The Shafarevich conjecture for hypersurfaces in abelian varieties

Abstract

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 Néron-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.

121 pages

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Cited by in corpus (5)

• The Shafarevich conjecture revisited: Finiteness of pointed families of polarized varieties
• Semisimplification for Subgroups of Reductive Algebraic Groups
• Siegel's theorem via the Lawrence-Venkatesh method
• Ax-Schanuel with derivatives for mixed period mappings
• Effective Methods for Diophantine Finiteness