Joint IAS/Princeton University Number Theory Seminar

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 a broad 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 sheaf convolution in algebraic geometry.