Supersingular p-adic L-functions, Maass-Shimura operators and Waldspurger formulas / Daniel J. Kriz.

Format:

Book

Author/Creator:

Kriz, Daniel J., 1992- author.

Series:

Annals of Mathematics Studies

Annals of Mathematics Studies ; v.402

Language:

English

Subjects (All):

p-adic analysis.

Physical Description:

1 online resource (277 pages)

Place of Publication:

Princeton, New Jersey : Princeton University Press, [2021]

Summary:

A groundbreaking contribution to number theory that unifies classical and modern resultsThis 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.

Contents:

Frontmatter

Contents

Preface

Acknowledgments

1 Introduction

2 Preliminaries: Generalities

3 Preliminaries: Geometry of the infinite-level modular curve

4 The fundamental de Rham periods

5 The p-adic Maass-Shimura operator

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

7 Bounding periods at supersingular CM points

8 Supersingular Rankin-Selberg p-adic L-functions

9 The p-adic Waldspurger formula

Bibliography

Index

Notes:

Description based on print version record.

ISBN:

9780691225739

0691225737

OCLC:

1272997061