Supersingular P-Adic L-Functions, Maass-Shimura Operators and Waldspurger Formulas: Ams-212 Kriz Daniel

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. 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. The main novelty is to move to infinite level and extend coefficients to period sheaves coming from relative p-adic Hodge theory.

He introduces the fundamental de Rham period, measuring the position of the Hodge filtration in relative de Rham cohomology. 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. This period can