Cyclic algebras over p-adic curves

In this paper we study division algebras over the function fields of curves over . The first and main tool is to view these fields as function fields over nonsingular S which are projective of relative dimension 1 over the p adic ring . A previous paper showed such division algebras had index bounded by n2 assuming the exponent was n and n was prime to p. In this paper we consider algebras of prime degree (and hence exponent) q≠p and show these algebras are cyclic. We also find a geometric criterion for a Brauer class to have index q.