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Class number zeta function of imaginary quadratic fields

Igor V. Nikolaev

Abstract

We introduce a zeta function counting imaginary quadratic number fields by their class numbers. It is proved that such a function is rational depending only on the eight roots of unity of degrees $1$ and $2$. As a corollary, one gets a lower bound $2p$ for the number of imaginary quadratic fields of the prime class number $p$. Our method is based on the study of periodic points of a dynamical system arising in the representation theory of the Drinfeld modules by the bounded linear operators on a Hilbert space.

Class number zeta function of imaginary quadratic fields

Abstract

We introduce a zeta function counting imaginary quadratic number fields by their class numbers. It is proved that such a function is rational depending only on the eight roots of unity of degrees and . As a corollary, one gets a lower bound for the number of imaginary quadratic fields of the prime class number . Our method is based on the study of periodic points of a dynamical system arising in the representation theory of the Drinfeld modules by the bounded linear operators on a Hilbert space.

Paper Structure

This paper contains 9 sections, 10 theorems, 24 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Noncommutative tori
  4. Non-abelian class field theory
  5. Dynamical zeta function
  6. Proofs
  7. Proof of Theorem \ref{['thm1.1']}
  8. Proof of Corollary \ref{['cor1.3']}: Part I
  9. Proof of Corollary \ref{['cor1.3']}: Part II

Key Result

Theorem 1.1

Figures (3)

  • Figure 1: Double cover of the Riemann sphere $\mathbf{C}\cup\infty$ by the disjoint union of four copies of complex tori $\mathbf{C}/(\mathbf{Z}+\mathbf{Z}\tau)$
  • Figure 2: Grössencharacter action on noncommutative tori
  • Figure 3: [Watkins 2004] Wat1

Theorems & Definitions (19)

  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.3
  • Remark 1.4
  • Theorem 2.1
  • Corollary 2.2
  • Theorem 2.3
  • Lemma 3.1
  • proof
  • Corollary 3.2
  • ...and 9 more