The Reduced Unitary Whitehead Groups over Function Fields of p-adic Curves Open Access
Pei, Zitong (Summer 2024)
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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