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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Acknowledgements
I would like to thank Bjorn Poonen for useful discussions and MIT for its hospitality during a visit of two weeks in May 2015, when these discussions took place. All computations were done using the computer algebra system Magma [3].
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Stoll, M. (2017). Chabauty Without the Mordell-Weil Group. In: Böckle, G., Decker, W., Malle, G. (eds) Algorithmic and Experimental Methods in Algebra, Geometry, and Number Theory. Springer, Cham. https://doi.org/10.1007/978-3-319-70566-8_28
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