Higher descents on an elliptic curve with a rational 2-torsion point

Fisher, Tom

doi:10.1090/mcom/3163

Higher descents on an elliptic curve with a rational 2-torsion point
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Math. Comp. 86 (2017), 2493-2518 Request permission

Abstract:

Let $E$ be an elliptic curve over a number field $K$. Descent calculations on $E$ can be used to find upper bounds for the rank of the Mordell-Weil group, and to compute covering curves that assist in the search for generators of this group. The general method of $4$-descent, developed in the PhD theses of Siksek, Womack and Stamminger, has been implemented in Magma (when $K=\mathbb {Q}$) and works well for elliptic curves with sufficiently small discriminant. By extending work of Bremner and Cassels, we describe the improvements that can be made when $E$ has a rational $2$-torsion point. In particular, when $E$ has full rational $2$-torsion, we describe a method for $8$-descent that is practical for elliptic curves $E/\mathbb {Q}$ with large discriminant.

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