Katz Exponential Sums and Differential Equations

1. Auflage 2016
ISBN: 978-1-4008-8243-4
Verlag: De Gruyter
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 448 Seiten

Reihe: Annals of Mathematics Studies

This book is concerned with two areas of mathematics, at first sight disjoint, and with some of the analogies and interactions between them. These areas are the theory of linear differential equations in one complex variable with polynomial coefficients, and the theory of one parameter families of exponential sums over finite fields. After reviewing some results from representation theory, the book discusses results about differential equations and their differential galois groups (G) and one-parameter families of exponential sums and their geometric monodromy groups (G). The final part of the book is devoted to comparison theorems relating G and G of suitably "corresponding" situations, which provide a systematic explanation of the remarkable "coincidences" found "by hand" in the hypergeometric case.

Autoren/Hrsg.

Katz, Nicholas M.

Fachgebiete

• Mathematik | Informatik Mathematik Mathematische Analysis Differentialrechnungen und -gleichungen

Inhaltsverzeichnis

Frontmatter, pg. i
Contents, pg. v
Introduction, pg. 1
CHAPTER 1. Results from Representation Theory, pg. 9
CHAPTER 2. D.E.'s and D-modules, pg. 31
CHAPTER 3. The Generalized Hypergeometric Equation, pg. 92
CHAPTER 4. Detailed Analysis of the Exceptional Cases, pg. 122
CHAPTER 5. Convolution of D-modules, pg. 161
CHAPTER 6. Fourier Transforms of Kummer Pullbacks of Hypergeometrics, pg. 178
CHAPTER 7. The l- adic Theory, pg. 193
CHAPTER 8. l-adic Hypergeometrics, pg. 251
CHAPTER 9. G2 Examples, Fourier Transforms, and Hypergeometrics, pg. 320
CHAPTER 10. l -adic Exceptional Cases, pg. 332
CHAPTER 11. Reductive Tannakian Categories, pg. 352
CHAPTER 12. Fourier Universality, pg. 363
CHAPTER 13. Stratifications and Convolution, pg. 384
CHAPTER 14. The Fundamental Comparison Theorems, pg. 402
References, pg. 425