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Supersingular P-Adic L-Functions, Maass-Shimura Operators and Waldspurger Formulas

Product Image: Supersingular P-Adic L-Functions, Maass-Shimura Operators and Waldspurger Formulas

Supersingular P-Adic L-Functions, Maass-Shimura Operators and Waldspurger Formulas

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"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." --
Product details
Number of Pages:
280
Release Date:
2021-11-09
Publication Date:
2021-11-09
Publisher:
Princeton University Press
Languages:
Original: English
ISBN10:
0691216479
ISBN13:
9780691216478
GPSR Manufacturer Reference:
[email protected]
Weight:
567 g
Height:
157 cm
Width:
236 cm
Thickness:
20 cm
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