Zero-Cycles on Torsors under Linear Algebraic Groups Público

Abstract

Let k be a field, let G be a smooth connected linear algebraic group over k, and let
X be a G-torsor. Totaro asked: if X admits a zero-cycle of positive degree d, does X have a
closed etale point of degree dividing d? We give a positive answer in two cases:
1. G is an algebraic torus of rank at most 2 and char(k) is arbitrary, and
2. G is an absolutely simple adjoint group of type A1 or A2n and char(k) is not 2.
We also give the first known examples where Totaro's question has a negative answer.
In particular, we exhibit failures via tori over number fields, p-adic fields, and complete
discrete valuation fields with global residue fields of characteristic not 2 and show that
Totaro's question has a negative answer in general for tori of all ranks at least 3.

Table of Contents

1 Introduction.........................................................................1

2 Fundamental Objects.............................................................4
2.1 Central Simple Algebras......................................................4
2.1.1 The Brauer Group............................................................4
2.1.2 Splitting Fields................................................................5
2.1.3 Involutions.....................................................................6
2.2 Linear Algebraic Groups......................................................6
2.2.1 Algebraic Tori.................................................................7
2.2.2 Adjoint Groups of Type An................................................8
2.2.3 Torsors and Zero-Cycles...................................................9

3 Galois Cohomology..............................................................10
3.1 Finite Group Cohomology...................................................10
3.2 Profinite Group Cohomology.................................................12
3.3 Some Important Maps.......................................................14
3.4 Some Important Computations...........................................15
3.5 Totaro's Question, Revisited...............................................17

4 Totaro's Question for Tori of Low Rank....................................18
4.1 Lemmata........................................................................19
4.2 Technical Results..............................................................24
4.3 Proof of Theorem 4.0.1.....................................................34
4.3.1 Rank 1 Tori...................................................................35
4.3.2 Rank 2 Tori...................................................................36
4.4 del Pezzo Surfaces...........................................................41

5 Totaro's Question for Adjoint Groups of Types A1 and A2n..........44
5.1 Lemmata........................................................................45
5.2 Proof of Theorem 5.0.4.....................................................48

6 Negative Answers to Totaro's Question...................................52
6.1 Examples over p-adic Fields..............................................53
6.2 Examples over Other Discrete Valuation Fields......................58

Bibliography.........................................................................63

About this Dissertation

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School	Laney Graduate School
Department	Mathematics
Degree	PhD
Submission	Dissertation
Language	English
Research Field	Mathematics
Palavra-chave	torsors zero-cycles galois cohomology linear algebraic groups
Committee Chair / Thesis Advisor	Raman, Parimala, Emory University
Committee Members	Zureick-Brown, David M, Emory University Venapally, Suresh , Emory University

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