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 $SK_1(D):=\frac{SL_1(D)}{\left[D^*, D^*\right]}$ is trivial.