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

This book provides a new theory of p-adic modular forms on modular curves, extending Katz's classical theory to the supersingular locus. It introduces generalized p-adic modular forms, fundamental de Rham periods, and splittings of the Hodge filtration, leading to new p-adic L-functions and Waldspurger formulas for imaginary quadratic fields.

Format: Hardback
Length: 280 pages
Publication date: 09 November 2021
Publisher: Princeton University Press

This groundbreaking work in number theory seamlessly unites classical and modern results, revolutionizing our understanding of p-adic modular forms on modular curves. By extending Katz's classical theory to the supersingular locus, the book introduces a novel theory of p-adic modular forms on modular curves. The key innovation lies in moving to infinite levels and extending coefficients to period sheaves derived from relative p-adic Hodge theory. This enables the simplification of the Hodge bundle on the infinite-level modular curve through a canonical differential that restricts to the Katz canonical differential on the ordinary Igusa tower.

In his definition, Daniel Kriz introduces generalized p-adic modular forms as sections of relative period sheaves that transform under the Galois group of the modular curve through weight characters. He also introduces the fundamental de Rham period, which quantifies the position of the Hodge filtration in relative de Rham cohomology. This period shares a similar relationship with Scholze's Hodge-Tate period, and they 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, providing 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.

This groundbreaking research not only deepens our understanding of p-adic modular forms but also has significant implications for various fields, including algebraic geometry, arithmetic, and theoretical physics. It opens up new avenues for exploring the mathematical beauty and complexity of number theory and has the potential to impact our understanding of fundamental mathematical structures in the broader context of modern mathematics.

Dimension: 235 x 156 (mm)
ISBN-13: 9780691216478