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

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