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

### Michael Stoll

Universität Bayreuth, Germany

## Abstract

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.

## Cite this article

Michael Stoll, Uniform bounds for the number of rational points on hyperelliptic curves of small Mordell–Weil rank. J. Eur. Math. Soc. 21 (2019), no. 3, pp. 923–956

DOI 10.4171/JEMS/857