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

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.