Failures of integral Springer’s Theorem

Daans, Nicolas; Kala, Vítězslav; Krásenský, Jakub; Yatsyna, Pavlo

doi:10.1090/proc/17141

Failures of integral Springer’s Theorem
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by Nicolas Daans, Vítězslav Kala, Jakub Krásenský and Pavlo Yatsyna;

Proc. Amer. Math. Soc. 153 (2025), 2369-2379

DOI: https://doi.org/10.1090/proc/17141

Published electronically: April 9, 2025

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Abstract:

We discuss the phenomenon where an element in a number field is not integrally represented by a given positive definite quadratic form, but becomes integrally represented by this form over a totally real extension of odd degree. We prove that this phenomenon happens infinitely often, and, conversely, establish finiteness results about the situation when the quadratic form is fixed.

References

Constantin N. Beli, Integral spinor norm groups over dyadic local fields, J. Number Theory 102 (2003), no. 1, 125–182. MR 1994477, DOI 10.1016/S0022-314X(03)00057-X
Constantin N. Beli, Representations of integral quadratic forms over dyadic local fields, Electron. Res. Announc. Amer. Math. Soc. 12 (2006), 100–112. MR 2237274, DOI 10.1090/S1079-6762-06-00165-X
Constantin N. Beli, A new approach to classification of integral quadratic forms over dyadic local fields, Trans. Amer. Math. Soc. 362 (2010), no. 3, 1599–1617. MR 2563742, DOI 10.1090/S0002-9947-09-04802-8
C. N. Beli, Representations of quadratic lattices over dyadic local fields, arXiv:1905.04552v2, 2022.
M. Bhargava and J. Hanke, Universal quadratic forms and the $290$-theorem, Preprint, 2011.
Manjul Bhargava, On the Conway-Schneeberger fifteen theorem, Quadratic forms and their applications (Dublin, 1999) Contemp. Math., vol. 272, Amer. Math. Soc., Providence, RI, 2000, pp. 27–37. MR 1803359, DOI 10.1090/conm/272/04395
Richard Elman, Nikita Karpenko, and Alexander Merkurjev, The algebraic and geometric theory of quadratic forms, American Mathematical Society Colloquium Publications, vol. 56, American Mathematical Society, Providence, RI, 2008. MR 2427530, DOI 10.1090/coll/056
Z. He, Arithmetic Springer theorem and $n$-universality under field extensions, arXiv:2312.09560, 2024.
Zilong He, Yong Hu, and Fei Xu, On indefinite $k$-universal integral quadratic forms over number fields, Math. Z. 304 (2023), no. 1, Paper No. 20, 26. MR 4581167, DOI 10.1007/s00209-023-03280-z
John S. Hsia, Yoshiyuki Kitaoka, and Martin Kneser, Representations of positive definite quadratic forms, J. Reine Angew. Math. 301 (1978), 132–141. MR 560499, DOI 10.1515/crll.1978.301.132
J. S. Hsia, Representations by spinor genera, Pacific J. Math. 63 (1976), no. 1, 147–152. MR 424685
Vítězslav Kala, Number fields without universal quadratic forms of small rank exist in most degrees, Math. Proc. Cambridge Philos. Soc. 174 (2023), no. 2, 225–231. MR 4545204, DOI 10.1017/S0305004122000214
Vítězslav Kala, Universal quadratic forms and indecomposables in number fields: a survey, Commun. Math. 31 (2023), no. 2, 81–114. MR 4621253
Myung-Hwan Kim, Recent developments on universal forms, Algebraic and arithmetic theory of quadratic forms, Contemp. Math., vol. 344, Amer. Math. Soc., Providence, RI, 2004, pp. 215–228. MR 2058677, DOI 10.1090/conm/344/06218
Daejun Kim and Seok Hyeong Lee, Lifting problem for universal quadratic forms over totally real cubic number fields, Bull. Lond. Math. Soc. 56 (2024), no. 3, 1192–1206. MR 4735613, DOI 10.1112/blms.12988
Jakub Krásenský, A cubic ring of integers with the smallest Pythagoras number, Arch. Math. (Basel) 118 (2022), no. 1, 39–48. MR 4371163, DOI 10.1007/s00013-021-01662-5
L. Kronecker, Zwei Sätze über Gleichungen mit ganzzahligen Coefficienten, J. Reine Angew. Math. 53 (1857), 173–175 (German). MR 1578994, DOI 10.1515/crll.1857.53.173
Vítězslav Kala and Pavlo Yatsyna, Lifting problem for universal quadratic forms, Adv. Math. 377 (2021), Paper No. 107497, 24. MR 4182914, DOI 10.1016/j.aim.2020.107497
Vítězslav Kala and Pavlo Yatsyna, On Kitaoka’s conjecture and lifting problem for universal quadratic forms, Bull. Lond. Math. Soc. 55 (2023), no. 2, 854–864. MR 4581328, DOI 10.1112/blms.12762
V. Kala and P. Yatsyna, Even better sums of squares over quintic and cyclotomic fields, arXiv:2402.03850, 2024.
D. G. Northcott, An inequality in the theory of arithmetic on algebraic varieties, Proc. Cambridge Philos. Soc. 45 (1949), 502–509. MR 33094, DOI 10.1017/s0305004100025202
O. Timothy O’Meara, Introduction to quadratic forms, Classics in Mathematics, Springer-Verlag, Berlin, 2000. Reprint of the 1973 edition. MR 1754311
Carl Ludwig Siegel, The trace of totally positive and real algebraic integers, Ann. of Math. (2) 46 (1945), 302–312. MR 12092, DOI 10.2307/1969025
Rainer Schulze-Pillot, Representation by integral quadratic forms—a survey, Algebraic and arithmetic theory of quadratic forms, Contemp. Math., vol. 344, Amer. Math. Soc., Providence, RI, 2004, pp. 303–321. MR 2060206, DOI 10.1090/conm/344/06226
Tonny Albert Springer, Sur les formes quadratiques d’indice zéro, C. R. Acad. Sci. Paris 234 (1952), 1517–1519 (French). MR 47021
Fei Xu, Arithmetic Springer theorem on quadratic forms under field extensions of odd degree, Integral quadratic forms and lattices (Seoul, 1998) Contemp. Math., vol. 249, Amer. Math. Soc., Providence, RI, 1999, pp. 175–197. MR 1732359, DOI 10.1090/conm/249/03757
Fei Xu and Yang Zhang, On indefinite and potentially universal quadratic forms over number fields, Trans. Amer. Math. Soc. 375 (2022), no. 4, 2459–2480. MR 4391724, DOI 10.1090/tran/8601

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Failures of integral Springer’s Theorem
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Abstract: