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 ${\mathbb{Q}}_{\mathit{p}}$. Let D be a finite dimensional division algebra over F, and let ${\mathrm{SL}}_1(\mathit{D})$ be the group of elements of reduced norm 1 in D. Prasad and Raghunathan proved that ${\mathit{H}}^2({\mathrm{SL}}_1(\mathit{D}),\mathbb{R}/\mathbb{Z})$ 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 ${\mathit{H}}^2({\mathrm{SL}}_1(\mathit{D}),\mathbb{R}/\mathbb{Z})$ for $\mathit{p}\ge 5$ and determine ${\mathit{H}}^2({\mathrm{SL}}_1(\mathit{D}),\mathbb{R}/\mathbb{Z})$ precisely when F is cyclotomic, $\mathit{p}\ge 19$, and the degree of D is not a power of p.