On the BDP Iwasawa main conjecture for modular forms

Abstract

Let p be a rational prime, and let X be a connected finite graph.In this article we study voltage covers X∞ of X attached to a voltage assignment α 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 Pic(Xn) of the intermediate voltage covers Xn, n ∈ N, and we prove one inclusion of a main conjecture for the projective limit of the Jacobians J(Xn).Moreover, we study the M H (G)-property of Zp G -modules and prove a necessary condition for this property which involves the µ-invariants of Zpsubcovers Y ⊆ X∞ of X. If the dimension of G is equal to 2, then this condition is also sufficient.