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arXiv · 1506.04286

Chabauty without the Mordell-Weil group

Michael Stoll
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Abstract

Based on ideas from recent joint work with Bjorn Poonen, we describe an algorithm that can in certain cases determine the set of rational points on a curve $C$, given only the $p$-Selmer group $S$ of its Jacobian (or some other abelian variety $C$ maps to) and the image of the $p$-Selmer set of $C$ in $S$. The method is more likely to succeed when the genus is large, which is when it is usually rather difficult to obtain generators of a finite-index subgroup of the Mordell-Weil group, which one would need to apply Chabauty's method in the usual way. We give some applications, for example to generalized Fermat equations of the form $x^5 + y^5 = z^p$.

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Michael Stoll. 2017-01-03. Chabauty without the Mordell-Weil group. https://doi.org/10.1007/978-3-319-70566-8_28

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