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

Let F be a p-adic field, that is, finite extension of Qp. D dimensional division algebra over F, and let SL1(D) the group elements reduced norm 1 in D. Prasad Raghunathan proved H2(SL1(D),R/Z) is cyclic p-group whose order bounded from below by number p-power roots unity unless quaternion Q2. In this paper we give an explicit upper bound for p≥5 determine precisely when cyclotomic, p≥19, degree not power p.