$L$-function of noncommutative tori

TL;DR

This paper introduces an $L$-function for noncommutative tori, showing it matches the Hasse-Weil $L$-function of elliptic curves with complex multiplication, leading to a localization formula for tori with real multiplication.

Contribution

It defines a new $L$-function for noncommutative tori and establishes its equivalence with classical $L$-functions of elliptic curves with complex multiplication.

Findings

The $L$-function for noncommutative tori coincides with the Hasse-Weil $L$-function of elliptic curves with complex multiplication.

A localization formula for noncommutative tori with real multiplication is derived.

The work bridges noncommutative geometry and classical number theory through $L$-functions.

Abstract

We introduce an analog of the $L$-function for noncommutative tori. It is proved that such a function coincides with the Hasse-Weil $L$-function of an elliptic curve with complex multiplication. As a corollary, one gets a localization formula for the noncommutative tori with real multiplication.