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

Abstract

AbstractFor 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.