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

TL;DR

This paper explores the growth of Jacobians in $Z_p^l$-voltage covers of graphs, proving an Iwasawa main conjecture and analyzing invariants, with numerous examples including non-trivial cases.

Contribution

It establishes an Iwasawa main conjecture for Jacobians in $Z_p^l$-covers and studies the variation of Iwasawa invariants across these graph covers.

Findings

Proved an Iwasawa main conjecture in this setting.

Constructed examples with non-trivial Iwasawa invariants.

Analyzed the variation of invariants over $Z_p^l$-covers.

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.