Zero-Cycles on Torsors under Linear Algebraic Groups Pubblico
Gordon-Sarney, Reed (2017)
Published
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
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