Chabauty without the Mordell-Weil group
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 , given only the -Selmer group of its Jacobian (or some other abelian variety maps to) and the image of the -Selmer set of in . 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 .
Cite
@article{arxiv.1506.04286,
title = {Chabauty without the Mordell-Weil group},
author = {Michael Stoll},
journal= {arXiv preprint arXiv:1506.04286},
year = {2019}
}
Comments
35 pages. v2: improved exposition in Section 2, promoted Propositions 2.1 and 2.6 to theorems; moved old Section 4 (now Section 3) before old Section 3 (now 4) and changed the new Section 3 to cover the general case (arbitrary $k$ and $p$ instead of $\mathbb Q$ and 2); improved the sufficient criterion for FLT in Section 7; minor edits throughout