 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 GroupTheoretic Facts about G_{geom} and G_{arith} 
Chapter 7 The Main Theorem 
Chapter 8 Isogenies, Connectedness, and LieIrreducibility 
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 GoursatKolchinRibet Type 
Chapter 14 The 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 G_{arith} 
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 G_{2} Examples: the Overall Strategy 
Chapter 26 G_{2} Examples: Construction in Characteristic Two 
Chapter 27 G_{2} Examples: Construction in Odd Characteristic 
Chapter 28 The Situation over ℤ: Results 
Chapter 29 The Situation over ℤ: Questions 
Chapter 30 Appendix: Deligne’s Fibre Functor  Bibliography
 Index
Isogenies, Connectedness, and LieIrreducibility
Isogenies, Connectedness, and LieIrreducibility
 Chapter:
 (p.45) Chapter 8Isogenies, Connectedness, and LieIrreducibility
 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, lieirreducibility, 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 GroupTheoretic Facts about G_{geom} and G_{arith} 
Chapter 7 The Main Theorem 
Chapter 8 Isogenies, Connectedness, and LieIrreducibility 
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 GoursatKolchinRibet Type 
Chapter 14 The 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 G_{arith} 
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 G_{2} Examples: the Overall Strategy 
Chapter 26 G_{2} Examples: Construction in Characteristic Two 
Chapter 27 G_{2} Examples: Construction in Odd Characteristic 
Chapter 28 The Situation over ℤ: Results 
Chapter 29 The Situation over ℤ: Questions 
Chapter 30 Appendix: Deligne’s Fibre Functor  Bibliography
 Index