An analog of the Iwasawa conjecture for a complete hyperbolic threefold of finite volume

TL;DR

This paper explores a geometric analog of the Iwasawa main conjecture by comparing invariants related to hyperbolic threefolds with number theory conjectures, revealing deep connections between geometry and algebraic number theory.

Contribution

It establishes a novel comparison between the Alexander invariant and the Ruelle-Selberg L-function for hyperbolic threefolds, extending Iwasawa theory analogies to geometric settings.

Findings

Relation between Alexander invariant and Ruelle-Selberg L-function at specific points

Geometric analog of the Iwasawa main conjecture demonstrated

Insights into the structure of hyperbolic threefolds with one cusp

Abstract

For a unitary local system of rank one on a complete hyperbolic threefold of finite volume which has only one cusp, we will compare the order of the Alexander invariant at t=1 and one of Ruelle-Selberg L-function at s=0. Our result may be considered as a geometric analog of the Iwasawa main conjecture in the algebraic number theory.