Let $k,n \geq 2$ be integers. A generalized Fermat curve of type $(k,n)$ is a compact Riemann surface $S$ that admits a subgroup of conformal automorphisms $H \leq \mbox{Aut}(S)$ isomorphic to $\mathbb{Z}_k^n$, such that the quotient surface $S/H$ is biholomorphic to the Riemann sphere $\hat{\mathbb{C}}$ and has $n+1$ branch points, each one of order $k$. There exists a good algebraic model for these objects, which makes them easier to study. Using tools from algebraic topology and integration theory on Riemann surfaces, we find a set of generators for the first homology group of a generalized Fermat curve. Finally, with this information, we find a set of generators for the period lattice of the associated Jacobian variety.

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广义费马曲线的周期

令 $k,n \geq 2$ 为整数。$(k,n)$ 类型的广义费马曲线是一个紧凑的黎曼曲面 $S$，它承认一个共形自同构的子群 $H \leq \mbox{Aut}(S)$ 同构于 $\mathbb{Z}_k ^n$，使得商面 $S/H$ 与黎曼球面 $\hat{\mathbb{C}}$ 是双全纯的，并且有 $n+1$ 个分支点，每个分支点的阶数为 $k$。这些对象存在一个很好的代数模型，这使得它们更容易研究。使用黎曼曲面上的代数拓扑和积分理论的工具，我们找到了广义费马曲线的第一个同调群的一组生成器。最后，根据这些信息，我们为相关的雅可比变体的周期格找到一组生成器。