Growth of points on hyperelliptic curves over number fields

Abstract

Fix a hyperelliptic curve C/ℚ of genus g, and consider the number fields K/ℚ generated by the algebraic points of C. In this paper, we study the number of such extensions with fixed degree n and discriminant bounded by X. We show that when g≥1 and n is sufficiently large relative to the degree of C, with n even if degC is even, there are ≫X c n such extensions, where c n is a positive constant depending on g which tends to 1/4 as n→∞. This result builds on work of Lemke Oliver and Thorne who, in the case where C is an elliptic curve, put lower bounds on the number of extensions with fixed degree and bounded discriminant over which the rank of C grows with specified root number.