Langlands reciprocity for the even dimensional noncommutative tori

TL;DR

This paper proposes a conjectural explicit formula linking higher-dimensional Dirichlet characters to the K-theory of noncommutative tori, establishing its validity in low dimensions and drawing parallels to classical reciprocity laws.

Contribution

It introduces a new conjecture connecting noncommutative geometry with number theory and proves it for specific low-dimensional cases, including a noncommutative Artin reciprocity law.

Findings

Conjectured explicit formula for higher-dimensional Dirichlet characters.

Proved the conjecture for 1D and 2D noncommutative tori.

Established a noncommutative analog of Artin reciprocity law.

Abstract

We conjecture an explicit formula for the higher dimensional Dirichlet character; the formula is based on the K-theory of the so-called noncommutative tori. It is proved, that our conjecture is true for the two-dimensional and one-dimensional (degenerate) noncommutative tori; in the second case, one gets a noncommutative analog of the Artin reciprocity law.