Idoneal genera and K3 surfaces covering an Enriques surface

Abstract

We introduce the notion of idoneal genera, which are a generalization of Euler's idoneal numbers. We prove that there exist only a finite number of idoneal genera, and we provide an algorithm to enumerate all idoneal genera of rank at least 3. As an application, we classify transcendental lattices of K3 surfaces covering an Enriques surface.