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Quantum Drinfeld Modules and Ray Class Fields of Real Quadratic Global Function Fields

L. Demangos, T. M. Gendron

Abstract

This is the second in a series of two papers presenting a solution to Hilbert's 12th problem for real quadratic function fields in positive characteristic, in the sense of proving an analog of the Theorem of Weber-Fueter. We also offer a conjectural treatment of the number field case using quasicrystal counterparts of the constructions used in function fields.

Quantum Drinfeld Modules and Ray Class Fields of Real Quadratic Global Function Fields

Abstract

This is the second in a series of two papers presenting a solution to Hilbert's 12th problem for real quadratic function fields in positive characteristic, in the sense of proving an analog of the Theorem of Weber-Fueter. We also offer a conjectural treatment of the number field case using quasicrystal counterparts of the constructions used in function fields.

Paper Structure

This paper contains 6 sections, 28 theorems, 195 equations.

Table of Contents

  1. Quantum Drinfeld Modules
  2. Relative and Absolute Ray Class Fields
  3. Generation of Ray Class Fields I: The Case $d=2$
  4. Generation of Ray Class Fields II: The Case $d=3$
  5. Generation of Ray Class Fields III: General Case
  6. Conjecture in Characteristic Zero

Key Result

Proposition 1.1

For all $i$, $\mathfrak{a}_{i}= \mathfrak{a}_{d-1}^{d-i}$.

Theorems & Definitions (56)

  • Proposition 1.1
  • proof
  • Theorem 1.1
  • Lemma 1.1
  • proof
  • Lemma 1.2
  • proof
  • Lemma 1.3
  • proof
  • proof : Proof of Theorem \ref{['qtexplimit']}
  • ...and 46 more