In this note, we give simple examples of sets of quadratic forms that have minimal -universality criteria of multiple cardinalities. This answers a question of Kim, Kim, and Oh [KKO05] in the negative.
Nous donnons des exemples simples d’ensembles de formes quadratiques qui ont des critères d’universalité minimaux de plusieurs cardinalités. Nous donnons ainsi une réponse négative à une question de Kim, Kim et Oh [KKO05].
Keywords: universality criteria, quadratic forms
@article{JTNB_2013__25_3_557_0,
author = {Noam D. Elkies and Daniel M. Kane and Scott Duke Kominers},
title = {Minimal $\mathcal{S}$-universality criteria may vary in size},
journal = {Journal de th\'eorie des nombres de Bordeaux},
pages = {557--563},
publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
volume = {25},
number = {3},
year = {2013},
doi = {10.5802/jtnb.848},
mrnumber = {3179676},
zbl = {1286.11046},
language = {en},
url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.848/}
}
TY - JOUR
AU - Noam D. Elkies
AU - Daniel M. Kane
AU - Scott Duke Kominers
TI - Minimal $\mathcal{S}$-universality criteria may vary in size
JO - Journal de théorie des nombres de Bordeaux
PY - 2013
SP - 557
EP - 563
VL - 25
IS - 3
PB - Société Arithmétique de Bordeaux
UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.848/
DO - 10.5802/jtnb.848
LA - en
ID - JTNB_2013__25_3_557_0
ER -
%0 Journal Article
%A Noam D. Elkies
%A Daniel M. Kane
%A Scott Duke Kominers
%T Minimal $\mathcal{S}$-universality criteria may vary in size
%J Journal de théorie des nombres de Bordeaux
%D 2013
%P 557-563
%V 25
%N 3
%I Société Arithmétique de Bordeaux
%U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.848/
%R 10.5802/jtnb.848
%G en
%F JTNB_2013__25_3_557_0
Noam D. Elkies; Daniel M. Kane; Scott Duke Kominers. Minimal $\mathcal{S}$-universality criteria may vary in size. Journal de théorie des nombres de Bordeaux, Volume 25 (2013) no. 3, pp. 557-563. doi : 10.5802/jtnb.848. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.848/
[Bha00] M. Bhargava, On the Conway-Schneeberger fifteen theorem. Quadratic forms and their applications: Proceedings of the Conference on Quadratic Forms and Their Applications, July 5–9, 1999, University College Dublin, Contemporary Mathematics, vol. 272, American Mathematical Society, 2000, pp. 27–37. | MR | Zbl
[Con00] J. H. Conway, Universal quadratic forms and the fifteen theorem. Quadratic forms and their applications: Proceedings of the Conference on Quadratic Forms and Their Applications, July 5–9, 1999, University College Dublin, Contemporary Mathematics, vol. 272, American Mathematical Society, 2000, pp. 23–26. | MR | Zbl
[Kim04] M.-H. Kim, Recent developments on universal forms. Algebraic and Arithmetic Theory of Quadratic Forms, Contemporary Mathematics, vol. 344, American Mathematical Society, 2004, pp. 215–228. | MR | Zbl
[KKO99] B. M. Kim, M.-H. Kim, and B.-K. Oh, -universal positive definite integral quinary quadratic forms. Integral quadratic forms and lattices: Proceedings of the International Conference on Integral Quadratic Forms and Lattices, June 15–19, 1998, Seoul National University, Korea, Contemporary Mathematics, vol. 249, American Mathematical Society, 1999, pp. 51–62. | MR | Zbl
[KKO05] —, A finiteness theorem for representability of quadratic forms by forms. Journal fur die Reine und Angewandte Mathematik 581 (2005), 23–30. | MR | Zbl
[Kom08a] S. D. Kominers, The -universality criterion is unique. Preprint, arXiv:0807.2099, 2008. | MR
[Kom08b] —, Uniqueness of the -universality criterion. Note di Matematica 28 (2008), no. 2, 203–206. | MR
[Oh00] B.-K. Oh, Universal -lattices of minimal rank. Proceedings of the American Mathematical Society 128 (2000), 683–689. | MR | Zbl
Cited by Sources: