Chabauty without the Mordell-Weil group

Titelangaben

Stoll, Michael:
Chabauty without the Mordell-Weil group.
In: Böckle, Gebhard ; Decker, Wolfram ; Malle, Gunter (Hrsg.): Algorithmic and Experimental Methods in Algebra, Geometry, and Number Theory. - Cham : Springer , 2017 . - S. 623-663
ISBN 978-3-319-70565-1
DOI: https://doi.org/10.1007/978-3-319-70566-8_28

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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.