Chabauty Without the Mordell-Weil Group

Michael Stoll ()
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Michael Stoll: Universität Bayreuth, Mathematisches Institut

A chapter in Algorithmic and Experimental Methods in Algebra, Geometry, and Number Theory, 2017, pp 623-663 from Springer

Abstract: 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 .

Keywords: Rational points on curves; Chabauty’s method; Selmer group; 11G30; 14G05; 14G25; 14H25; 11Y50; 11D41 (search for similar items in EconPapers)
Date: 2017
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-319-70566-8_28

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DOI: 10.1007/978-3-319-70566-8_28

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