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

TL;DR

This paper proves the triviality of the reduced unitary Whitehead group for certain central simple algebras over function fields of p-adic curves, under specific conditions.

Contribution

It establishes the triviality of the reduced unitary Whitehead group for period 2 algebras with involution over these function fields, extending understanding in algebraic K-theory.

Findings

Proves $SK_1U( A, au)$ is trivial when $p eq 2$.

Focuses on algebras of period 2 over function fields of p-adic curves.

Provides new results in algebraic K-theory related to involutions and central simple algebras.

Abstract

Let $F_0$ be the function field of a curve over a $p$-adic field $K,$ and let $F$ be a quadratic extension over $F_0$. Let $A$ be a central simple algebra over $F$ of period $2,$ and let $\tau$ be a $F/F_0$-involution on $A$. We show the triviality of the reduced unitary Whitehead group $SK_1U( A, \tau)$ if $p\neq 2$.