K3 Surfaces Associated with Curves of Genus Two

@article{Kumar2007K3SA,
  title={K3 Surfaces Associated with Curves of Genus Two},
  author={Abhinav Kumar},
  journal={International Mathematics Research Notices},
  year={2007},
  volume={2008}
}
  • Abhinav Kumar
  • Published 24 January 2007
  • Mathematics
  • International Mathematics Research Notices
It is known ([10, 27]) that there is a unique K3 surface X which corresponds to a genus 2 curve C such that X has a Shioda-Inose structure with quotient birational to the Kummer surface of the Jacobian of C. In this paper we give an explicit realization of X as an elliptic surface over ℙ 1 with specified singular fibers of type II* and III*. We describe how the Weierstrass coefficients are related to the Igusa-Clebsch invariants of C. 
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TLDR
The techniques are applied to compute the full Mordell–Weil group in several examples of arithmetic interest, arising from isogenous elliptic curves with complex multiplication, for which these K3 surfaces are singular.
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