Convolution and Equidistribution
eBook - ePub

Convolution and Equidistribution

Sato-Tate Theorems for Finite-Field Mellin Transforms

  1. 208 pages
  2. English
  3. ePUB (mobile friendly)
  4. Available on iOS & Android
eBook - ePub

Convolution and Equidistribution

Sato-Tate Theorems for Finite-Field Mellin Transforms

About this book

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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Information

Publisher
Princeton University Press
Year
2012
Print ISBN
9780691153315
9780691153308
eBook ISBN
9781400842704
Topic
Mathematics
Subtopic
Algebraic Geometry
Index
Mathematics

Table of contents

  1. Cover Page
  2. Title Page
  3. Copyright
  4. Contents
  5. Introduction
  6. Chapter 1. Overview
  7. Chapter 2. Convolution of Perverse Sheaves
  8. Chapter 3. Fibre Functors
  9. Chapter 4. The Situation over a Finite Field
  10. Chapter 5. Frobenius Conjugacy Classes
  11. Chapter 6. Group-Theoretic Facts about Ggeom and Garith
  12. Chapter 7. The Main Theorem
  13. Chapter 8. Isogenies, Connectedness, and Lie-Irreducibility
  14. Chapter 9. Autodualities and Signs
  15. Chapter 10. A First Construction of Autodual Objects
  16. Chapter 11. A Second Construction of Autodual Objects
  17. Chapter 12. The Previous Construction in the Nonsplit Case
  18. Chapter 13. Results of Goursat-Kolchin-Ribet Type
  19. Chapter 14. The Case of SL(2); the Examples of Evans and Rudnick
  20. Chapter 15. Further SL(2) Examples, Based on the Legendre Family
  21. Chapter 16. Frobenius Tori and Weights; Getting Elements of Garith
  22. Chapter 17. GL(n) Examples
  23. Chapter 18. Symplectic Examples
  24. Chapter 19. Orthogonal Examples, Especially SO(n) Examples
  25. Chapter 20. GL(n) × GL(n) × … × GL(n) Examples
  26. Chapter 21. SL(n) Examples, for n an Odd Prime
  27. Chapter 22. SL(n) Examples with Slightly Composite n
  28. Chapter 23. Other SL(n) Examples
  29. Chapter 24. An O(2n) Example
  30. Chapter 25. G2 Examples: the Overall Strategy
  31. Chapter 26. G2 Examples: Construction in Characteristic Two
  32. Chapter 27. G2 Examples: Construction in Odd Characteristic
  33. Chapter 28. The Situation over Z: Results
  34. Chapter 29. The Situation over Z: Questions
  35. Chapter 30. Appendix: Deligne’s Fibre Functor
  36. Bibliography
  37. Index

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Yes, you can access Convolution and Equidistribution by Nicholas M. Katz in PDF and/or ePUB format, as well as other popular books in Mathematics & Algebraic Geometry. We have over 1.5 million books available in our catalogue for you to explore.