Isogenies of certain K3 surfaces of rank 18

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 o...

Full description

Saved in:
Bibliographic Details
Published in:arXiv.org 2021-09
Main Authors: Braeger, Noah, Clingher, Adrian, Malmendier, Andreas, Spatig, Shantel
Format: Article
Language:English
Subjects:
Curves
Jacobians
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary: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.
ISSN:2331-8422
DOI:10.48550/arxiv.2109.03189