Convolution and Equidistribution: Sato-Tate Theorems for Finite-Field Mellin Transforms (AM-180)

Overview

Convolution and Equidistribution explores an important aspect of number theory--the theory of exponential sums over finite fields and their Mellin transforms--from a new, categorical point of view. The book presents fundamentally important results and a plethora of examples, opening up new directions in the subject.

The finite-field Mellin transform (of a function on the multiplicative group of a finite field) is defined by summing that function against variable multiplicative ...

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Convolution and Equidistribution: Sato-Tate Theorems for Finite-Field Mellin Transforms (AM-180)

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Overview

Convolution and Equidistribution explores an important aspect of number theory--the theory of exponential sums over finite fields and their Mellin transforms--from a new, categorical point of view. The book presents fundamentally important results and a plethora of examples, opening up new directions in the subject.

The finite-field Mellin transform (of a function on the multiplicative group of a finite field) is defined by summing that function against variable multiplicative characters. The basic question considered in the book is how the values of the Mellin transform are distributed (in a probabilistic sense), in cases where the input function is suitably algebro-geometric. This question is answered by the book's main theorem, using a mixture of geometric, categorical, and group-theoretic methods.

By providing a new framework for studying Mellin transforms over finite fields, this book opens up a new way for researchers to further explore the subject.

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Editorial Reviews

Mathematical Reviews Clippings - Florent Jouve
The book is written in a clear and enlightening style. The author provides the reader with many examples that are developed throughout a dozen chapters. These examples help understand and clarify the depth and the variety of applications of the beautiful main equidistribution statement that relies on rather complicated and subtle algebrageometric arguments.
From the Publisher
"The book is written in a clear and enlightening style. The author provides the reader with many examples that are developed throughout a dozen chapters. These examples help understand and clarify the depth and the variety of applications of the beautiful main equidistribution statement that relies on rather complicated and subtle algebrageometric arguments."—Florent Jouve, Mathematical Reviews Clippings

"The book provides the reader with much material around the question of the equidistribution of the angles if one fixes f and varies over the multiplicative character x. More than one hundred pages of examples provide the reader with great insight in the different applications of the main theorem. This turns the book into a very good basis for research in this area."—Manfred G. Madritsch, Zentralblatt MATH

"Once a certain basic understanding is reached, this book, like the others written by N. Katz, reveals itself to be very precisely and sharply written, and to be full of riches. And finally, this theory shows spectacularly how some of the most abstract ideas of algebra and algebraic geometry may be essential to solving extremely concrete problems."—Emmanuel Kowalski, Bulletin of the American Mathematical Society

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Product Details

  • ISBN-13: 9780691153308
  • Publisher: Princeton University Press
  • Publication date: 2/26/2012
  • Series: Annals of Mathematics Studies Series
  • Pages: 212
  • Product dimensions: 6.40 (w) x 9.40 (h) x 0.70 (d)

Meet the Author

Nicholas M. Katz is professor of mathematics at Princeton University. He is the author or coauthor of six previous titles in the Annals of Mathematics Studies: "Arithmetic Moduli of Elliptic Curves "(with Barry Mazur); "Gauss Sums, Kloosterman Sums, and Monodromy Groups; Exponential Sums and Differential Equations; Rigid Local Systems; Twisted L-Functions and Monodromy;" and "Moments, Monodromy, and Perversity."

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Table of Contents

Introduction 1

Chapter 1 Overview 7

Chapter 2 Convolution of Perverse Sheaves 19

Chapter 3 Fibre Functors 21

Chapter 4 The Situation over a Finite Field 25

Chapter 5 Frobenius Conjugacy Classes 31

Chapter 6 Group-Theoretic Facts about Ggeom and Garith 33

Chapter 7 The Main Theorem 39

Chapter 8 Isogenies, Connectedness, and Lie-Irreducibility 45

Chapter 9 Autodualities and Signs 49

Chapter 10 A First Construction of Autodual Objects 53

Chapter 11 A Second Construction of Autodual Objects 55

Chapter 12 The Previous Construction in the Nonsplit Case 61

Chapter 13 Results of Goursat-Kolchin-Ribet Type 63

Chapter 14 The Case of SL(2); the Examples of Evans and Rudnick 67

Chapter 15 Further SL(2) Examples, Based on the Legendre Family 73

Chapter 16 Frobenius Tori and Weights; Getting Elements of Garith 77

Chapter 17 GL(n) Examples 81

Chapter 18 Symplectic Examples 89

Chapter 19 Orthogonal Examples, Especially SO(n) Examples 103

Chapter 20 GL(n) × GL(n) × ... × GL(n) Examples 113

Chapter 21 SL(n) Examples, for n an Odd Prime 125

Chapter 22 SL(n) Examples with Slightly Composite n 135

Chapter 23 Other SL(n) Examples 141

Chapter 24 An O(2n) Example 145

Chapter 25 G2 Examples: the Overall Strategy 147

Chapter 26 G2 Examples: Construction in Characteristic Two 155

Chapter 27 G2 Examples: Construction in Odd Characteristic 163

Chapter 28 The Situation over Z: Results 173

Chapter 29 The Situation over Z: Questions 181

Chapter 30 Appendix: Deligne's Fibre Functor 187

Bibliography 193

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