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There are no universal ternary quadratic forms over biquadratic fields

Part of: Forms and linear algebraic groups Algebraic number theory: global fields

Published online by Cambridge University Press:  16 September 2020

Jakub Krásenský ,
Magdaléna Tinková  and
Kristýna Zemková
Jakub Krásenský
Affiliation:
Charles University, Faculty of Mathematics and Physics, Department of Algebra, Sokolovská 83, 18600Praha 8, Czech Republic (krasensky@seznam.cz; tinkova.magdalena@gmail.com)
Magdaléna Tinková
Affiliation:
Charles University, Faculty of Mathematics and Physics, Department of Algebra, Sokolovská 83, 18600Praha 8, Czech Republic (krasensky@seznam.cz; tinkova.magdalena@gmail.com)
Kristýna Zemková*
Affiliation:
Fakultät für Mathematik, Technische Universität Dortmund, D-44221Dortmund, Germany (zemk.kr@gmail.com)
*
*Corresponding author.
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Abstract

We study totally positive definite quadratic forms over the ring of integers $\mathcal {O}_K$ of a totally real biquadratic field $K=\mathbb {Q}(\sqrt {m}, \sqrt {s})$. We restrict our attention to classic forms (i.e. those with all non-diagonal coefficients in $2\mathcal {O}_K$) and prove that no such forms in three variables are universal (i.e. represent all totally positive elements of $\mathcal {O}_K$). Moreover, we show the same result for totally real number fields containing at least one non-square totally positive unit and satisfying some other mild conditions. These results provide further evidence towards Kitaoka's conjecture that there are only finitely many number fields over which such forms exist. One of our main tools are additively indecomposable elements of $\mathcal {O}_K$; we prove several new results about their properties.

Keywords

Indecomposable integeruniversal quadratic formternary quadratic formbiquadratic number field

MSC classification

Primary: 11E20: General ternary and quaternary quadratic forms; forms of more than two variables
Secondary: 11E12: Quadratic forms over global rings and fields 11R04: Algebraic numbers; rings of algebraic integers 11R80: Totally real fields
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 63 , Issue 3 , August 2020 , pp. 861 - 912
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Copyright
Copyright © The Authors, 2020. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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