Division Algebras Over p-Adic Curves

Let K be a p-adic field, C a curve defined over K, and n a positive integer prime to p. Let L = K (C) be the function field. In this paper we show that every Brauer group element in Br (L) of order n has index dividing n². More generally, we prove a similar result when K is a local field of characteristic p for n prime to p, or in fact any field of the following form. Let A be an Henselian discrete valuation ring with field of fractions K and residue field k. Suppose n is an integer prime to the characteristic of k. Assume every projective curve over any finite extension of k has 0 Brauergroup. Then if C is a curve over K, every element ofBr (K{C)) of order n has Schur index a divisor of n². Suppose on the other hand that that A, K are as above, but the residue field k has the following properties. First, every finite extention of k has zero Brauer group. Second, if I is the function field of a curve over/c, every element of the Brauer group of I has exponent equal to index. Then if C is a curve over K, every element of Br {K{C)) of order n has Schur index a divisor of n³. This paper also has an appendix, by William Jacob and J.-P Tignol, with an example of a division of degree n² and exponent n over Q p(t).