On the growth of the Jacobians in $\mathbb{Z}_p^l$-voltage covers of graphs

Kleine, Sören; Müller, Katharina

doi:10.5802/alco.366

On the growth of the Jacobians in

ℤ_{p}^{l}

-voltage covers of graphs

Kleine, Sören ¹; Müller, Katharina ²

¹ Institut für Anwendungssicherheit Universität der Bundeswehr München Werner-Heisenberg-Weg 39 85577 Neubiberg Germany
² Institut für Theoretische Informatik Mathematik und Operations Research Universität der Bundeswehr München Werner-Heisenberg-Weg 39 85577 Neubiberg Germany

Algebraic Combinatorics, Volume 7 (2024) no. 4, pp. 1011-1038

Abstract

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 $l \geq 2$ , 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.

Received: 2023-06-27
Revised: 2024-01-19
Accepted: 2024-02-22
Published online: 2024-09-03

MR Zbl

DOI: 10.5802/alco.366

Classification: 11C30, 11R23, 05C25, 05C40, 05C50, 11C20
Keywords: Voltage cover of a graph, Greenberg’s conjecture, Iwasawa main conjecture, (generalised) Iwasawa invariants

Author's affiliations:

Kleine, Sören ¹; Müller, Katharina ²

¹ Institut für Anwendungssicherheit Universität der Bundeswehr München Werner-Heisenberg-Weg 39 85577 Neubiberg Germany
² Institut für Theoretische Informatik Mathematik und Operations Research Universität der Bundeswehr München Werner-Heisenberg-Weg 39 85577 Neubiberg Germany

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Kleine, S\"oren and M\"uller, Katharina},
     title = {On the growth of the {Jacobians} in $\mathbb{Z}_p^l$-voltage covers of graphs},
     journal = {Algebraic Combinatorics},
     pages = {1011--1038},
     year = {2024},
     publisher = {The Combinatorics Consortium},
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Kleine, Sören; Müller, Katharina. On the growth of the Jacobians in $\mathbb{Z}_p^l$-voltage covers of graphs. Algebraic Combinatorics, Volume 7 (2024) no. 4, pp. 1011-1038. doi: 10.5802/alco.366

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