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

TL;DR

This paper demonstrates that most totally real fields lack universal quadratic forms and the Northcott property, showing these features are rare in the space of all such fields.

Contribution

The authors introduce a new theorem on square classes of totally positive units, proving the scarcity of fields with universal forms or Northcott property.

Findings

The set of fields with universal quadratic forms is meager.

Most totally real fields do not have the Northcott property.

A new theorem on square classes of units is established.

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.