Convolution and Equidistribution
Sato-Tate Theorems for Finite-Field Mellin Transforms
1st Edition
Published on 24. January 2012
208 pages
978-1-4008-4270-4 (ISBN)
Description
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.
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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Nicholas M. Katz
Convolution and Equidistribution
Sato-Tate Theorems for Finite-Field Mellin Transforms
Book
01/2012
Princeton University Press
€83.20
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Nicholas M. Katz
Convolution and Equidistribution
Sato-Tate Theorems for Finite-Field Mellin Transforms
Book
01/2012
Princeton University Press
€263.69
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Person
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.
Content
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
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
