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Counting rational points on Kummer surfaces

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Abstract

We consider the problem of counting the number of rational points on the family of Kummer surfaces associated with two non-isogenous elliptic curves. For this two-parameter family we prove Manin’s unity, using the presentation of the Kummer surfaces as isotrivial elliptic fibration and as double cover of the modular elliptic surface of level two. By carrying out the rational point-count with respect to either of the two elliptic fibrations explicitly, we obtain an interesting new identity between two-parameter counting functions.

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Notes

  1. A Fuchsian (differential) equation is a linear homogeneous ordinary differential equation with analytic coefficients in the complex domain whose singular points are all regular singular points.

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  1. Department of Mathematics and Statistics, Utah State University, Logan, UT, 84322, USA

    Andreas Malmendier & Yih Sung

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  1. Andreas Malmendier
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  2. Yih Sung
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Correspondence to Andreas Malmendier.

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Malmendier, A., Sung, Y. Counting rational points on Kummer surfaces. Res. number theory 5, 27 (2019). https://doi.org/10.1007/s40993-019-0166-x

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