An analogue of the Herbrand-Ribet theorem in graph theory

2025/04/07 by Vallières, Daniel, Wilson, Chase A.
#05C25 #11M41 #20C11 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2504.05529

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 \mathbbF_p and 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 \mathbbF_p^×. Given a 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.

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