Exponential Sums, Hypergeometric Sheaves, and Monodromy Groups.

Format:

Book

Author/Creator:

Katz, Nicholas M.

Contributor:

Tiep, Pham Huu.

Series:

Annals of Mathematics Studies

Annals of Mathematics Studies ; v.220

Language:

English

Subjects (All):

Exponential sums.

Sheaf theory.

Monodromy groups.

Physical Description:

1 online resource (0 pages)

Edition:

1st ed.

Place of Publication:

Princeton : Princeton University Press, 2025.

System Details:

Mode of access: World Wide Web.

Summary:

"This is a monograph in the Annals of Mathematics series that studies the monodromy groups, associated to certain local systems arising from hypergeometric sheaves (and with some relationship with exponential sums). In the book, the authors first deduce various conditions on the monodromy group associated to the local system to get some quite powerful classification theorems. Some features of the book include: (a) introductory material on monodromy groups; (b) various results translating the information on hypergeometric sheaves and local system to group theory; (c) a summary of some results in representation theory which are not available in book form; (d) new results on irreducible subgroups of classical groups containing special elements, which will have applications in many different areas; and (e) results on moments and deciding when the monodromy group is infinite and estimating its size when it is finite"-- Provided by publisher.

"An examination of some of the remarkable connections between group theory and arithmetic algebraic geometry over finite fieldsExponential sums have been of great interest ever since Gauss, and their importance in analytic number theory goes back a century to Kloosterman. Grothendieck's creation of the machinery of l-adic cohomology led to the understanding that families of exponential sums give rise to local systems, while Deligne, who gave his general equidistribution theorem after proving the Riemann hypothesis part of the Weil conjectures, established the importance of the monodromy groups of these local systems. Deligne's theorem shows that the monodromy group of the local system incarnating a given family of exponential sums determines key statistical properties of the family of exponential sums in question. Despite the apparent simplicity of this relation of monodromy groups to statistical properties, the actual determination of the monodromy group in any particular situation is highly nontrivial and leads to many interesting questions.This book is devoted to the determination of the monodromy groups attached to various explicit families of exponential sums, especially those attached to hypergeometric sheaves, arguably the simplest local systems on G_m, and to some simple (in the sense of simple to write down) one-parameter families of one-variable sums. These last families turn out to have surprising connections to hypergeometric sheaves. One of the main technical advances of this book is to bring to bear a group-theoretic condition (S+), which, when it applies, implies very strong structural constraints on the monodromy group, and to show that (S+) does indeed apply to the monodromy groups of most hypergeometric sheaves"-- Provided by publisher.

Contents:

Cover

Contents

Introduction

Chapter 1. The basic (S-), (S), and (S+) settings

1.1. Conditions (S-), (S), and (S+) for local systems

1.2. Kloosterman and hypergeometric sheaves

1.3. More on condition (S+) forhypergeometric sheaves

1.4. Moments and monodromy

Chapter 2. Some basic facts about monodromy groups

2.1. Arithmetic semisimplicity

2.2. Finiteness of Ggeom and Garith

2.3. Geometric and arithmetic determinants

2.4. Infinite monodromy groups

2.5. Estimating the size of Ggeom when it is finite

2.6. Limsup formula for moments

Chapter 3. Representations of reductive groups containing elements with special spectra

3.1. Almost quasisimple groups containing elements with simple spectra

3.2. Modules with small weight multiplicities

3.3. Regular spectrum and simple spectrum elements

Chapter 4. Hypergeometric sheaves with wild part of dimension one

4.1. General situation

4.2. Further analysis

Chapter 5. Tensor induced local systems

5.1. 2-tensor induced sheaves

5.2. Tensor induced sheaves: General case

Chapter 6. (Non-)existence results

6.1. Type A

6.2. Other types

Chapter 7. Extraspecial normalizers and local systems in odd characteristic

7.1. A supersingularity result

7.2. Linear groups in characteristic p &gt

2

7.3. Local systems in characteristic p &gt

Chapter 8. Extraspecial normalizers and local systems in characteristic 2

8.1. Squared traces in characteristic 2

8.2. Traces of elements in normalizersof extraspecial 2-groups

8.3. Linear groups in characteristic 2

8.4. Unitary-type subgroups

8.5. Local systems in characteristic p=2

Chapter 9. Two further kinds of local system in characteristic 2

9.1. Another kind of hypergeometric sheaf in characteristic 2.

9.2. Local systems in characteristic 2 with Witt vectors: The F2 story

9.3. Local systems with Witt vectors: The Fq story

Chapter 10. One-parameter families of exponential sums

10.1. Generalities

10.2. The (A,1)-case

10.3. The (A,B)-case

Chapter 11. Multi-parameter families of exponential sums

11.1. Preliminaries

11.2. The general case

Chapter 12. Local systems with non-monomial coefficients

12.1. Local systems of the first kind

12.2. Local systems of the second kind

12.3. Local systems of the third kind

12.4. Another approach to M2,2

12.5. Some applications of almost injectivity

Appendices

Appendix A1: The Magma program usedin Lemma 6.2.6

Appendix A2: The Magma program usedin Lemma 10.3.17

Acknowledgements

Bibliography

Symbols

Index.

Notes:

Includes bibliographical references and index.

Description based on publisher supplied metadata and other sources.

ISBN:

0-691-27224-7

OCLC:

1517397499