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

Abstract

We show that there is a bound depending only on $g,r$ and [$K:\mathbb{Q}$] 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 $K=\mathbb{Q}$, 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 $\mathbb{Q}$ 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.