Isogenies of certain K3 surfaces of rank 18

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.

arXiv comment67 pages

References in corpus (9)

• Elliptic fibrations on a generic Jacobian Kummer surface
• K3 surfaces, modular forms, and non-geometric heterotic compactifications
• The Satake sextic in elliptic fibrations on K3
• Kummer surfaces associated with Seiberg-Witten curves
• A Universal Genus-Two Curve from Siegel Modular Forms
• Kummer sandwiches and Greene-Plesser construction
• Moonshine and Donaldson invariants of CP^2
• Nikulin involutions and the CHL string
• Counting Rational Points on Kummer surfaces