On the non-commutative Iwasawa main conjecture for voltage covers of graphs

TL;DR

This paper establishes an Iwasawa main conjecture for voltage covers of graphs with p-adic Lie group symmetries, linking algebraic invariants of graph covers to p-adic L-functions, and explores module properties related to these invariants.

Contribution

It formulates and proves an Iwasawa main conjecture for voltage covers of graphs in the non-commutative p-adic setting, extending algebraic graph theory into Iwasawa theory.

Findings

Proves an Iwasawa main conjecture for the projective limit of Picard groups.

Establishes one inclusion of the main conjecture for Jacobians.

Provides a necessary condition involving μ-invariants for the alculus property of modules.

Abstract

Let $p$ be a rational prime, and let $X$ be a connected finite graph. In this article we study voltage covers $X_\infty$ of $X$ attached to a voltage assignment ${\alpha}$ which takes values in some uniform $p$-adic Lie group $G$. We formulate and prove an Iwasawa main conjecture for the projective limit of the Picard groups $\text{Pic}(X_n)$ of the intermediate voltage covers $X_n$, ${n \in \mathbb{N}}$, and we prove one inclusion of a main conjecture for the projective limit of the Jacobians $J(X_n)$. Moreover, we study the $\mathfrak{M}_H(G)$-property of $\mathbb{Z}_p[[G]]$-modules and prove a necessary condition for this property which involves the $\mu$-invariants of $\mathbb{Z}_p$-subcovers ${Y \subseteq X_\infty}$ of $X$. If the dimension of $G$ is equal to 2, then this condition is also sufficient.