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...

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Bibliographic Details
Published in:Research in the mathematical sciences 2021-12, Vol.8 (4), Article 57
Main Authors: Braeger, Noah, Clingher, Adrian, Malmendier, Andreas, Spatig, Shantel
Format: Article
Language:English
Subjects:
Applications of Mathematics
Computational Mathematics and Numerical Analysis
Curves
Jacobians
Mathematics
Mathematics and Statistics
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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-isogenous 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 (Bull Lond Math Soc 38(2):209–223, 2006).
ISSN:2522-0144
2197-9847
DOI:10.1007/s40687-021-00293-0