The Reduced Unitary Whitehead Groups over Function Fields of p-adic Curves Öffentlichkeit

Abstract

The study of the Whitehead groups of semi-simple simply connected groups is classical with an abundance of new open questions concerning the triviality of these groups. The Kneser-Tits conjecture on the triviality of these groups was answered in the negative by Platanov for general fields. There is a relation between reduced Whitehead groups and R-equivalence classes in algebraic groups.

Let G be an algebraic group over a field F. Let RG(F) be the equivalence class of the identity element in G(F). Then RG(F) is a normal subgroup of G(F) and the quotient G(F)/RG(F) is called the group of R-equivalence classes of G(F). It is well known that for the semi-simple simply connected isotropic group G over F, the Whitehead group W(G, F) is isomorphic to the group of R-equivalence classes.

Suppose that D_0 is a central division F_0-algebra for a field F_0. If the group G(F_0) of rational points is given by SL_n(D_0) for an integer n > 1, then W(G, F_0) is the reduced Whitehead group of D_0. Let F be a quadratic field extension of F_0 and D be a central division F-algebra. Suppose that D has an involution of second kind τ such that F^τ = F_0. If the Hermitian form h_τ of τ is isotropic and the group G(F_0) is given by SU(D, h_τ ), then W(G, F_0) is isomorphic to the reduced unitary Whitehead group of D.

Let F_0 be the function field of a p-adic curve. Let F/F_0 be a quadratic field extension. Let A be a central simple algebra over F. Assume that the period of A is 2 and A has a unitary F/F_0 involution. We provide a proof for the triviality of the reduced unitary Whitehead group of A.

Table of Contents

1 Introduction 1

1.1 An Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Algebras with Involutions 4

2.1 Central Simple Algebras: An Introduction . . . . . . . . . . . . . . . 4

2.2 Unitary Involutions: Existence Criterion and Properties . . . . . . . . 13

3 Whitehead Groups 20

3.1 Reduced Whitehead Group: SK1 . . . . . . . . . . . . . . . . . . . . 20

3.2 Reduced Unitary Whitehead Group: SK1U . . . . . . . . . . . . . . 22

3.3 R-Equivalence and Whitehead Groups of Algebraic Groups . . . . . . 25

4 Patching 28

4.1 Patching for Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . 28

4.2 Local-Global Principles over Arithmetic Curves . . . . . . . . . . . . 30

5 SK1U over Function Fields of p-adic Curves 37

5.1 The Plan of the proof . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.3 Weak Approximations over Global Fields . . . . . . . . . . . . . . . . 41

5.4 Complete Discretely Valued Fields . . . . . . . . . . . . . . . . . . . . 44

5.5 Two Dimensional Complete Fields . . . . . . . . . . . . . . . . . . . . 47

5.6 Choice at Nodal Points . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.7 Choices at Codimension One Points and Curve Points . . . . . . . . . 54

5.8 Choice of U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.9 The main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

About this Dissertation

Rights statement

• Permission granted by the author to include this thesis or dissertation in this repository. All rights reserved by the author. Please contact the author for information regarding the reproduction and use of this thesis or dissertation.

School	Laney Graduate School
Department	Mathematics
Degree	Ph.D.
Submission	Dissertation
Language	English
Research Field	Mathematics
Stichwort	Patching Number theory Involution Whitehead group
Committee Chair / Thesis Advisor	Raman Parimala, Emory University
Committee Members	Victoria Powers, Emory University Suresh Venapally, Emory University

Zuletzt geändert

Primary PDF

Thumbnail	Title	Date Uploaded	Actions
	The Reduced Unitary Whitehead Groups over Function Fields of p-adic Curves ()	2024-06-26 12:42:09 -0400	Press to Select an action Download

Supplemental Files

Thumbnail	Title	Date Uploaded	Actions