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

Daniel Kriz

Published by Princeton University Press 2021

Supersingular p-adic L-functions, Maass-Shimura Operators and Waldspurger Formulas

(AMS-212)

Daniel Kriz

Volume 212 in the series Annals of Mathematics Studies

https://doi.org/10.1515/9780691225739

Overview
Contents

About this book

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.

Author / Editor information

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

Topics

Frontmatter Publicly Available Download PDF	i
Contents Publicly Available Download PDF	vii
Preface Publicly Available Download PDF	xi
Acknowledgments Publicly Available Download PDF	xv
1 Introduction Requires Authentication Unlicensed Licensed Download PDF	1
2 Preliminaries: Generalities Requires Authentication Unlicensed Licensed Download PDF	27
3 Preliminaries: Geometry of the infinite-level modular curve Requires Authentication Unlicensed Licensed Download PDF	63
4 The fundamental de Rham periods Requires Authentication Unlicensed Licensed Download PDF	83
5 The p-adic Maass-Shimura operator Requires Authentication Unlicensed Licensed Download PDF	118
6 P-adic analysis of the p-adic Maass-Shimura operators Requires Authentication Unlicensed Licensed Download PDF	162
7 Bounding periods at supersingular CM points Requires Authentication Unlicensed Licensed Download PDF	197
8 Supersingular Rankin-Selberg p-adic L-functions Requires Authentication Unlicensed Licensed Download PDF	216
9 The p-adic Waldspurger formula Requires Authentication Unlicensed Licensed Download PDF	236
Bibliography Requires Authentication Unlicensed Licensed Download PDF	251
Index Requires Authentication Unlicensed Licensed Download PDF	257

Language: English
Publisher: Princeton University Press
Audience: Professional and scholarly;College/higher education;
Pages
- Main content: 280
Keywords: p-adic modular forms; Waldspurger formula; Hodge theory; number theory; arithmetic geometry; unified number theory; mathematical function; mathematical proof; (pre)perfectoid spaces; Hodge-Tate filtration; Hasse invariant; Katz's theory; supersingular locus

eBook

Published: November 9, 2021
ISBN: 9780691225739