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arXiv · 2102.11737

A parametrized set of explicit elements of $\Sha(E/\Q)[3]$

Steven R. Groen·Jaap Top
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Abstract

In this paper, we give explicit equations for homogeneous spaces corresponding to a rational isogeny of degree $3$. An explicit set of elliptic curves with elements of order $3$ in their Tate-Shafarevich group is constructed. Combining this gives explicit examples of plane cubics over $\Q$ that have a point everywhere locally, but not globally.

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Steven R. Groen, Jaap Top. 2021-02-23. A parametrized set of explicit elements of $\Sha(E/\Q)[3]$. https://arxiv.org/abs/2102.11737

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