The Shafarevich conjecture for hypersurfaces in abelian varieties

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