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

Author/Creator:
Katz, Nicholas M.
Publication:
Princeton : Princeton University Press, 2012.
Format/Description:
Book
1 online resource (193 pages)
Series:
Annals of Mathematics Studies
Annals of Mathematics Studies ; v.180
Status/Location:
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Subjects:
Mellin transform.
Convolutions (Mathematics).
Sequences (Mathematics).
Form/Genre:
Electronic books.
Summary:
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.
Contents:
Intro
Title Page
Copyright
Contents
Introduction
Chapter 1. Overview
Chapter 2. Convolution of Perverse Sheaves
Chapter 3. Fibre Functors
Chapter 4. The Situation over a Finite Field
Chapter 5. Frobenius Conjugacy Classes
Chapter 6. Group-Theoretic Facts about Ggeom and Garith
Chapter 7. The Main Theorem
Chapter 8. Isogenies, Connectedness, and Lie-Irreducibility
Chapter 9. Autodualities and Signs
Chapter 10. A First Construction of Autodual Objects
Chapter 11. A Second Construction of Autodual Objects
Chapter 12. The Previous Construction in the Nonsplit Case
Chapter 13. Results of Goursat-Kolchin-Ribet Type
Chapter 14. The Case of SL(2)
the Examples of Evans and Rudnick
Chapter 15. Further SL(2) Examples, Based on the Legendre Family
Chapter 16. Frobenius Tori and Weights
Getting Elements of Garith
Chapter 17. GL(n) Examples
Chapter 18. Symplectic Examples
Chapter 19. Orthogonal Examples, Especially SO(n) Examples
Chapter 20. GL(n) × GL(n) × … × GL(n) Examples
Chapter 21. SL(n) Examples, for n an Odd Prime
Chapter 22. SL(n) Examples with Slightly Composite n
Chapter 23. Other SL(n) Examples
Chapter 24. An O(2n) Example
Chapter 25. G2 Examples: the Overall Strategy
Chapter 26. G2 Examples: Construction in Characteristic Two
Chapter 27. G2 Examples: Construction in Odd Characteristic
Chapter 28. The Situation over Z: Results
Chapter 29. The Situation over Z: Questions
Chapter 30. Appendix: Deligne's Fibre Functor
Bibliography
Index.
Notes:
Description based on publisher supplied metadata and other sources.
Local notes:
Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2021. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.
Other format:
Print version: Katz, Nicholas M. Convolution and Equidistribution
ISBN:
9781400842704
9780691153308
OCLC:
769343171