Most totally real fields do not have universal forms or Northcott property

TL;DR

The paper investigates how often totally real infinite extensions admit universal quadratic forms or possess the Northcott property, using the constructible topology on the space of totally real subfields \(\mathcal{X}\) of \(\mathbb{Q}^{\operatorname{tr}}\). A key technical development is a bound on the number of square classes of totally positive units represented by a positive definite quadratic lattice of rank \(n\), which yields obstructions to universality and informs Kitaoka's conjecture. The authors prove that, in this topology, the set of totally real fields carrying a universal lattice or having the Northcott property is meager, and they provide explicit constructions of fields with neither property. They also show that non-totally real fields always admit a universal form of rank \(4\). Together, these results illuminate the scarcity of universal forms in the infinite totally real setting and connect unit-group structure with universality in a topological-arithmetic framework.

Abstract

We show that, in the space of all totally real fields equipped with the constructible topology, the set of fields that admit a universal quadratic form, or have the Northcott property, is meager. The main tool is a new theorem on the number of square classes of totally positive units represented by a quadratic lattice of a given rank.

Theorems & Definitions (33)

• Theorem 1: see \ref{['prp:uqf-meager']}
• Theorem 2: see \ref{['thm:rscl']}
• Theorem 3: see \ref{['prp:Northcott-meager']}
• Lemma 2.1
• proof
• Lemma 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof