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.

29 pages, minor changes to previous version