$K_n$循环电压覆盖的Jacobian
The Jacobian of Cyclic Voltage Covers of $K_n$
摘要：
本文证明了基图$X$（$G$称为电压群，$Y$称为导出图）的覆盖图族$Y$的雅可比行列式.特别感兴趣的是，当通过将阶$d$的循环群的生成元映射到$X$的单个边缘（所有其他边缘被分配单位元）来给出电压分配时，称为单个电压分配。当基图$X$是$n$顶点上的完全图时，对每个$n$和$d$，确定了单电压分配导出图的雅可比矩阵的阶数和阿贝尔群结构.使用zeta函数，一般的产品公式，涉及的阶的$Y$的雅可比矩阵的$X$的开发，这些公式变得非常简单和明确的特殊情况下，单电压覆盖的$X$。
英文摘要：
This paper proves results about the Jacobians of a certain family of covering graphs, $Y$, of a base graph $X$, that is constructed from an assignment of elements from a group $G$ to the edges of $X$ ($G$ is called the voltage group and $Y$ is called the derived graph). Of particular interest is when the voltage assignment is given by mapping a generator of the cyclic group of order $d$ to a single edge of $X$ (all other edges are assigned the identity), called a single voltage assignment. Both the order and abelian group structure of the Jacobian of single voltage assignment derived graphs are determined when the base graph $X$ is the complete graph on $n$ vertices, for every $n$ and $d$. Using zeta-functions, general product formulas that relate the order of the Jacobian of $Y$ to that of $X$ are developed; these formulas become very simple and explicit in the special case of single voltage covers of $X$.
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