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

• Superior document: Title is part of eBook package: De Gruyter Princeton Annals of Mathematics eBook-Package 1940-2020
• VerfasserIn: Katz, Nicholas M.,
• Place / Publishing House: Princeton, NJ : : Princeton University Press, , [2012]; ©2012
• Year of Publication: 2012
• Edition: Course Book
• Language: English
• Series: Annals of Mathematics Studies ; 180
• Physical Description: 1 online resource (208 p.)
• Other title: Frontmatter --; 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) x GL(n) x ... x 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
• Format: Mode of access: Internet via World Wide Web.
• ISBN: 9781400842704; 9783110494914; 9783110442502
• DOI: 10.1515/9781400842704?locatt=mode:legacy
• Access: restricted access
• Hierarchical level: Monograph
• Statement of Responsibility: 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.