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

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Daniel Kriz

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

AMS-212

Daniel Kriz

$135

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English

Princeton University Press
19 November 2021

Algebra; Number theory

Series: Annals of Mathematics Studies

"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."

By: Daniel Kriz
Imprint: Princeton University Press
Country of Publication: United States
Dimensions: Height: 235mm, Width: 156mm,
ISBN: 9780691216461
ISBN 10: 0691216460
Series: Annals of Mathematics Studies
Pages: 280
Publication Date: 19 November 2021
Audience: College/higher education , Professional and scholarly , Further / Higher Education , Undergraduate
Format: Paperback
Publisher's Status: Active

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