P-adic Aspects Of Modular Forms

Baskar Balasubramanyam, Haruzo Hida, A Raghuram;Jacques Tilouine

• Publisher: WSPC
• Year: 2016
• Topic: Mathematics
• eBook ISBN:
  9789814719247
• Subtopic: Applied Mathematics
• Index: Mathematics

The aim of this book is to give a systematic exposition of results in some important cases where p -adic families and p -adic L -functions are studied. We first look at p -adic families in the following cases: general linear groups, symplectic groups and definite unitary groups. We also look at applications of this theory to modularity lifting problems. We finally consider p -adic L -functions for GL(2), the p -adic adjoint L -functions and some cases of higher GL( n ).

Contents:

• An Overview of Serre's p -Adic Modular Forms (Miljan Brakočević and R Sujatha)
• p -Adic Families of Ordinary Siegel Cusp Forms (Jacques Tilouine)
• Ordinary Families of Automorphic Forms on Definite Unitary Groups (Baskar Balasubramanyam and Dipramit Majumdar)
• Notes on Modularity Lifting in the Ordinary Case (David Geraghty)
• p -Adic L -Functions for Hilbert Modular Forms (Mladen Dimitrov)
• Arithmetic of Adjoint L -Values (Haruzo Hida)
• p -Adic L -Functions for GL n (Debargha Banerjee and A Raghuram)
• Non-Triviality of Generalised Heegner Cycles Over Anticyclotomic Towers: A Survey (Ashay A Burungale)
• The Euler System of Heegner Points and p -Adic L -Functions (Ming-Lun Hsieh)
• Non-Commutative q -Expansions (Mahesh Kakde)

Readership: Researchers in algebra and number theory.

Chapter 6

Arithmetic of adjoint L-values

Haruzo Hida

Department of Mathematics,
UCLA, Los Angeles, CA 90095-1555, USA

Contents

6.1.Introduction

6.2.Some ring theory

6.2.1.Differentials

6.2.2.Congruence and differential modules

6.3.Deformation rings

6.3.1.One dimensional case

6.3.2.Congruence modules for group algebras

6.3.3.Proof of Tate’s theorem

6.3.4.Two dimensional cases

6.3.5.Adjoint Selmer groups

6.3.6.Selmer groups and differentials

6.4.Hecke algebras

6.4.1.Finite level

6.4.2.Ordinary of level Np^∞

6.4.3.Modular Galois representation

6.4.4.Hecke algebra is universal

6.5.Analytic and topological methods

6.5.1.Analyticity of adjoint L-functions

6.5.2.Integrality of adjoint L-values

6.5.3.Congruence and adjoint L-values

6.5.4.Adjoint non-abelian class number formula

6.5.5.p-Adic adjoint L-functions

References

6.1.Introduction

In this lecture note of the mini-course, we discuss the following five topics:

(1)Introduction to the ordinary (i.e. slope 0) Hecke algebras (the so-called “big Hecke algebra”);

(2)Some basic ring theory to deal with Hecke algebras;

(3)“R = T” theorem of Wiles–Taylor, and its consequence for the adjoint Selmer groups;

(4)Basics of adjoint L-function (analytic continuation, rationality of the value at s = 1);

(5)Relation of adjoint L-values to congruence of a modular form f and the Selmer group of the adjoint Galois representation Ad(ρ_f) (the adjoint main conjectures).

Let us describe some history of these topics along with the content of the note. Doi and the author started the study of the relation between congruence among cusp forms and L-values in 1976 (the L-value governing the congruence is now called the adjoint L-value L(1, Ad(f))). How it was started was described briefly in a later paper [DHI98]. Here is a quote from the introduction of this old article:

“It was in 1976 when Doi found numerically non-trivial congruence among Hecke eigenforms of a given conductor for a fixed weight κ [DO77]. Almost immediately after his discovery, Doi and Hida started, from...

Table of contents

• Cover
• Halftitle
• Title
• Copyright
• Preface
• Contents
• An overview of Serre’s p-adic modular forms
• p-adic families of ordinary Siegel cusp forms
• Ordinary families of automorphic forms on definite unitary groups
• Notes on modularity lifting in the ordinary case
• p-adic L-functions for Hilbert modular forms
• Arithmetic of adjoint L-values
• p-adic L-functions for GLn
• Non-triviality of generalised Heegner cycles over anticyclotomic towers: a survey
• The Euler system of Heegner points and p-adic L-functions
• Non-commutative q-expansions