Reduced Whitehead groups of prime exponent algebras 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 $S{K}_1(D):=S{L}_1(D)/[D^{\ast },D^{\ast }]$ is trivial.