- 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

# Isogenies, Connectedness, and Lie-Irreducibility

# Isogenies, Connectedness, and Lie-Irreducibility

- Chapter:
- (p.45) Chapter 8Isogenies, Connectedness, and Lie-Irreducibility
- Source:
- Convolution and Equidistribution
- Author(s):
### Nicholas M. Katz

- Publisher:
- Princeton University Press

This chapter takes up the proofs of Theorems 8.1 and 8.2. For each prime to *p* integer *n*, we have the *n*'th power homomorphism [*n*] : G → G. Formation of the direct image is an exact functor from *Perv* to itself, which maps *Neg* to itself, in Ƿ to itself, and which (because a homomorphism) is compatible with middle convolution. So for a given object *N* in G_{arith}, [*n*]_{*} allows us to view 〈*N*〉_{arith} as a Tannakian subcategory of 〈[n]_{*}*N*〉_{arith}, and 〈*N*〉_{geom} as a Tannakian subcategory of 〈[*n*]_{*}*N*〉_{geom}.

*Keywords:*
number theory, isogenies, connectedness, lie-irreducibility, Tannakian groups

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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