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Lifting problem for universal quadratic forms over totally real cubic number fields

Daejun Kim, Seok Hyeong Lee

Abstract

Lifting problem for universal quadratic forms asks for totally real number fields $K$ that admit a positive definite quadratic form with coefficients in $\mathbb{Z}$ that is universal over the ring of integers of $K$. In this paper, we show that $K=\mathbb{Q}(ζ_7+ζ_7^{-1})$ is the only such totally real cubic field. Moreover, we show that there is no such biquadratic field.

Lifting problem for universal quadratic forms over totally real cubic number fields

Abstract

Lifting problem for universal quadratic forms asks for totally real number fields that admit a positive definite quadratic form with coefficients in that is universal over the ring of integers of . In this paper, we show that is the only such totally real cubic field. Moreover, we show that there is no such biquadratic field.
Paper Structure (10 sections, 11 theorems, 36 equations, 1 figure)

This paper contains 10 sections, 11 theorems, 36 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations and terminologies
  4. Lattice structure on ${\mathcal{O}_K^{}}$ and $\rho({\mathcal{O}_K^{}})$
  5. Some lemmas
  6. When $d=2$: real quadratic number field case
  7. An algebraic proof
  8. Geometric interpretation
  9. When $d=3$: real cubic number field case
  10. Real biquadratic field case

Key Result

Theorem 1.1

There does not exist a totally real number field $K$ of degree $1,2,3,4,5,$ or $7$ which has principal codifferent ideal and a universal ${\mathbb Z}$-form defined over ${\mathcal{O}_K^{}}$, unless $K={\mathbb Q}, {\mathbb Q}(\sqrt{5}),$ or ${\mathbb Q}(\zeta_7+\zeta_7^{-1})$. The ${\mathbb Z}$-form

Figures (1)

  • Figure 1: Description of cones, points, and a line on plane $H$

Theorems & Definitions (19)

  • Theorem 1.1: KY
  • Remark 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2: KY
  • Lemma 2.3
  • proof
  • Theorem 3.1: Theorem 1.1 of KY
  • Theorem 4.1
  • ...and 9 more