A Buium--Coleman bound for the Mordell--Lang conjecture

Abstract

For $X$ a hyperbolic curve of genus $g$ with good reduction at $p\geq 2g$, we give an explicit bound on the Mordell--Lang locus $X(\mathbb{C})\cap \Gamma $, when $\Gamma \subset J(\mathbb{C})$ is the divisible hull of a subgroup of $J(\mathbb{Q} _p ^{\mathrm{nr}})$ of rank less than $g$. Without any assumptions on the rank (but with all the other assumptions) we show that $X(\mathbb{C})\cap \Gamma $ is unramified at $p$, and bound the size of its image in $X(\overline{\mathbb{F} }_p )$. As a corollary, we obtain a new proof that Mordell implies Mordell--Lang for curves.