On the average number of rational points of bounded height on hyperelliptic curves

Abstract

In this note, an estimate for the average number of rational points of height at most H on genus g ≥ 2 curves is obtained. It gives the asymptotics of the average over curves of size up to N with error term \({O(N^{-1}H^2 + N^{-\frac{3}{2}}H^3)}\). Lower asymptotic density of rational points of arbitrary height is described.