The Shafarevich Conjecture for Hypersurfaces in Abelian Varieties
The Shafarevich Conjecture for Hypersurfaces in Abelian Varieties

Will Sawin, Columbia University
Zoom link: https://princeton.zoom.us/j/97126136441
Passcode: the three digit integer that is the cube of the sum of its digits
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 padic Hodge theory techniques from work of Lawrence and Venkatesh as well as sheaf convolution in algebraic geometry.