Journal of the Ramanujan Mathematical Society

David J. Saltman ¹

Affiliations
1 Department of Mathematics, The University of Texas Austin, Texas 78712, US

Abstract

There are two problems with the paper in the title, which is reference [l]. One is a genuine gap, and one is a difficulty with an argument that was not actually needed. Let me acknowledge this author's deep gratitude to Ofer Gabber for spotting these two problems, and suggesting fixes. Let me also thank J.-L. Colliot-Thelene for his help. We adopt all the terminology and notation of [l].

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David J. Saltman ¹

Affiliations
1 Department of Mathematics, The University of Texas Austin, Texas-78712, US

Abstract

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).

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