The Jacobian of Cyclic Voltage Covers of $K_n$

Abstract

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 $X$.