Explicit Chabauty over number fields

Abstract

Let $C$ be a smooth projective absolutely irreducible curve of genus $g\ge 2$ over a number field $K$ of degree $d$, and let $J$ denote its Jacobian. Let $r$ denote the Mordell–Weil rank of $J\left(K\right)$. We give an explicit and practical Chabauty-style criterion for showing that a given subset $\mathcal{K}\subseteq C\left(K\right)$ is in fact equal to $C\left(K\right)$. This criterion is likely to be successful if $r\le d\left(g-1\right)$. We also show that the only solution to the equation $x^2+y^3=z^{10}$ in coprime nonzero integers is $\left(x,y,z\right)=\left(\pm 3,-2,\pm 1\right)$. This is achieved by reducing the problem to the determination of $K$-rational points on several genus-$2$ curves where $K=\mathbb{Q}$ or $\mathbb{Q}\left(\sqrt[3]{2}\right)$ and applying the method of this paper.