Show Summary Details

- Title Pages
- Introduction
- Chapter 1 Overview
- Chapter 2 Convolution of Perverse Sheaves
- Chapter 3 Fibre Functors
- Chapter 4 The Situation over a Finite Field
- Chapter 5 Frobenius Conjugacy Classes
- Chapter 6 Group-Theoretic Facts about Ggeom and Garith
- Chapter 7 The Main Theorem
- Chapter 8Isogenies, Connectedness, and Lie-Irreducibility
- Chapter 9 Autodualities and Signs
- Chapter 10 A First Construction of Autodual Objects
- Chapter 11 A Second Construction of Autodual Objects
- Chapter 12 The Previous Construction in the Nonsplit Case
- Chapter 13 Results of Goursat-Kolchin-Ribet Type
- Chapter 14The Case of SL(2); the Examples of Evans and Rudnick
- Chapter 15 Further SL(2) Examples, Based on the Legendre Family
- Chapter 16 Frobenius Tori and Weights; Getting Elements of Garith
- Chapter 17 GL(n) Examples
- Chapter 18 Symplectic Examples
- Chapter 19 Orthogonal Examples, Especially SO(n) Examples
- Chapter 20 GL(n) × GL(n) × … × GL(n) Examples
- Chapter 21 SL(n) Examples, for n an Odd Prime
- Chapter 22 SL(n) Examples with Slightly Composite n
- Chapter 23 Other SL(n) Examples
- Chapter 24 An O(2n) Example
- Chapter 25 G2 Examples: the Overall Strategy
- Chapter 26 G2 Examples: Construction in Characteristic Two
- Chapter 27 G2 Examples: Construction in Odd Characteristic
- Chapter 28 The Situation over ℤ: Results
- Chapter 29The Situation over ℤ: Questions
- Chapter 30Appendix: Deligne’s Fibre Functor
- Bibliography
- Index

# The Situation over a Finite Field

# The Situation over a Finite Field

- Chapter:
- (p.25) Chapter 4 The Situation over a Finite Field
- Source:
- Convolution and Equidistribution
- Author(s):
### Nicholas M. Katz

- Publisher:
- Princeton University Press

DOI:10.23943/princeton/9780691153308.003.0005

This chapter deals with the case of a finite field *k*, and a group scheme 𝓖/*k*. which is a form of 𝔾ₘ.

*Keywords:*
number theory, finite field, group scheme

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- Title Pages
- Introduction
- Chapter 1 Overview
- Chapter 2 Convolution of Perverse Sheaves
- Chapter 3 Fibre Functors
- Chapter 4 The Situation over a Finite Field
- Chapter 5 Frobenius Conjugacy Classes
- Chapter 6 Group-Theoretic Facts about Ggeom and Garith
- Chapter 7 The Main Theorem
- Chapter 8Isogenies, Connectedness, and Lie-Irreducibility
- Chapter 9 Autodualities and Signs
- Chapter 10 A First Construction of Autodual Objects
- Chapter 11 A Second Construction of Autodual Objects
- Chapter 12 The Previous Construction in the Nonsplit Case
- Chapter 13 Results of Goursat-Kolchin-Ribet Type
- Chapter 14The Case of SL(2); the Examples of Evans and Rudnick
- Chapter 15 Further SL(2) Examples, Based on the Legendre Family
- Chapter 16 Frobenius Tori and Weights; Getting Elements of Garith
- Chapter 17 GL(n) Examples
- Chapter 18 Symplectic Examples
- Chapter 19 Orthogonal Examples, Especially SO(n) Examples
- Chapter 20 GL(n) × GL(n) × … × GL(n) Examples
- Chapter 21 SL(n) Examples, for n an Odd Prime
- Chapter 22 SL(n) Examples with Slightly Composite n
- Chapter 23 Other SL(n) Examples
- Chapter 24 An O(2n) Example
- Chapter 25 G2 Examples: the Overall Strategy
- Chapter 26 G2 Examples: Construction in Characteristic Two
- Chapter 27 G2 Examples: Construction in Odd Characteristic
- Chapter 28 The Situation over ℤ: Results
- Chapter 29The Situation over ℤ: Questions
- Chapter 30Appendix: Deligne’s Fibre Functor
- Bibliography
- Index