Reduced Whitehead groups of prime exponent algebras over p-adic curves

TL;DR

This paper proves that for certain prime exponent division algebras over p-adic curve function fields, the reduced Whitehead group SK_1 is trivial, extending understanding of algebraic K-theory in this context.

Contribution

It establishes the triviality of SK_1 for prime exponent division algebras over p-adic curves under specific roots of unity conditions, a new result in algebraic K-theory.

Findings

SK_1(D) is trivial for the given algebras

Conditions on roots of unity are crucial for the result

Advances understanding of algebraic structures over p-adic curves

Abstract

Let F be the function field of a curve over a p-adic field. Let D/F be a central division algebra of prime exponent $\ell$ which is different from p. Assume that F contains a primitive ${\ell}^2$-th root of unity. Then the group $SK_1(D):=SL_1(D)/[D^*, D^*]$ is trivial.