Exponential Sums and Differential Equations. (AM-124), Volume 124 / Nicholas M. Katz.

Format:

Book

Author/Creator:

Katz, Nicholas M., author.

Series:

Annals of mathematics studies ; no. 124.

Annals of Mathematics Studies ; 305

Language:

English

Subjects (All):

Exponential sums.

Differential equations.

Physical Description:

1 online resource (445 pages) : illustrations.

Place of Publication:

Princeton, NJ : Princeton University Press, [2016]

Language Note:

English

Summary:

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.

Contents:

Frontmatter

Contents

Introduction

CHAPTER 1. Results from Representation Theory

CHAPTER 2. D.E.'s and D-modules

CHAPTER 3. The Generalized Hypergeometric Equation

CHAPTER 4. Detailed Analysis of the Exceptional Cases

CHAPTER 5. Convolution of D-modules

CHAPTER 6. Fourier Transforms of Kummer Pullbacks of Hypergeometrics

CHAPTER 7. The ℓ- adic Theory

CHAPTER 8. ℓ-adic Hypergeometrics

CHAPTER 9. G2 Examples, Fourier Transforms, and Hypergeometrics

CHAPTER 10. ℓ -adic Exceptional Cases

CHAPTER 11. Reductive Tannakian Categories

CHAPTER 12. Fourier Universality

CHAPTER 13. Stratifications and Convolution

CHAPTER 14. The Fundamental Comparison Theorems

References

Notes:

Bibliographic Level Mode of Issuance: Monograph

Includes bibliographical references.

Description based on online resource; title from PDF title page (publisher's Web site, viewed 08. Jul 2019)

ISBN:

1-4008-8243-5

OCLC:

979633762