An analogue of the Herbrand-Ribet theorem in graph theory
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| author | Vallières, Daniel Wilson, Chase A. |
| author_facet | Vallières, Daniel Wilson, Chase A. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_05529 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An analogue of the Herbrand-Ribet theorem in graph theory Vallières, Daniel Wilson, Chase A. Number Theory Combinatorics 05C25, 20C11, 11M41 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. |
| title | An analogue of the Herbrand-Ribet theorem in graph theory |
| topic | Number Theory Combinatorics 05C25, 20C11, 11M41 |
| url | https://arxiv.org/abs/2504.05529 |