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Artin $L$-functions and noncommutative tori

Igor V. Nikolaev

Abstract

Using the ideas of Deninger, we prove that the Artin $L$-functions coincide with such of the noncommutative tori. This result can be viewed as the Langlands reciprocity for noncommutative tori.

Artin $L$-functions and noncommutative tori

Abstract

Using the ideas of Deninger, we prove that the Artin -functions coincide with such of the noncommutative tori. This result can be viewed as the Langlands reciprocity for noncommutative tori.
Paper Structure (13 sections, 7 theorems, 19 equations)

This paper contains 13 sections, 7 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Deninger's foliated spaces
  4. Foliated spaces
  5. Dedekind zeta function
  6. Analogy Den1
  7. Noncommutative tori
  8. $C^*$-algebras
  9. K-theory of $C^*$-algebras
  10. Noncommutative tori
  11. Drinfeld modules
  12. Proof of theorem \ref{['thm1.5']}
  13. Remarks

Key Result

Theorem 1.5

(N) For each finite Galois extension $\mathbf{K}$ of the field $\mathbf{Q}$ and each irreducible representation $\sigma_{n+1}: Gal~(\mathbf{K} | \mathbf{Q})\to GL_{n+1}({\Bbb C})$, there exists a noncommutative torus $\mathscr{A}_{RM}^{2n}$, such that:

Theorems & Definitions (24)

  • Definition 1.1
  • Remark 1.2
  • Definition 1.3
  • Definition 1.4
  • Theorem 1.5
  • Remark 1.6
  • Definition 2.1
  • Remark 2.2
  • Theorem 2.3
  • Theorem 2.4
  • ...and 14 more