# On certain isogenies between K3 surfaces

## TL;DR

This paper constructs special non-Galois isogenies between K3 surfaces using cyclic Galois covers induced by symplectic automorphisms, identifying families where such constructions are possible and analyzing their properties.

## Contribution

It introduces a novel method to construct non-Galois isogenies between K3 surfaces via compositions of cyclic Galois covers, expanding understanding of their automorphism groups.

## Key findings

- Existence of infinitely many families of K3 surfaces with symplectic automorphisms
- Explicit description of transcendental lattices for 2^n:1 isogenous K3 surfaces
- Analysis of Galois closures of specific isogenies and their geometric properties

## Abstract

The aim of this paper is to construct "special" isogenies between K3 surfaces, which are not Galois covers between K3 surfaces, but are obtained by composing cyclic Galois covers, induced by quotients by symplectic automorphisms. We determine the families of K3 surfaces for which this construction is possible. To this purpose we will prove that there are infinitely many big families of K3 surfaces which both admit a finite symplectic automorphism and are (desingularizations of) quotients of other K3 surfaces by a symplectic automorphism. In the case of involutions, for any $n\in\mathbb{N}_{>0}$ we determine the transcendental lattices of the K3 surfaces which are $2^n:1$ isogenous (by a non Galois cover) to other K3 surfaces. We also study the Galois closure of the $2^2:1$ isogenies and we describe the explicit geometry on an example.