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.

- Cover Page
- 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