06728nam 2201429 a 450 991078973090332120210519213339.01-283-37996-197866133799621-4008-4270-010.1515/9781400842704(CKB)2670000000133403(EBL)827807(OCoLC)769343171(SSID)ssj0000575757(PQKBManifestationID)11408626(PQKBTitleCode)TC0000575757(PQKBWorkID)10553428(PQKB)10618809(MiAaPQ)EBC827807(StDuBDS)EDZ0000406946(DE-B1597)447783(OCoLC)979905293(DE-B1597)9781400842704(Au-PeEL)EBL827807(CaPaEBR)ebr10521872(CaONFJC)MIL337996(PPN)199244731(PPN)187959633(EXLCZ)99267000000013340320110826d2012 uy 0engurun#---|u||utxtccrConvolution and equidistribution[electronic resource] Sato-Tate theorems for finite-field Mellin transforms /Nicholas M. KatzCourse BookPrinceton ;Oxford Princeton University Pressc20121 online resource (213 p.)Annals of mathematics studies ;no. 180Description based upon print version of record.0-691-15330-2 0-691-15331-0 Includes bibliographical references and index.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 --IndexConvolution 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.Annals of mathematics studies ;no. 180.Mellin transformConvolutions (Mathematics)Sequences (Mathematics)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.Mellin transform.Convolutions (Mathematics)Sequences (Mathematics)515/.723SI 830rvkKatz Nicholas M.1943-59374MiAaPQMiAaPQMiAaPQBOOK9910789730903321Convolution and equidistribution854265UNINA