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Drinfeld modules as noncommutative tori

Igor V. Nikolaev

Abstract

The Drinfeld module is a tool of the explicit class field theory for the function fields. We first observe a similarity of such modules with the noncommutative tori, and then use it to develop an explicit class field theory for the number fields. The case of the imaginary quadratic number fields is treated in detail.

Drinfeld modules as noncommutative tori

Abstract

The Drinfeld module is a tool of the explicit class field theory for the function fields. We first observe a similarity of such modules with the noncommutative tori, and then use it to develop an explicit class field theory for the number fields. The case of the imaginary quadratic number fields is treated in detail.
Paper Structure (16 sections, 11 theorems, 37 equations)

This paper contains 16 sections, 11 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Noncommutative tori
  4. $C^*$-algebras
  5. K-theory of $C^*$-algebras
  6. Noncommutative tori
  7. AF-algebras
  8. Mundici algebra
  9. Cluster $C^*$-algebras
  10. Cluster $C^*$-algebra $\mathbb{A}(S_{g,n})$
  11. Mundici algebra $\mathbb{A}(S_{1,1})$
  12. Drinfeld modules
  13. Proof
  14. Proof of theorem \ref{['thm1.1']}
  15. Proof of corollary \ref{['cor1.2']}
  16. ...and 1 more sections

Key Result

Theorem 1.1

The following is true: (i) $\bigoplus_p C^*(k \langle\tau_p\rangle)\cong \mathbb{A}(S_{1,1})$; (ii) the Drinfeld module $Drin_A(k)$ is equivalent to an inclusion $Spec~(A) \subset\overline{\mathbf{Q}}/\mathbf{Z}$, such that $Drin_A(k)$ is trivial if and only if $Spec~(A)\subset\mathbf{Q}/\mathbf{Z}$

Theorems & Definitions (22)

  • Theorem 1.1
  • Corollary 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 2.1
  • Theorem 2.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 12 more