An Equivariant Main Conjecture for Branched ${\mathbb{Z}}_p$-Towers of Finite Graphs and Applications

Let $p$ be a prime and let $Y\mathrm{/}X$ be a finite abelian branched cover of finite connected graphs. We consider a compatible branched ${\mathbb{Z}}_p$-tower above $Y$ whose total Galois group over $X$ is abelian. We construct an equivariant Picard Iwasawa module over the completed group ring of the total Galois group and give an explicit presentation of this module. On the analytic side, we construct an equivariant Artin–Ihara $L$-element at the top of the tower and isolate a modified factor whose value at $u=1$ is well defined. Our main result computes the Fitting ideal of the equivariant Picard Iwasawa module in terms of these special values and augmentation ideals determined by the inertia groups. As applications, we recover the main conjecture for branched ${\mathbb{Z}}_p$-towers from our earlier work and obtain finite-level Fitting-ideal and annihilation formulas which can be viewed as graph-theoretic analogues of the conjectures of Kurihara and Brumer–Stark.