Isogenies of certain K3 surfaces of rank 18

TL;DR

This paper constructs geometric isogenies between specific families of K3 surfaces with Picard rank 18, revealing algebraic correspondences and isomorphic Galois representations, extending previous work in the field.

Contribution

It introduces new geometric isogenies between three types of K3 surface families of rank 18, generalizing prior research by van Geemen and Top.

Findings

Existence of algebraic correspondences between the K3 surfaces.

Isomorphism of associated four-dimensional Galois representations.

Application to subfamilies of Picard rank 19.

Abstract

We construct geometric isogenies between three types of two-parameter families of K3 surfaces of Picard rank 18. One is the family of Kummer surfaces associated with Jacobians of genus-two curves admitting an elliptic involution, another is the family of Kummer surfaces associated with the product of two non-isogeneous elliptic curves, and the third is the twisted Legendre pencil. The isogenies imply the existence of algebraic correspondences between these K3 surfaces and prove that the associated four-dimensional Galois representations are isomorphic. We also apply our result to several subfamilies of Picard rank 19. The result generalizes work of van Geemen and Top.