Local Conjugacy of Irreducible Hyperbolic Toral Automorphisms

TL;DR

This paper explores the relationship between conjugacy in integer matrices, algebraic number theory, and local conjugacy over p-adic integers, focusing on hyperbolic toral automorphisms and ideal equivalence.

Contribution

It clarifies the connection between weak ideal equivalence and local conjugacy in $GL(n, ext{Z}_p)$ for hyperbolic automorphisms.

Findings

Established the link between weak ideal equivalence and local conjugacy.

Provided conditions under which local conjugacy implies global conjugacy.

Connected algebraic number theory concepts with matrix conjugacy problems.

Abstract

This paper is dedicated to the conjugacy problem in $GL(n,\Z)$ and its connection with algebraic number theory. This connection may be summed up in the Latimer-MacDuffee-Taussky Theorem, which, in a very broad sense, identifies the conjugacy relation in $GL(n,\Z)$ with the relation of arithmetic equivalence between certain ideals of algebraic numbers. The main purpouse is to clarify the link between a weaker relation between ideals, weak equivalence, and local conjugacy in $GL(n,\Z)$, i.e., the conjugacy of $GL(n,\Z)$ matrices in the groups $GL(n,\Z_{p})$ of invertible matrices over the $p$-adic integers.