An analogue of the Herbrand-Ribet theorem in graph theory
Daniel Vallières, Chase A. Wilson
TL;DR
This work constructs a graph-theoretic analogue of the Herbrand–Ribet theorem by studying Galois covers $Y/X$ with $\Delta\simeq \mathbb{F}_{p}^{\times}$ and connecting the $p$-primary Picard components to the $p$-adic special values of the equivariant Ihara $L$-function at $u=1$ via $h_{Y/X}(1,\psi)=\psi(\eta_{Y/X}(1))$. The authors prove that $\mathrm{Fit}_{\mathbb{Z}_{p}}(e_{\psi}A)=(h_{Y/X}(1,\psi))$ and deduce the size relation $\# e_{\psi}A=|h_{Y/X}(1,\psi)|_{p}^{-1}$ for nontrivial $\psi$, while reducing modulo $p$ yields $e_{\psi}C\neq 0$ iff $h_{Y/X}(1,\psi)=0$ in $\mathbb{F}_{p}$; these are complemented by a trivial-character identification and a Mazur–Wiles–style refinement via Fitting ideals. The approach blends explicit determinant descriptions of the equivariant Ihara zeta function with algebraic machinery (idempotents, Fitting ideals) to mirror Iwasawa-theoretic arguments in graph theory, and is illustrated by concrete voltage-graph examples validating the main statements. Overall, the work deepens the bridge between arithmetic phenomena and graph zeta theory, suggesting broader Iwasawa-like structure in combinatorial settings.
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 $Δ$ isomorphic to $\mathbb{F}_{p}^{\times}$. Given a $\mathbb{Z}_{p}$-valued character of $Δ$, 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.