Uniform Bounds for the Number of Rational Points on Hyperelliptic Curves of Small Mordell-weil Rank

We show that there is a bound depending only on g, r and [K : Q] for the number of K-rational points on a hyperelliptic curve C of genus g over a number field K such that the Mordell-Weil rank r of its Jacobian is at most g − 3. If K = Q, an explicit bound is 8rg + 33(g − 1) + 1. The proof is based on Chabauty’s method; the new ingredient is an estimate for the number of zeros of an abelian logarithm on a p-adic ‘annulus’ on the curve, which generalizes the standard bound on disks. The key observation is that for a p-adic field k, the set of k-points on C can be covered by a collection of disks and annuli whose number is bounded in terms of g (and k). We also show, strengthening a recent result by Poonen and the author, that the lower density of hyperelliptic curves of odd degree over Q whose only rational point is the point at infinity tends to 1 uniformly over families defined by congruence conditions, as the genus g tends to infinity.