An arithmetic topos for integer matrices

JOURNAL OF NUMBER THEORY

卷:204

Article

Hemelaer, Jens^1,2 

[1] Univ Antwerp, Dept Math, Middelheimlaan 1, B-2020 Antwerp, Belgium

[2] Res Fdn Flanders FWO, Brussels, Belgium

关键词: Arithmetic site;    Monoid;    Topos;    Topos automorphism;    Adele ring;    Topos-theoretic point;    Torsion-free abelian group;    Zeta function;    Goormaghtigh conjecture;    Big picture;   

DOI  :  10.1016/j.jnt.2019.03.023

来源: Elsevier

【 摘 要 】

We study the topos of sets equipped with an action of the monoid of regular 2 x 2 matrices over the integers. In particular, we show that the topos-theoretic points are given by the double quotient GL(2)((Z) over cap)\ M-2(A(f))/GL(2)(Q), so they classify the groups Z(2) subset of A subset of 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 Q by Z up to isomorphism relates to Goormaghtigh conjecture. (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

Free