A local-global principle for adjoint groups over function fields of p-adic curves Open Access

Abstract

Let k be a number field and G a semisimple simply connected linear algebraic group

over k. The Kneser conjecture states that the Hasse principle holds for principal

homogeneous spaces under G. Kneser’s conjecture is a theorem due to Kneser for all

classical groups, Harder for exceptional groups other than E8, and Chernousov for

E8. It has also been proved by Sansuc that if G is an adjoint linear algebraic group

over k, then the Hasse principle holds for principal homogeneous spaces under G.

Now let p ∈ N be a prime with p ̸= 2, and let K be a p-adic field. Let F be the function

field of a curve over K. Let ΩF be the set of all divisorial discrete valuations of F.

It is a conjecture of Colliot-Th´el`ene, Parimala and Suresh that if G is a semisimple

simply connected linear algebraic group over F, then the Hasse principle holds for

principal homogeneous spaces under G. This conjecture has been proved for all groups

of classical type. In this thesis, we ask whether the Hasse principle holds for adjoint

groups over F, motivated by the number field case. We give a positive answer to this

question for a class of adjoint classical groups.

Table of Contents

1 Introduction 1

1.1 The Plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Central Simple Algebras, Involutions and Hermitian Forms 3

2.1 Central Simple Algebras . . . . . . . . . . . . . . . . . . . . . 3

2.1.1 The Brauer Group . . . . . . . . . . . . . . . . . . . . 4

2.1.2 Quaternion Algebras . . . . . . . . . . . . . . . . . . . 6

2.1.3 Ramifications of Central Simple Algebras . . . . . . . 6

2.2 Involutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.1 Similitudes of Algebras with Involution . . . . . . . . 9

2.3 Hermitian Forms . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3.1 Hermitian Forms over Division Algebras and

Quadratic Forms . . . . . . . . . . . . . . . . . . . . . 14

2.3.2 Quadratic Forms over Complete Discretely Valuated

Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3 Galois Cohomology 18

3.1 Profinite Groups and Galois Groups . . . . . . . . . . . . . . 18

3.2 Cohomology of Profinite Groups . . . . . . . . . . . . . . . . 19

3.3 Principal Homogeneous Spaces . . . . . . . . . . . . . . . . . 20

4 Linear Algebraic Groups and Patching Techniques 22

4.1 First Definitions . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.2 Classification of Absolutely Simple, Adjoint, Classical Linear

Algebraic Groups . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.3 Semi-Global Fields and Patching . . . . . . . . . . . . . . . . 27

4.4 Local-Global Principles for Linear Algebraic Groups . . . . . 28

5 Main Theorems 30

5.1 Quadratic Forms Over Two Dimensional

Complete Fields . . . . . . . . . . . . . . . . . . . . . . . . . 30

5.2 Semi-Global Fields - Quadratic Forms Case . . . . . . . . . . 35

5.3 Quaternion Division Algebras Over Two

Dimensional Complete Fields . . . . . . . . . . . . . . . . . . 39

5.4 Semi-Global Fields - Symplectic Involution

Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5.5 The Main Theorems . . . . . . . . . . . . . . . . . . . . . . . 42

About this Dissertation

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School	Laney Graduate School
Department	Mathematics
Degree	Ph.D.
Submission	Dissertation
Language	English
Research Field	Mathematics
Keyword	Algebra
Committee Chair / Thesis Advisor	Parimala Raman, Emory University
Committee Members	Suresh Venapally, Emory University David Zureick-Brown, Emory University

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