Exponential Sums, Hypergeometric Sheaves, and Monodromy Groups
1st Edition
Published on 2. May 2025
352 pages
978-0-691-27224-5 (ISBN)
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Nicholas M. Katz | Pham Huu Tiep
Exponential Sums, Hypergeometric Sheaves, and Monodromy Groups
Book
06/2025
Princeton University Press
€77.00
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Nicholas M. Katz | Pham Huu Tiep
Exponential Sums, Hypergeometric Sheaves, and Monodromy Groups
Book
06/2025
Princeton University Press
€167.50
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Persons
Nicholas M. Katz is professor of mathematics at Princeton University. His books include Convolution and Equidistribution: Sato-Tate Theorems for Finite-Field Mellin Transforms (Princeton). Pham Huu Tiep is the Joshua Barlaz Professor and Distinguished Professor of Mathematics at Rutgers University.
Content
- 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 & 2
- 7.3. Local systems in characteristic p & 2
- 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
