On Selmer Groups And Factoring P Adic L Functions
Download On Selmer Groups And Factoring P Adic L Functions PDF/ePub or read online books in Mobi eBooks. Click Download or Read Online button to get On Selmer Groups And Factoring P Adic L Functions book now. This website allows unlimited access to, at the time of writing, more than 1.5 million titles, including hundreds of thousands of titles in various foreign languages.
On Selmer Groups and Factoring P-adic L-functions
Samit Dasgupta has proved a formula factoring a certain restriction of a 3-variable Rankin-Selberg p-adic L-function as a product of a 2-variable p-adic L-function related to the adjoint representation of a Hida family and a Kubota-Leopoldt p-adic L-function. We prove a result involving Selmer groups that along with Dasgupta's result is consistent with the main conjectures associated to the 4-dimensional representation (to which the 3-variable p-adic L-function is associated), the 3-dimensional representation (to which the 2-variable p-adic L-function is associated) and the 1-dimensional representation (to which the Kubota-Leopoldt p-adic L-function is associated). Under certain additional hypotheses, we indicate how one can use work of Urban to deduce main conjectures for the 3-dimensional representation and the 4-dimensional representation. Dasgupta's method of proof is based on an earlier work of Gross in 1980. Gross's work involved factoring a certain restriction of a 2-variable p-adic L-function associated to an imaginary quadratic field (constructed by Katz) into a product of two Kubota-Leopoldt p-adic L-functions. In 1982, Greenberg proved the corresponding result on the algebraic side involving classical Iwasawa modules, as predicted by the main conjectures for imaginary quadratic fields and Q. Our methods are inspired by this work of Greenberg. One key technical input to our methods is studying the behavior of Selmer groups under specialization.
$p$-adic $L$-Functions and $p$-adic Representations
Author: Bernadette Perrin-Riou
language: en
Publisher: American Mathematical Soc.
Release Date: 2000
Traditionally, p-adic L-functions have been constructed from complex L-functions via special values and Iwasawa theory. In this volume, Perrin-Riou presents a theory of p-adic L-functions coming directly from p-adic Galois representations (or, more generally, from motives). This theory encompasses, in particular, a construction of the module of p-adic L-functions via the arithmetic theory and a conjectural definition of the p-adic L-function via its special values. Since the original publication of this book in French (see Astérisque 229, 1995), the field has undergone significant progress. These advances are noted in this English edition. Also, some minor improvements have been made to the text.
Supersingular p-adic L-functions, Maass-Shimura Operators and Waldspurger Formulas
Author: Daniel Kriz
language: en
Publisher: Princeton University Press
Release Date: 2021-11-09
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.