TITLE: K3 surfaces with good reduction away from finitely many places
QUESTION [15 upvotes]: Let S be a finite set of primes in Q.  What, if anything, do we know about K3 surfaces over Q with good reduction away from S?  (To be more precise, I suppose I mean schemes over Spec Z[1/S] whose geometric fibers are (smooth) K3 surfaces, endowed with polarization of some fixed degree.)  Are there only finitely many isomorphism classes, as would be the case for curves of fixed genus?  If one doesn't know (or expect) finiteness, does one have an upper bound for the number of such K3 surfaces X/Q of bounded height?

REPLY [8 votes]: Yves Andre has proved the finiteness of the number of K3 surfaces over a number field $K$ with a polarisation of fixed degree $d$ and having good reduction (as a polarised variety) outside a fixed finite set of primes. 
The reference is: "On the Shafarevich and Tate conjectures for hyper-Kähler varieties".Math. Ann. 305 (1996), no. 2, 205–248.