On the second cohomology of the norm one group of a p-adic division algebra

Abstract

Let $F$ be a $p$-adic field, that is, a finite extension of ${\mathbb{Q}}_p$. Let $D$ be a finite-dimensional central division algebra over $F$ and let $S{L}_1(D)$ be the group of elements of reduced norm $1$ in $D$. Prasad and Raghunathan proved that $H^2(S{L}_1(D),\mathbb{R}/\mathbb{Z})$ is a cyclic $p$-group whose order is bounded from below by the number of $p$-power roots of unity in $F$, unless $D$ is a quaternion algebra over ${\mathbb{Q}}_2$. In this paper we give an explicit upper bound for the order of $H^2(S{L}_1(D),\mathbb{R}/\mathbb{Z})$ for $p\ge 5$ and determine $H^2(S{L}_1(D),\mathbb{R}/\mathbb{Z})$ precisely when $F$ is cyclotomic, $p\ge 19$ and the degree of $D$ is not a power of $p$.

67 pages. This is a paper by the first author with an appendix by both authors. To appear in the Michigan Mathematical Journal