Noncommutative Geometry and Arithmetic

TL;DR

This paper surveys recent developments in noncommutative geometry, specifically focusing on noncommutative tori with real multiplication, and explores their connections to Stark numbers and modular forms in number theory.

Contribution

It introduces a geometric framework for noncommutative tori with real multiplication, paralleling classical elliptic curve theory for quadratic fields.

Findings

Relation between Stark numbers and noncommutative tori geometry

Shadows of modular forms on noncommutative boundaries

Insights into the moduli space of noncommutative tori

Abstract

This is an overview of recent results aimed at developing a geometry of noncommutative tori with real multiplication, with the purpose of providing a parallel, for real quadratic fields, of the classical theory of elliptic curves with complex multiplication for imaginary quadratic fields. This talk concentrates on two main aspects: the relation of Stark numbers to the geometry of noncommutative tori with real multiplication, and the shadows of modular forms on the noncommutative boundary of modular curves, that is, the moduli space of noncommutative tori. To appear in Proc. ICM 2010.