The argument that I had is flawed and I do not know how to fix it. Here is a short description of one part that I cannot address. We are given two polarized K3 surfaces *X* and *Y* of degrees *x* and *y* respectively. It is clear that we can check if *X* and *Y* are isomorphic as polarized surfaces, but we would like to check if they are isomorphic as surfaces. If we could prove that there were finitely many polarizations on *X* having degree *y* and this finite set were computable, then we would be done. Unfortunately, there could be infinitely many such polarizations on *X*.

For Abelian surfaces the situation is slightly better, since the number of polarisations with a given degree is finite (by a theorem of Narasimhan and Nori). To find the polarisations on an Abelian variety *A*, one possibility is to compute the orbits of the automorphism group of *A* on the Neron-Severi group of *A*. While this could be difficult over $\overline{\mathbb{Q}}$, it might be replaced by a similar statement modulo a prime of good reduction, where at least Picard numbers can be computed by the Tate conjecture. Also in this case, I do not know if there are further problems with this approach.

I believe that this kills completely my previous post!