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Mazur's conjecture and an unexpected rational curve on Kummer surfaces and their superelliptic generalisations

Tom 187 / 2019

Damián Gvirtz Acta Arithmetica 187 (2019), 189-200 MSC: Primary 14J27; Secondary 11G05. DOI: 10.4064/aa180201-1-10 Opublikowany online: 30 November 2018


We prove the following special case of Mazur’s conjecture on the topology of rational points. Let $E$ be an elliptic curve over $\mathbb{Q}$ with $j$-invariant $1728$. For a class of elliptic pencils which are quadratic twists of $E$ by quartic polynomials, the rational points on the projective line with positive rank fibres are dense in the real topology. This extends results obtained by Rohrlich and Kuwata–Wang for quadratic and cubic polynomials.

For the proof, we investigate a highly singular rational curve on the Kummer surface $K$ associated to a product of two elliptic curves over $\mathbb{Q}$, which previously appeared in publications by Mestre, Kuwata–Wang and Satgé. We produce this curve in a simpler manner by finding algebraic equations which give a direct proof of rationality. We find that the same equations give rise to rational curves on a class of more general surfaces extending the Kummer construction. This leads to further applications apart from Mazur’s conjecture, for example the existence of rational points on simultaneous twists of superelliptic curves.

Finally, we give a proof of Mazur’s conjecture for the Kummer surface $K$ without any restrictions on the $j$-invariants of the two elliptic curves.


  • Damián GvirtzDepartment of Mathematics
    Imperial College London
    South Kensington Campus
    London, SW7 2AZ, United Kingdom

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