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

TL;DR

This paper establishes explicit uniform bounds on the number of rational points on algebraic curves of small Mordell--Weil rank, extending previous results to all curves and providing unconditional bounds on torsion points.

Contribution

It provides the first explicit uniform bounds for rational points on arbitrary curves with Mordell--Weil rank at most g-3, generalizing prior hyperelliptic results.

Findings

Explicit uniform bound on rational points for curves with rank ≤ g-3

Unconditional bound on torsion points on Jacobians

Bounds applicable to highly degenerate reduction types

Abstract

Let $X$ be a curve of genus $g\geq 2$ over a number field $F$ of degree $d = [F:Q]$. The conjectural existence of a uniform bound $N(g,d)$ on the number $\#X(F)$ of $F$-rational points of $X$ is an outstanding open problem in arithmetic geometry, known by [CHM97] to follow from the Bombieri--Lang conjecture. A related conjecture posits the existence of a uniform bound $N_{{\rm tors},\dagger}(g,d)$ on the number of geometric torsion points of the Jacobian $J$ of $X$ which lie on the image of $X$ under an Abel--Jacobi map. For fixed $X$ this quantity was conjectured to be finite by Manin--Mumford, and was proved to be so by Raynaud [Ray83]. We give an explicit uniform bound on $\#X(F)$ when $X$ has Mordell--Weil rank $r\leq g-3$. This generalizes recent work of Stoll on uniform bounds on hyperelliptic curves of small rank to arbitrary curves. Using the same techniques, we give an explicit, unconditional uniform bound on the number of $F$-rational torsion points of $J$ lying on the image of $X$ under an Abel--Jacobi map. We also give an explicit uniform bound on the number of geometric torsion points of $J$ lying on $X$ when the reduction type of $X$ is highly degenerate. Our methods combine Chabauty--Coleman's $p$-adic integration, non-Archimedean potential theory on Berkovich curves, and the theory of linear systems and divisors on metric graphs.