The "standard" definition of a K3 surface is field independent (unless you are a physicist):

p_g=0, q=1, 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 surface is finite characteristics are reductions mod p.

What obviously you don't get is all these spaces sitting 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.

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.