An arithmetic topos for integer matrices

TL;DR

This paper explores a topos-theoretic framework for integer matrices, linking algebraic structures with number theory, automorphisms, and geometric interpretations, and relates these to conjectures in number theory.

Contribution

It introduces a topos for regular 2x2 integer matrices, characterizes its points and automorphisms, and connects these to number theory conjectures and geometric models.

Findings

Topos points correspond to certain group classifications.

Automorphisms relate to symmetries in the topos structure.

Connections established with Conway's big picture and the Arithmetic Site.

Abstract

We study the topos of sets equipped with an action of the monoid of regular $2 \times 2$ matrices over the integers. In particular, we show that the topos-theoretic points are given by the double quotient $\left. GL_2(\hat{\mathbb{Z}}) ~\middle\backslash~ M_2(\mathbb{A}_f)~\middle/~GL_2(\mathbb{Q})\right.$, so they classify the groups $\mathbb{Z}^2 \subseteq A \subseteq \mathbb{Q}^2$ up to isomorphism. We determine the topos automorphisms and then point out the relation with Conway's big picture and the work of Connes and Consani on the Arithmetic Site. As an application to number theory, we show that classifying extensions of $\mathbb{Q}$ by $\mathbb{Z}$ up to isomorphism relates to Goormaghtigh conjecture.