Title Exponential sums and differential equations / by Nicholas M. Katz

Published Princeton, N.J. : Princeton University Press, 1990

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Description 1 online resource Series Annals of mathematics studies ; no. 124 Annals of mathematics studies ; no. 124. Contents Cover; Title; Copyright; Contents; Introduction; Chapter 1- Results from Representation Theory; 1.0-1.6 Statements of the main results; 1.7 The proofs; 1.8 Appendix: direct sums and tensor products; Chapter 2- D.E.'s and D-modules; 2.1 The basic set-up; 2.2 Torsors and lifting problems; 2.3 Relation to transcendence; 2.4 Behavior of Ggal under specialization; 2.5 Specialization of morphisms; 2.6 Direct sums and tensor products; 2.7 A basic trichotomy; 2.8 The main D.E. theorem; 2.9 Generalities on D-modules on curves; 2.10 Some equations on Â1, with a transition to Gm 2.10.16 Location of the singularities of a Fourier Transform2.11 Systematic study of equations on Gm; Chapter 3- The generalized hypergeometric equation; 3.1 Basic definitions; 3.2-3.3 Basic results about irreducibility and contiguity; 3.4 Duality; 3.5 The case n= m; Lie irreducibility, rigidity, Belyi and Kummer induction; 3.6 The case n ̀"m; 3.7 Intrinsic characterization; rigidity for n ̀"m; 3.8 Direct sums, tensor products, and Kummer inductions; Chapter 4 Detailed analysis of the exceptional cases; 4.0 Eliminating a few cases; 4.1 The G2 case 4.2 The Spin(7), PSL(3) and SL(2)xSL(2)xSL(2) cases4.3 The PSL(3) case: detailed analysis; 4.4 The Spin(7) case: detailed analysis; 4.5 The SL(2)xSL(2)xSL(2) case; 4.6 The SL(3)xSL(3) case; Chapter 5- Convolution of D-modules; 5.1 Generalities; 5.2 Convolution on Gm and Fourier Transform on Â1; 5.3 Convolution of hypergeometrics on Gm; 5.4 Motivic interpretation of hypergeometrics of type (n, n); 5.5 Application to Grothendieck's p-curvature conjecture; Chapter 6- Fourier transforms of Kummer pullbacks of hypergeometrics; 6.1 Some D.E.'s on Â1 as Kummer pullbacks of hypergeometrics 6.2 Fourier transforms of Kummer pullbacks of hypergeometrics: a remarkable stability6.3 convolution of hypergeometrics with nondisjoint exponents, via a modified sort of hypergeometric; 6.4 Applications to Fourier transforms of Kummer pullbacks of hypergeometrics; Chapter 7- The l-adic theory; 7.1 Exceptional sets of primes; 7.2 l-adic analogue of the main D.E. theorem 2.8.1; 7.3 Construction of irreducible sheaves via Fourier transform; 7.4 Local monodromy of Fourier transforms d'apres Laumon; 7.5 ""Numerical"" explicitation of Lauman's results 7.6 Pseudoreflection examples and applications7.7 A highest slope application; 7.8 Fourier transform-stable classes of sheaves; 7.9 Fourier transforms of tame pseudoreflection sheaves; 7.10 Examples; 7.11 Sato-Tate laws for one-variable exponential sums; 7.12 Special linear examples; 7.13 Symplectic examples; 7.14 Orthogonal examples; Chapter 8- l-adic hypergeometrics; 8.1 Rapid review of perversity, Fourier transform, and convolution; 8.2 Definition of hypergeometric complexes and hypergeometric sums over finite fields; 8.3 Variant: hypergeometric complexes over algebraically closed fields Analysis Differential equations Exponential sums Bibliography Includes bibliographical references (pages 425-430) Subject Exponential sums. Differential equations. MATHEMATICS -- Differential Equations -- General. Differential equations Exponential sums Form Electronic book ISBN 0691085986 9780691085982 0691085994 9780691085999 1400882435 9781400882434

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