Hostname: page-component-6bf8c574d5-pdxrj Total loading time: 0 Render date: 2025-03-08T06:36:49.844Z Has data issue: false hasContentIssue false
  • English
  • Français

Number fields without universal quadratic forms of small rank exist in most degrees

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

Published online by Cambridge University Press:  06 May 2022

VÍTĚZSLAV KALA
VÍTĚZSLAV KALA*
Affiliation:
Charles University, Faculty of Mathematics and Physics, Department of Algebra, Sokolovská 49/83, 18675 Praha 8, Czech Republic. e-mails: [email protected], [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove that in each degree divisible by 2 or 3, there are infinitely many totally real number fields that require universal quadratic forms to have arbitrarily large rank.

MSC classification

Primary: 11E12: Quadratic forms over global rings and fields
Secondary: 11E20: General ternary and quaternary quadratic forms; forms of more than two variables 11R21: Other number fields 11R80: Totally real fields
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 174 , Issue 2 , March 2023 , pp. 225 - 231
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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 author was supported by Czech Science Foundation GAČR, grant 21-00420M, and by Charles University, projects PRIMUS/20/SCI/002 and UNCE/SCI/022.

References

Bhargava, M.. On the Conway–Schneeberger fifteen theorem. Contemp. Math. 272 (1999), 27–37.CrossRefGoogle Scholar
Bhargava, M. and Hanke, J.. Universal quadratic forms and the 290-theorem. Preprint.Google Scholar
Bhargava, M., Shankar, A. and Wang, X.. Squarefree values of polynomial discriminants I. Invent. Math. (to appear), https://arxiv.org/abs/1611.09806.Google Scholar
Blomer, V. and Kala, V.. Number fields without universal n-ary quadratic forms. Math. Proc. Camb. Phil. Soc. 159 (2015), 239–252.Google Scholar
Blomer, V., Kala, V.. On the rank of universal quadratic forms over real quadratic fields. Doc. Math. 23 (2018), 15–34.CrossRefGoogle Scholar
Čech, M., Lachman, D., Svoboda, J., Tinková, M. and Zemková, K.. Universal quadratic forms and indecomposables over biquadratic fields. Math. Nachr. 292 (2019), 540–555.Google Scholar
Chan, W. K., Kim, M.-H. and Raghavan, S.. Ternary universal integral quadratic forms. Japan. J. Math. 22 (1996), 263–273.CrossRefGoogle Scholar
Chan, W. K. and Oh, B.-K.. Can we recover an integral quadratic form by representing all its subforms?. Preprint, https://arxiv.org/abs/2201.08957.Google Scholar
Deutsch, J. I.. Universality of a non-classical integral quadratic form over $\mathbb Q(\sqrt 5)$ . Acta Arith. 136 (2009), 229–242.Google Scholar
DoleƝálek, M.. Subfields of number field extensions and quadratic forms. Bachelor’s thesis. Charles University (2022, to appear).Google Scholar
Earnest, A. G. and Khosravani, A.. Universal positive quaternary quadratic lattices over totally real number fields. Mathematika 44 (1997), 342–347.CrossRefGoogle Scholar
Gras, M.-N.. Non monogénéité de l’anneau des entiers des extensions cycliques de $\mathbb{Q}$ de degré premier $l\geq 5$ . J. Number Theory 23 (1986), 347–353.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.. Universal quadratic forms and elements of small norm in real quadratic fields. Bull. Aust. Math. Soc. 94 (2016), 7–14.Google Scholar
Kala, V. and Svoboda, J.. Universal quadratic forms over multiquadratic fields. Ramanujan J. 48 (2019), 151–157.Google Scholar
Kala, V. and Tinková, M.. Universal quadratic forms, small norms and traces in families of number fields. Int. Math. Res. Not. IMRN (to appear), https://arxiv.org/abs/2005.12312.Google Scholar
Kala, V. and Yatsyna, P.. Lifting problem for universal quadratic forms. Adv. Math. 377 (2021), 107497, 24 pp.CrossRefGoogle Scholar
Kalyanswamy, S.. Inverse Galois Problem for Totally Real Number Fields. Cornell University Mathematics Department Senior Thesis, 2012. http://pi.math.cornell.edu/files/Research/SeniorTheses/kalyanswamyThesis.pdf.Google Scholar
Kedlaya, K. S.. A construction of polynomials with squarefree discriminants. Proc. Amer. Math. Soc. 140 (2012), 3025–3033.Google Scholar
Kim, B. M.. Universal octonary diagonal forms over some real quadratic fields. Comment. Math. Helv. 75 (2000), 410–414.Google Scholar
Kim, B. M.. Kim, M.-H. and Oh, B.-K.. A finiteness theorem for representability of quadratic forms by forms. J. Reine Angew. Math. 581 (2005), 23–30.Google Scholar
Kim, B. M., Kim, M.-H. and Park, D.. Real quadratic fields admitting universal lattices of rank 7. J. Number Theory 233 (2022), 456–466.CrossRefGoogle Scholar
Krásenský, J., Tinková, M. and Zemková, K.. There are no universal ternary quadratic forms over biquadratic fields. Proc. Edinb. Math. Soc. 63 (2020), 861–912.Google 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
Neukirch, J.. Algebraic Number Theory (Springer-Verlag, Berlin, 1999).CrossRefGoogle Scholar
O’Meara, O. T.. Introduction to Quadratic forms (Springer-Verlag, Berlin, 1973).CrossRefGoogle Scholar
Schur, I.. Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten. Math. Z. 1 (1918), 377–402.Google Scholar
Shanks, D.. The simplest cubic number fields. Math. Comp. 28 (1974), 1137–1152.CrossRefGoogle Scholar
Siegel, C. L.. Sums of m th powers of algebraic integers. Ann. of Math. 46 (1945), 313–339.CrossRefGoogle Scholar
Sun, L.. A finiteness theorem for quadratic forms. Preprint.Google Scholar
Yatsyna, P.. A lower bound for the rank of a universal quadratic form with integer coefficients in a totally real field. Comment. Math. Helv. 94 (2019), 221–239.Google Scholar