Uniform bounds for the number of rational points on hyperelliptic curves of small Mordell-Weil rank

TL;DR

This paper establishes uniform bounds on the number of rational points on hyperelliptic curves of small Mordell-Weil rank, using p-adic methods and covering techniques, with implications for the distribution of rational points.

Contribution

It introduces a new estimate for zeros of logarithms in p-adic annuli and applies it to bound rational points on hyperelliptic curves, extending previous results.

Findings

Bound depends only on genus g and degree [K:Q]

Explicit bound for Q: 8 r g + 33 (g - 1) + 1

Density of hyperelliptic curves with only point at infinity tends to 1

Abstract

We show that there is a bound depending only on g 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 8 r g + 33 (g - 1) + 1. The proof is based on Chabauty's method; the new ingredient is an estimate for the number of zeros of a logarithm in 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.