Hostname: page-component-848d4c4894-75dct Total loading time: 0 Render date: 2024-05-17T23:34:38.423Z Has data issue: false hasContentIssue false
  • English
  • Français

THE LIFTING PROBLEM FOR UNIVERSAL QUADRATIC FORMS OVER SIMPLEST CUBIC FIELDS

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

Published online by Cambridge University Press:  06 October 2023

DANIEL GIL-MUÑOZ Open the ORCID record for DANIEL GIL-MUÑOZ [Opens in a new window]  and
MAGDALÉNA TINKOVÁ Open the ORCID record for MAGDALÉNA TINKOVÁ [Opens in a new window]
DANIEL GIL-MUÑOZ
Affiliation:
Department of Algebra, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 00 Praha 8, Czech Republic e-mail: daniel.gil-munoz@mff.cuni.cz
MAGDALÉNA TINKOVÁ*
Affiliation:
Faculty of Information Technology, Czech Technical University in Prague, Thákurova 9, 160 00 Praha 6, Czech Republic and Institute of Analysis and Number Theory, TU Graz, Kopernikusgasse 24/II, 8010 Graz, Austria
*
e-mail: tinkova.magdalena@gmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

The lifting problem for universal quadratic forms over a totally real number field K consists of determining the existence or otherwise of a quadratic form with integer coefficients (or $\mathbb {Z}$-form) that is universal over K. We prove the nonexistence of universal $\mathbb {Z}$-forms over simplest cubic fields for which the integer parameter is big enough. The monogenic case is already known. We prove the nonexistence in the nonmonogenic case by using the existence of a totally positive nonunit algebraic integer in K with minimal (codifferent) trace equal to one.

Keywords

universal quadratic formtotally real number fieldadditively indecomposable integer

MSC classification

Primary: 11R16: Cubic and quartic extensions
Secondary: 11E12: Quadratic forms over global rings and fields 11R04: Algebraic numbers; rings of algebraic integers 11R80: Totally real fields
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , First View , pp. 1 - 13
Check for updates
Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

The first author was supported by Czech Science Foundation GAČR, grant 21-00420M, and by Charles University Research Centre program UNCE/SCI/022. The second author was supported by Czech Science Foundation GAČR, grant 22-11563O.

References

Alaca, S., ‘ $p$ -integral bases of a cubic field’, Proc. Amer. Math. Soc. 126 (1998), 1949–1953.CrossRefGoogle Scholar
Gil-Muñoz, D. and Tinková, M., ‘Additive structure of non-monogenic simplest cubic fields’, Preprint, 2022, arXiv:2212.00364.Google Scholar
Hashimoto, Y. and Aoki, M., ‘Normal integral bases and Gaussian periods in the simplest cubic fields’, Ann. Math. Qué., to appear. Published online (19 July 2022).CrossRefGoogle Scholar
Hsia, J. S., Kitaoka, Y. and Kneser, M., ‘Representations of positive definite quadratic forms’, J. reine angew. Math. 301 (1978), 132–141.Google Scholar
Kala, V. and Tinková, M., ‘Universal quadratic forms, small norms and traces in families of number fields’, Int. Math. Res. Not. IMRN 2023 (2023), 7541–7577.CrossRefGoogle Scholar
Kala, V. and Yatsyna, P., ‘Lifting problem for universal quadratic forms’, Adv. Math. 377 (2021), Article no. 107497.CrossRefGoogle Scholar
Kashio, T. and Sekigawa, R., ‘The characterization of cyclic cubic fields with power integral bases’, Kodai Math. J. 44 (2021), 290–306.CrossRefGoogle Scholar
Kim, D. and Lee, S. H., ‘Lifting problem for universal quadratic forms over totally real cubic number fields’, Preprint, 2023, arXiv:2307.07118.Google Scholar
Lemmermeyer, F. and Pethö, A., ‘Simplest cubic fields’, Manuscripta Math. 88 (1995), 53–58.CrossRefGoogle Scholar
Maaß, H., ‘Über die Darstellung total positiver Zahlen des Körpers $R(\sqrt{5})$ als Summe von drei Quadraten’, Abh. Math. Sem. Univ. Hamburg 14 (1941), 185–191.Google Scholar
Narkiewicz, W., Elementary and Analytic Theory of Algebraic Numbers (Polish Scientific Publishers, Warszawa, 1974).Google Scholar
Shanks, D., ‘The simplest cubic fields’, Math. Comput. 28 (1974), 1137–1152.CrossRefGoogle Scholar
Siegel, C. L., ‘Sums of $m$ -th powers of algebraic integers’, Ann. of Math. (2) 46 (1945), 313–339.CrossRefGoogle Scholar
Thomas, E. and Vasquez, A. T., ‘On the resolution of cusp singularities and the Shintani decomposition in totally real cubic number fields’, Math. Ann. 247 (1980), 1–20.CrossRefGoogle Scholar