Local Conjugacy of Irreducible Hyperbolic Toral Automorphisms

Abstract

This paper is dedicated to the conjugacy problem in $GL(n,\mathbb{Z})$ 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 $GL(n,\mathbb{Z})$ 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 $GL(n,\mathbb{Z})$, i.e., the conjugacy of $GL(n,\mathbb{Z})$ matrices in the groups $GL(n,{\mathbb{Z}}_p)$ of invertible matrices over the $p$-adic integers.

13 pages