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arXiv 1010.2603math.NT
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数域上的显式Chabauty

Explicit Chabauty over Number Fields

Samir Siksek

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摘要：

设$C$是$d$度数域$K$上亏格为$g \geq 2$的光滑射影绝对不可约曲线，用$J$表示其雅可比矩阵.用$r$表示$J(K)$的Mordell-Weil等级。我们给出了一个明确的和实用的Chabauty式的标准表明，一个给定的子集$\cK \subseteq C(K)$实际上等于$C(K)$。如果是$r \leq d(g-1)$，则该标准可能成功。我们还证明了方程$x^2+y^3=z^{10}$在互质非零整数中的唯一解是$(x,y,z)=(\pm 3, -2, \pm 1)$。这是通过将问题归结为确定几个亏格$2$曲线上的$K$ -有理点来实现的，其中$K=\Q$或$\Q(\sqrt[3]{2})$，并应用本文的方法。

英文摘要：

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.

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