  Faltings proved that there are finitely many abelian varieties of genus $g$
over a number field $K$, with good reduction outside a finite set of primes
$S$. Fixing one of these abelian varieties $A$, we prove that there are
finitely many smooth hypersurfaces in $A$, with good reduction outside $S$,
representing a given ample class in the N\'eron-Severi group of $A$, up to
translation, as long as the dimension of $A$ is at least $4$. Our approach
builds on the approach of arXiv:1807.02721 which studies $p$-adic variations of
Hodge structure to turn finiteness results for $p$-adic Galois representations
into geometric finiteness statements. A key new ingredient is an approach to
proving big monodromy for the variations of Hodge structure arising from the
middle cohomology of these hypersurfaces using the Tannakian theory of sheaf
convolution on abelian varieties.
