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Let $A$ be an abelian scheme of dimension at least four over a$\mathbb{Z}$-finitely generated integral domain $R$ of characteristic zero, andlet $L$ be an ample line bundle on $A$. We prove that the set of smoothhypersurfaces $D$ in $A$ representing $L$ is finite by showing that the modulistack of such hypersurfaces has only finitely many $R$-points. We accomplishthis by using level structures to interpolate finiteness results between thismoduli stack and the stack of canonically polarized varieties.

Smooth hypersurfaces in abelian varieties over arithmetic rings