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

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.

author accepted manuscript, 13 pages

References in corpus (10)

• Number fields without n-ary universal quadratic forms
• On certain infinite extensions of the rationals with Northcott property
• On Kitaoka's conjecture and lifting problem for universal quadratic forms
• Universal quadratic forms and indecomposables in number fields: A survey
• Minimal rank of universal lattices and number of indecomposable elements in real multiquadratic fields
• Undecidability of ${\mathbb{Q}}^{(2)}$
• Real quadratic fields with a universal quadratic form of given rank have density zero
• Definability and decidability for rings of integers in totally imaginary fields
• On the Northcott property for infinite extensions
• Universal quadratic forms and Northcott property of infinite number fields

Cited by in corpus (1)

• Pythagoras numbers for infinite algebraic fields