Publication at Faculty of Mathematics and Physics |
2023
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(logD)d-1, where d is the degree of K over 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.