On Kitaoka's conjecture and lifting problem for universal quadratic forms

Abstract

For a totally positive definite quadratic form over the ring of integers of a totally real number field $K$, we show that there are only finitely many totally real field extensions of $K$ of a fixed degree over which the form is universal (namely, those that have a short basis in a suitable sense). Along the way we give a general construction of a universal form of rank bounded by $D(\log D{)}^{d-1}$, where $d$ is the degree of $K$ over $\mathbb{Q}$ and $D$ is its discriminant. Furthermore, for any fixed degree we prove (weak) Kitaoka's conjecture that there are only finitely many totally real number fields with a universal ternary quadratic form.

8 pages, to appear in Bull. LMS

References in corpus (2)

• Pythagoras numbers of orders in biquadratic fields
• On the Pythagoras number of the simplest cubic fields

Cited by in corpus (10)

• Pythagoras numbers of orders in biquadratic fields
• Universal quadratic forms and indecomposables in number fields: A survey
• Lifting problem for universal quadratic forms over totally real cubic number fields
• Universal quadratic forms and Northcott property of infinite number fields
• Most totally real fields do not have universal forms or Northcott property
• Arithmetic of cubic number fields: Jacobi-Perron, Pythagoras, and indecomposables
• Sails for universal quadratic forms
• Additive structure of non-monogenic simplest cubic fields
• Universal quadratic forms and Dedekind zeta functions
• Failures of integral Springer's Theorem