Convolution and equidistribution [[electronic resource] ] : Sato-Tate theorems for finite-field Mellin transforms / / Nicholas M. Katz

• Autore: Katz Nicholas M. <1943->
• Pubblicazione: Princeton ; ; Oxford, : Princeton University Press, c2012
• Edizione: Course Book
• Descrizione fisica: 1 online resource (213 p.)
• Disciplina: 515/.723
• Soggetto topico: Mellin transform; Convolutions (Mathematics); Sequences (Mathematics)
• Soggetto non controllato: ArtinГchreier reduced polynomial; Emanuel Kowalski; EulerАoincar formula; Frobenius conjugacy class; Frobenius conjugacy; Frobenius tori; GoursatЋolchinВibet theorem; Kloosterman sheaf; Laurent polynomial; Legendre; Mellin transform; Pierre Deligne; Ron Evans; Tannakian category; Tannakian groups; Zeeev Rudnick; algebro-geometric; autodual objects; autoduality; characteristic two; connectedness; dimensional objects; duality; equidistribution; exponential sums; fiber functor; finite field Mellin transform; finite field; finite fields; geometrical irreducibility; group scheme; hypergeometric sheaf; interger monic polynomials; isogenies; lie-irreducibility; lisse; middle convolution; middle extension sheaf; monic polynomial; monodromy groups; noetherian connected scheme; nonsplit form; nontrivial additive character; number theory; odd characteristic; odd prime; orthogonal case; perverse sheaves; polynomials; pure weight; semisimple object; semisimple; sheaves; signs; split form; supermorse; theorem; theorems
• Classificazione: SI 830
• Note generali: Description based upon print version of record.
• Nota di bibliografia: Includes bibliographical references and index.
• Nota di contenuto: Front matter -- 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
• Sommario/riassunto: 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.
• Titolo autorizzato: Convolution and equidistribution
• ISBN: 1-283-37996-1; 9786613379962; 1-4008-4270-0
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 9910789730903321
• Lo trovi qui: Univ. Federico II
• Serie: Annals of mathematics studies ; ; no. 180.