On the growth of the Jacobian in $Z_p^l$-voltage covers of graphs

Abstract

We investigate the growth of the $p$-part of the Jacobians in voltage covers of finite connected multigraphs, where the voltage group is isomorphic to ${\mathbb{Z}}_p^l$ for some $l\ge 2$, and we study analogues of a conjecture of Greenberg on the growth of class numbers in multiple ${\mathbb{Z}}_p$-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 ${\mathbb{Z}}_p^l$-covers of a fixed finite graph $X$. We discuss many examples; in particular, we construct examples with non-trivial Iwasawa invariants.