  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.
