Convolution and Equidistribution Sato Tate Theorems for Finite Field Mellin Transforms 1st Edition by Nicholas M Katz ISBN 1400842700 9781400842704

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Product details:

ISBN 10: 1400842700

ISBN 13: 9781400842704

Author: Nicholas M. Katz

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.

Table of contents:

1. Overview

2. Convolution of Perverse Sheaves

3. Fibre Functors

4. The Situation over a Finite Field

5. Frobenius Conjugacy Classes

6. Group-Theoretic Facts about G_geom and G_arith

7. The Main Theorem

8. Isogenies, Connectedness and Lie-Irreducibility

9. Autodualities and Signs

10. A First Construction of Autodual Objects

11. A Second Construction of Autodual Objects

12. The Previous Construction in the Nonsplit Case

13. Results of Goursat–Kolchin–Ribet Type

14. The Case of SL₂: the Examples of Evans and Rudnick

15. Further SL₂ Examples Based on the Legendre Family

16. Frobenius Tori and Weights: Getting Elements of G_arith

17. GLₙ Examples

18. Symplectic Examples

19. Orthogonal Examples, Especially SOₙ Examples

20. GLₙ × GLₙ × GLₙ Examples

21. SLₙ Examples for n an Odd Prime

22. SLₙ Examples with Slightly Composite n

23. Other SLₙ Examples

24. An O₂ₙ Example

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Tags: Nicholas M Katz, Convolution, Equidistribution, Sato, Tate, Theorems, Finite, Field, Mellin, Transforms