Explicit Chabauty over Number Fields

Abstract

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