A Chabauty-Coleman bound for surfaces

TL;DR

This paper extends Coleman's explicit bound for rational points from curves to hyperbolic surfaces embedded in abelian varieties, using novel p-adic differential equations techniques.

Contribution

It introduces a new method to bound rational points on surfaces by generalizing Coleman's approach for curves, involving overdetermined differential systems in positive characteristic.

Findings

First explicit bound for rational points on surfaces

Method applicable to hyperbolic surfaces in abelian varieties

Uses p-adic differential equations to analyze intersections

Abstract

Building on work by Chabauty from 1941, Coleman proved in 1985 an explicit bound for the number of rational points of a curve $C$ of genus $g\ge 2$ defined over a number field $F$, with Jacobian of rank at most $g-1$. Namely, in the case $F=\mathbb{Q}$, if $p>2g$ is a prime of good reduction, then the number of rational points of $C$ is at most the number of $\mathbb{F}_p$-points plus a contribution coming from the canonical class of $C$. We prove a result analogous to Coleman's bound in the case of a hyperbolic surface $X$ over a number field, embedded in an abelian variety $A$ of rank at most one, under suitable conditions on the reduction type at the auxiliary prime. This provides the first extension of Coleman's explicit bound beyond the case of curves. The main innovation in our approach is a new method to study the intersection of a $p$-adic analytic subgroup with a subvariety of $A$ by means of overdetermined systems of differential equations in positive characteristic.