#### Vol. 7, No. 4, 2013

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Explicit Chabauty over number fields

### Samir Siksek

Vol. 7 (2013), No. 4, 765–793
##### 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 $\mathsc{K}\subseteq C\left(K\right)$ 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 $\left(x,y,z\right)=\left(±3,-2,±1\right)$. This is achieved by reducing the problem to the determination of $K$-rational points on several genus-$2$ curves where $K=ℚ$ or $ℚ\left(\sqrt[3]{2}\right)$ and applying the method of this paper.

##### Keywords
Chabauty, Coleman, jacobian, divisor, abelian integral, Mordell–Weil sieve, generalized Fermat, rational points
##### Mathematical Subject Classification 2010
Primary: 11G30
Secondary: 14K20, 14C20