Supersingular p-adic L-functions, Maass-Shimura Operators and Waldspurger Formulas: (AMS-212) (Annals of Mathematics Studies, 402)

**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. "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." -- Publisher's description