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 $C$, 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 $x^5+y^5=z^p$.

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

References in corpus (3)

• Two-cover descent on hyperelliptic curves
• Independence of rational points on twists of a given curve
• Odd degree number fields with odd class number

Cited by in corpus (2)

• Division by 2 on odd degree hyperelliptic curves and their jacobians
• On some Generalized Fermat Equations of the form $x^2+y^{2n}=z^p$