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 ${\ell }^{2th}$ root of unity. Then the abstract group ${\mathrm{SK}}_1(D):=\frac{{\mathrm{SL}}_1(D)}{\left[D^{\ast },D^{\ast }\right]}$ is trivial.