The Unpolarized Shafarevich Conjecture for K3 Surfaces
The Unpolarized Shafarevich Conjecture for K3 Surfaces

Yiwei She , Columbia University
Fine Hall 214
Shafarevich made the following conjecture for higher genus curves: the set of isomorphism classes of genus g curves over K (a number field) with good reduction outside S (a fixed finite set of primes) is finite. Faltings proved this and the analogous conjecture for abelian varieties of given degree. Zarhin proved this finiteness across all degrees. Using Faltings’ theorem, Andre proved the finiteness of K3 surfaces (over K, S) of a given degree. We prove the analog of Zarhin’s theorem, i.e. there are still finitely many K3 surfaces across all degrees.