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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-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).

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Notes

  1. 1.

    By definition, \({\mathbb {H}}_2\) is the set of two-by-two symmetric matrices over \({\mathbb {C}}\) whose imaginary part is positive definite

  2. 2.

    We corrected two minor typos in the statement of the main theorem.

  3. 3.

    They were given explicitly in [14]

  4. 4.

    All surfaces in Theorem 3.31 have been twisted by \((-1)\) compared to [60] This is due to a different sign choice in the definition of the twisted Legendre pencil in Eq. (3.2).

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Acknowledgements

The authors would like to thank the referees for their thoughtful comments and effort toward improving the manuscript. The authors also thank Dr. Muhammad Arjumand Masood for some helpful discussions about Jacobian elliptic functions during an early stage of this project. N.B. and S.S. would like to acknowledge the support from the Office of Research and Graduate Studies at Utah State University. A.C. acknowledges support from a UMSL Mid-Career Research Grant. A.M. acknowledges support from the Simons Foundation through Grant No. 202367.

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Correspondence to Andreas Malmendier.

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Braeger, N., Clingher, A., Malmendier, A. et al. Isogenies of certain K3 surfaces of rank 18. Res Math Sci 8, 57 (2021). https://doi.org/10.1007/s40687-021-00293-0

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Keywords

  • Isogenies
  • Kummer surfaces
  • Elliptic K3 surfaces

Mathematics Subject Classification

  • 14J27
  • 14J28
  • 14G25