An analogue of the Herbrand-Ribet theorem in graph theory

Abstract

We study an analogue of the Herbrand-Ribet theorem, and its refinement by Mazur and Wiles, in graph theory. For an odd prime number $p$, we let ${\mathbb{F}}_p$ and ${\mathbb{Z}}_p$ denote the finite field with $p$ elements and the ring of $p$-adic integers, respectively. We consider Galois covers $Y/X$ of finite graphs with Galois group $\Delta$ isomorphic to ${\mathbb{F}}_p^{\times }$. Given a ${\mathbb{Z}}_p$-valued character of $\Delta$, we relate the cardinality of the corresponding character component of the $p$-primary subgroup of the degree zero Picard group of $Y$ to the $p$-adic absolute value of the special value at $u=1$ of the corresponding Artin-Ihara $L$-function.