Zero Cycles of Degree One on Principal Homogeneous Spaces Público

Abstract

Abstract
Zero Cycles of Degree One on Principal Homogeneous Spaces
By Jodi A. Black

Let k be a field and let G be a connected linear algebraic group over k. Let X be a principal homogeneous space under G over k. Jean-Pierre Serre has asked the following: "If X admits a zero cycle of degree one, does X have a k-rational point?"

We give a positive answer to the question in two settings:
1. The field k is of characteristic different from 2 and the group G is simply connected or adjoint and of classical type.
2. The field k is perfect and of virtual cohomological dimension at most 2 and the
simply connected group associated to G satisfies a Hasse principle over k.

Zero Cycles of Degree One on Principal Homogeneous Spaces
By
Jodi A. Black
B.A., Wesleyan University, 2006
Advisor: R. Parimala, Ph.D.
A dissertation submitted to the Faculty of the
James T. Laney School of Graduate Studies of Emory University
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
in Mathematics
2011

Table of Contents

1 Introduction 1
2 Galois Cohomology 3
2.1 Finite Group Cohomology . . . . . . . . . . . . . . . . . . . . . . . . 3
2.1.1 Defining the Cohomology Sets . . . . . . . . . . . . . . . . . . 3
2.1.2 Functoriality . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Profinite Group Cohomology . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.2 Virtual Cohomological Dimension . . . . . . . . . . . . . . . . 10
2.3 Galois Cohomology of Algebraic Groups . . . . . . . . . . . . . . . . 10
2.3.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3.2 Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3.3 Fundamental Results . . . . . . . . . . . . . . . . . . . . . . . 12
3 Algebras with Involution 14
3.1 Central Simple Algebras . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.2 Splitting Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.3 The Brauer group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.4 Involutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.5 Hermitian Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.6 Groups Associated to an Algebra with Involution . . . . . . . . . . . 24

4 Linear Algebraic Groups 26
4.1 Semisimple Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.2 Classification of Absolutely Simple Semisimple Groups . . . . . . . . 27
4.2.1 Classical Groups . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.2.2 Groups of Type G2 . . . . . . . . . . . . . . . . . . . . . . . . 29
4.2.3 Groups of Type F4 . . . . . . . . . . . . . . . . . . . . . . . . 30
4.3 The Homological Torsion Primes . . . . . . . . . . . . . . . . . . . . 31
4.4 Unipotent Groups and Reductive Groups . . . . . . . . . . . . . . . . 32
4.5 Norm Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.6 The Rost Invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5 The Hasse principle 38
5.1 The Hasse Principle over a Number Field . . . . . . . . . . . . . . . . 38
5.2 The Hasse Principle over a Field of Virtual Cohomological Dimension
at most 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

6 A question of Serre 42
6.1 Zero Cycles on Principal Homogeneous Spaces . . . . . . . . . . . . . 42
6.2 The Kernel of the Restriction Map . . . . . . . . . . . . . . . . . . . 43
6.3 Known Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
7 Results under Semisimple Groups 46
7.1 Absolutely Simple Simply Connected Groups of Classical Type . . . . 46
7.2 Absolutely Simple Adjoint Groups of Classical Type . . . . . . . . . . 51
7.3 Exceptional Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
7.4 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
8 Results over Virtual Cohomological Dimension 2 Fields 65
8.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
8.2 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
9 Conclusion 72
Bibliography 74

About this Dissertation

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School	Laney Graduate School
Department	Math and Computer Science
Degree	PhD
Submission	Dissertation
Language	English
Research Field	Mathematics
Palabra Clave	Zero Cycles Linear Algebraic Groups Galois Cohomology
Committee Chair / Thesis Advisor	Raman, Parimala, Emory University
Committee Members	Ono, Ken, Emory University Garibaldi, Skip, Emory University

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