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

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Journal Title
Journal ISSN
Volume Title
A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä
Date
2023-04
Major/Subject
Mcode
Degree programme
Language
en
Pages
11
854-864
Series
BULLETIN OF THE LONDON MATHEMATICAL SOCIETY, Volume 55, issue 2
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.
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Keywords
TOTALLY POSITIVE NUMBERS, LATTICES, RANK, ORDERS
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Citation
Kala, V & Yatsyna, P 2023, ' On Kitaoka's conjecture and lifting problem for universal quadratic forms ', Bulletin of the London Mathematical Society, vol. 55, no. 2, pp. 854-864 . https://doi.org/10.1112/blms.12762