Convolution and Equidistribution: Sato-Tate Theorems for Finite-Field Mellin Transforms

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.

1110926377
Convolution and Equidistribution: Sato-Tate Theorems for Finite-Field Mellin Transforms

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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Convolution and Equidistribution: Sato-Tate Theorems for Finite-Field Mellin Transforms

Convolution and Equidistribution: Sato-Tate Theorems for Finite-Field Mellin Transforms

by Nicholas M. Katz
Convolution and Equidistribution: Sato-Tate Theorems for Finite-Field Mellin Transforms
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Convolution and Equidistribution: Sato-Tate Theorems for Finite-Field Mellin Transforms

Convolution and Equidistribution: Sato-Tate Theorems for Finite-Field Mellin Transforms

by Nicholas M. Katz

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.

Product Details

ISBN-13: 9781400842704
Publisher: Princeton University Press
Publication date: 01/24/2012
Series: Annals of Mathematics Studies , #180
Sold by: Barnes & Noble
Format: eBook
Pages: 208
File size: 4 MB

About 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.

Table of Contents

  • FrontMatter, pg. i
  • Contents, pg. vi
  • Introduction, pg. 1
  • CHAPTER 1. Overview, pg. 7
  • CHAPTER 2. Convolution of Perverse Sheaves, pg. 19
  • CHAPTER 3. Fibre Functors, pg. 21
  • CHAPTER 4. The Situation over a Finite Field, pg. 25
  • CHAPTER 5. Frobenius Conjugacy Classes, pg. 31
  • CHAPTER 6. Group-Theoretic Facts about Ggeom and Garith, pg. 33
  • CHAPTER 7. The Main Theorem, pg. 39
  • CHAPTER 8. Isogenies, Connectedness, and Lie-Irreducibility, pg. 45
  • CHAPTER 9. Autodualities and Signs, pg. 49
  • CHAPTER 10. A First Construction of Autodual Objects, pg. 53
  • CHAPTER 11. A Second Construction of Autodual Objects, pg. 55
  • CHAPTER 12. The Previous Construction in the Nonsplit Case, pg. 61
  • CHAPTER 13. Results of Goursat-Kolchin-Ribet Type, pg. 63
  • CHAPTER 14. The Case of SL(2); the Examples of Evans and Rudnick, pg. 67
  • CHAPTER 15. Further SL(2) Examples, Based on the Legendre Family, pg. 73
  • CHAPTER 16. Frobenius Tori and Weights; Getting Elements of Garith, pg. 77
  • CHAPTER 17. GL(n) Examples, pg. 81
  • CHAPTER 18. Symplectic Examples, pg. 89
  • CHAPTER 19. Orthogonal Examples, Especially SO(n) Examples, pg. 103
  • CHAPTER 20. GL(n) x GL(n) x ... x GL(n) Examples, pg. 113
  • CHAPTER 21. SL(n) Examples, for n an Odd Prime, pg. 125
  • CHAPTER 22. SL(n) Examples with Slightly Composite n, pg. 135
  • CHAPTER 23. Other SL(n) Examples, pg. 141
  • CHAPTER 24. An O(2n) Example, pg. 145
  • CHAPTER 25. G2 Examples: the Overall Strategy, pg. 147
  • CHAPTER 26. G2 Examples: Construction in Characteristic Two, pg. 155
  • CHAPTER 27. G2 Examples: Construction in Odd Characteristic, pg. 163
  • CHAPTER 28. The Situation over ℤ: Results, pg. 173
  • CHAPTER 29. The Situation over ℤ: Questions, pg. 181
  • CHAPTER 30. Appendix: Deligne's Fibre Functor, pg. 187
  • Bibliography, pg. 193
  • Index, pg. 197

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