How to classify **K3 surfaces** over an arbitrary field *k*?



K3 over fields other than C?
The "standard" definition of a K3 surface is field independent (unless you are a physicist):


$p\_g=1, q=0$, and trivial canonical class.


Some results:


* Mumford and Bombieri showed that you get (just as in the complex case) a 19 dimensional family of K3 surfaces for any degree (the 19 dimensional thingy is a deformation theory argument which is completely algebraic).
* Deligne showed that all the K3 surfaces in finite characteristics are reductions mod p.


What you obviously don't get is the fact that all these spaces sit together in a nice 20 dimensional complex ball. I also don't know if you can carry over any of the recent Kodaira dimension computation of these moduli (which are very analytic in nature).


Reference: Complex algebraic surfaces (Beauville): Chapter VIII and Appendix A.


