The Shafarevich conjecture for hypersurfaces in abelian varieties

Abstract: Let $K$ be a number field, $S$ a finite set of primes of $O_K$, and $g$ a positive integer. Shafarevich conjectured, and Faltings proved, that there are only finitely many curves of genus $g$, defined over $K$ and having good reduction outside $S$. Analogous results have been proven for other families, replacing "curves of genus $g$" with "K3 surfaces", "del Pezzo surfaces" etc.; these results are also called Shafarevich conjectures. There are good reasons to expect the Shafarevich conjecture to hold for many families of varieties: the moduli space should have only finitely many integral points.

Will Sawin and I prove this for hypersurfaces in abelian varieties of dimension not equal to 3.

number theory

Audience: researchers in the topic

( slides )

UCLA Number Theory Seminar