The Shafarevich conjecture for hypersurfaces in abelian varieties

Abstract: Faltings proved the statement, previously conjectured by Shafarevich, that there are finitely many abelian varieties of dimension $n$, defined over a fixed number field, with good reduction outside a fixed finite set of primes, up to isomorphism. In joint work with Brian Lawrence, we prove an analogous finiteness statement for hypersurfaces in a fixed abelian variety with good reduction outside a finite set of primes. I will give an introduction to some of the ideas in the proof, which builds on $p$-adic Hodge theory techniques from work of Lawrence and Venkatesh as well as a little-known area of algebraic geometry

number theory

Audience: researchers in the topic

( paper | slides | video )

Rational Points and Galois Representations

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