Some arithmetic properties of Weierstrass points: Hyperelliptic curves

Abstract

The set of Weierstrass points for pluricanonical linear systems on an algebraic curve C have been extensively studied from a geometric viewpoint. If the curve is defined over a number fieldk, then thesen th order Weierstrass points are defined over an algebraic extensionk n ofk, and it is an interesting question to ask for the arithmetic properties of the points and the extension that they generate. In this paper we begin the study of the arithmetic properties of higher order Weierstrass points in the special case of hyperelliptic curves. We give an upper bound for the average height of these points, and we show that for sufficiently large primesp, the first order Weierstrass points and then th order Weierstrass points remain distinct modulop. This limits to some extent the ramification that can occur in the extensionk n /k. We also present two numerical examples which indicate that a complete description of the ramification is likely to be complicated.