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

2022/12/07 by Vítězslav Kala, Pavlo Yatsyna · 5 citations
Mathematics · #Analytic Number Theory Research #Algebraic Geometry and Number Theory #Advanced Algebra and Geometry

paper · doi:10.1112/blms.12762

Abstract

Abstract For a totally positive definite quadratic form over the ring of integers of a totally real number field , we show that there are only finitely many totally real field extensions of 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 , where is the degree of over and 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.

Cited by

• Universality criterion sets for quadratic forms over number fields
• Asymptotics of n-universal lattices over number fields
• Universal quadratic forms and Dedekind zeta functions
• Kitaoka's Conjecture and sums of squares
• Extending bounds on minimal ranks of universal quadratic lattices to larger number fields

Related

• Universality lifting from a general base field
• Kitaoka's Conjecture for quadratic fields
• A lower bound for the rank of a universal quadratic form with integer\n coefficients in a totally real number field
• Kitaoka's conjecture and sums of squares
• Kitaoka's Conjecture and sums of squares