The special value $u=1$ of Artin-Ihara $L$-functions

TL;DR

This paper investigates the special value u=1 of Artin-Ihara L-functions for abelian graph covers, establishing an annihilation result similar to Brumer's conjecture and analyzing related algebraic structures.

Contribution

It introduces an annihilation statement for Artin-Ihara L-functions at u=1 and computes the index of an ideal akin to the Stickelberger ideal in this context.

Findings

Proves an annihilation statement analogous to Brumer's conjecture.

Calculates the index of an ideal similar to the Stickelberger ideal.

Provides observations on spanning trees in abelian multigraph coverings.

Abstract

We study the special value $u=1$ of Artin-Ihara $L$-functions associated to characters of the automorphism group of abelian covers of multigraphs. In particular, we show an annihilation statement analogous to a classical conjecture of Brumer on annihilation of class groups for abelian extensions of number fields and we also calculate the index of an ideal analogous to the classical Stickelberger ideal in algebraic number theory. Along the way, we make some observations about the number of spanning trees in abelian multigraph coverings that may be of independent interest.