On certain isogenies between K3 surfaces

Abstract

We will prove that there are infinitely many families of K3 surfaces which both admit a finite symplectic automorphism and are (desingularizations of) quotients of other K3 surfaces by a symplectic automorphism. These families have an unexpectedly high dimension. We apply this result to construct “special” isogenies between K3 surfaces which are not Galois covers between K3 surfaces but are obtained by composing cyclic Galois covers. In the case of involutions, for any n ∈ N > 0 n\in \mathbb {N}_{>0} we determine the transcendental lattices of the K3 surfaces which are 2 n : 1 2^n:1 isogenous (by the mentioned “special” isogeny) to other K3 surfaces.