The Weierstrass Subgroup of a Curve Has Maximal Rank

We show that the Weierstrass points of the generic curve of genus g over an algebraically closed field of characteristic 0 generate a group of maximal rank in the Jacobian. The Weierstrass points are a set of distinguished points on curves, which are geometrically intrinsic. In particular, the group these points generate in the Jacobian is a geometric invariant of the curve. A natural question is to determine the structure of this group. For some particular curves with large automorphisms groups (for instance, Fermat curves [15]), these groups have been found to be torsion. The first author provided the first examples where this group has positive rank ([7], [8]) and obtained a lower bound of 11 on the rank of the generic genus 3 curve. The motivation of this paper was to bridge the gap between this bound and the expected bound of 23 – meaning that there are no relations between the Weierstrass points on the generic genus 3 curve. The result we obtain is valid for generic curves of any genus. More precisely, let the Weierstrass subgroup of a curve C be the group generated by the Weierstrass points in the Jacobian of the curve C. We show that Theorem 1. The Weierstrass subgroup of the generic curve of genus g ≥ 3 is Zg(g 2−1)−1. As a consequence of this theorem, we deduce the following corollaries. Corollary 2. For any field K of characteristic zero, the group generated by the Weierstrass points of a curve over K in its Jacobian is Zg(g 2−1)−1, outside of a set of curves whose moduli lie in a thin set in Mg(K). Corollary 3. For every g ≤ 13 there exist infinitely many curves of genus g defined over Q, up to isomorphism over Q̄, for which the group generated by the Weierstrass points in its Jacobian is isomorphic to Zg(g 2−1)−1. We start by recalling some basic definitions and properties of Weierstrass points, then some results concerning the behaviour of Weierstrass points under specialisation. We then describe the fundamental tools in our study, which are the natural Galois module structure of the subgroup of divisors with support on the Weierstrass points and the geometric characterisation of the Galois group. Using the specialisation of this Galois module in families, we obtain the main result. The second author thanks the ARC for financial support, and the third author thanks the School of Mathematics and Statistics at the University of Sydney for its warm hospitality. 2000 Mathematics Subject Classification 11G30, 14H40, 14H10, 14H55, 14Q05