In honour of George Szekeres on his 90th birthday
If X is a variety over a number field K, the set of K-rational points on X is contained in the subset of the adelic points cut out by the Brauer group; we call this set the set of Brauer points on the variety. If S is a set of valuations of K, we also define S-Brauer points in a natural way. It is natural to ask how good a bound on the rational points is provided by the Brauer (or S-Brauer) points.
Let p > 3 be a prime number, and let X be the Fermat curve of degree p, xp + yp = 1. Let K be the field of p-th roots of unity, and let r be the p-rank of the class group of K. In this paper we show that if r < (p + 3)/8, then the set of p-Brauer points on X has cardinality at most p. We construct elements of the Brauer group of X by relating it to the Weil-Chatelet group of the jacobian of X, then use the method of Coleman and Chabauty to bound the points cut out by these elements.