Convolution and equidistribution : Sato-Tate theorems for finite-field Mellin transforms
著者
書誌事項
Convolution and equidistribution : Sato-Tate theorems for finite-field Mellin transforms
（Annals of mathematics studies, no. 180）
Princeton University Press, 2012
- : hardcover
- : pbk
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注記
Bibliography: p. 193-195
Includes index
内容説明・目次
内容説明
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.
目次
- *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
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