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

Pei, Zitong (Summer 2024)

Permanent URL: https://etd.library.emory.edu/concern/etds/gm80hw80f?locale=en
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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

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