On the growth of the Jacobians in ℤ p l -voltage covers of graphs

We investigate the growth of the p-part of the Jacobians in voltage covers of finite connected graphs, where the voltage group is isomorphic to ℤ p l for some l≥2, and we study analogues of a conjecture of Greenberg on the growth of class numbers in multiple ℤ p -extensions of number fields. Moreover we prove an Iwasawa main conjecture in this setting, and we study the variation of (generalised) Iwasawa invariants as one runs over the ℤ p l -covers of a fixed finite graph X. We discuss many examples; in particular, we construct examples with non-trivial Iwasawa invariants.