Real Multiplication and noncommutative geometry

TL;DR

This paper explores the use of two-dimensional quantum tori as a new approach to understanding real quadratic fields, inspired by classical complex multiplication theory and Stark conjectures.

Contribution

It proposes a novel framework using quantum tori to extend the concept of complex multiplication to real quadratic fields.

Findings

Basic constructions of quantum tori are discussed.

The approach offers a potential pathway to describe real quadratic fields.

Connections to Stark conjectures are highlighted.

Abstract

Classical theory of Complex Multiplication (CM) shows that all abelian extensions of a complex quadratic field $K$ are generated by the values of appropriate modular functions at the points of finite order of elliptic curves whose endomorphism rings are orders in $K$. For real quadratic fields, a similar description is not known. However, the relevant (still unproved) case of Stark conjectures ([St1]) strongly suggests that such a description must exist. In this paper we propose to use two--dimensional quantum tori corresponding to real quadratic irrationalities as a replacement of elliptic curves with complex multiplication. We discuss some basic constructions of the theory of quantum tori from the perspective of this Real Multiplication (RM) research project.