On the Second Cohomology of the Norm One Group of a p-Adic Division Algebra

Let F be a p-adic field, that is, a finite extension of Qp. Let D be a finite dimensional division algebra over F and let SL1(D) be the group of elements of reduced norm 1 in D. Prasad and Raghunathan proved that H(SL1(D), R/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 Q2. In this paper we give an explicit upper bound for the order of H(SL1(D), R/Z) for p ≥ 5, and determine H(SL1(D), R/Z) precisely when F is cyclotomic, p ≥ 19 and the degree of D is not a power of p.