.1 Interpolation formula 228

8.3.2 Interpolation formula 234

9 The p-adic Waldspurger formula 236

9.1 Coleman integration 237

9.2 Coleman primitives in our situation 237

9.3 The p-adic Waldspurger formula 243

9.4 p-adic Kronecker limit formula 248

Bibliography 251

Index 257