Almost regular quaternary quadratic forms

Jacek Bochnak[1]; Byeong-Kweon Oh[2]

  • [1] Vrije Universiteit Department of Mathematics 1081 HV Amsterdam De Boelelaan 1081 A (The Netherlands)
  • [2] Sejong University Department of Applied Mathematics Seoul, 143-747 (Korea)

Annales de l’institut Fourier (2008)

  • Volume: 58, Issue: 5, page 1499-1549
  • ISSN: 0373-0956

Abstract

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We investigate the almost regular positive definite integral quaternary quadratic forms. In particular, we show that every such form is p -anisotropic for at most one prime number p . Moreover, for a prime p there is an almost regular p -anisotropic quaternary quadratic form if and only if p ≤ 37 . We also study the genera containing some almost regular p -anisotropic quaternary form. We show several finiteness results concerning the families of these genera and give effective criteria for almost regularity.

How to cite

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Bochnak, Jacek, and Oh, Byeong-Kweon. "Almost regular quaternary quadratic forms." Annales de l’institut Fourier 58.5 (2008): 1499-1549. <http://eudml.org/doc/10355>.

@article{Bochnak2008,
abstract = {We investigate the almost regular positive definite integral quaternary quadratic forms. In particular, we show that every such form is $p$-anisotropic for at most one prime number $p$. Moreover, for a prime $p$ there is an almost regular $p$-anisotropic quaternary quadratic form if and only if $p \le 37$. We also study the genera containing some almost regular $p$-anisotropic quaternary form. We show several finiteness results concerning the families of these genera and give effective criteria for almost regularity.},
affiliation = {Vrije Universiteit Department of Mathematics 1081 HV Amsterdam De Boelelaan 1081 A (The Netherlands); Sejong University Department of Applied Mathematics Seoul, 143-747 (Korea)},
author = {Bochnak, Jacek, Oh, Byeong-Kweon},
journal = {Annales de l’institut Fourier},
keywords = {Quadratic equations; almost regular quadratic forms; quadratic equations},
language = {eng},
number = {5},
pages = {1499-1549},
publisher = {Association des Annales de l’institut Fourier},
title = {Almost regular quaternary quadratic forms},
url = {http://eudml.org/doc/10355},
volume = {58},
year = {2008},
}

TY - JOUR
AU - Bochnak, Jacek
AU - Oh, Byeong-Kweon
TI - Almost regular quaternary quadratic forms
JO - Annales de l’institut Fourier
PY - 2008
PB - Association des Annales de l’institut Fourier
VL - 58
IS - 5
SP - 1499
EP - 1549
AB - We investigate the almost regular positive definite integral quaternary quadratic forms. In particular, we show that every such form is $p$-anisotropic for at most one prime number $p$. Moreover, for a prime $p$ there is an almost regular $p$-anisotropic quaternary quadratic form if and only if $p \le 37$. We also study the genera containing some almost regular $p$-anisotropic quaternary form. We show several finiteness results concerning the families of these genera and give effective criteria for almost regularity.
LA - eng
KW - Quadratic equations; almost regular quadratic forms; quadratic equations
UR - http://eudml.org/doc/10355
ER -

References

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  1. A. N. Andrianov, Quadratic Forms and Hecke operators, (1987), Springer-Verlag, Berlin Zbl0613.10023MR884891
  2. M. Bhargava, On the Conway-Schneeberger fifteen theorem, in ‘Quadratic Forms and Their Applications’ (Dublin), pp.27–37, 272 (2000), Amer. Math. Soc., Providence, RI Zbl0987.11027
  3. M. Bhargava, J. Hanke, Universal quadratic forms and the 290 theorem 
  4. J. Bochnak, B.-K. Oh, Almost universal quadratic forms: an effective solution of a problem of Ramanujan Zbl1242.11028
  5. J. W. S. Cassels, Rational Quadratic Forms, (1978), Academic Press, London Zbl0395.10029MR522835
  6. W. K. Chan, A. Earnest, B.-K. Oh, Regularity properties of positive definite integral quadratic forms. Algebraic and arithmetic theory of quadratic forms, pp.59–71, 344 (2004), Amer. Math. Soc., Providence, RI Zbl1142.11326MR2058667
  7. W. K. Chan, B.-K. Oh, Finiteness theorems for positive definite n -regular quadratic forms, Trans. Amer. Math. Soc. 355 (2003), 2385-2396 Zbl1026.11046MR1973994
  8. W. K. Chan, B.-K. Oh, Positive ternary quadratic forms with finitely many exceptions, Proc. Amer. Math. Soc. 132 (2004), 1567-1573 Zbl1129.11310MR2051115
  9. L. Gerstein, The growth of class numbers of quadratic forms, Amer. J. Math. 94 (1972), 221-236 Zbl0252.10018MR319889
  10. J. Hanke, Universal quadratic forms and the 290 theorem 
  11. W. C. Jagy, I. Kaplansky, A. Schiemann, There are 913 regular ternary forms, Mathematika 44 (1997), 332-341 Zbl0923.11060MR1600553
  12. H. D. Kloosterman, On the representation of numbers in the form a x 2 + b y 2 + c z 2 + d t 2 , Acta Math. 49 (1926), 407-464 Zbl53.0155.01
  13. J. Martinet, Perfect Lattices in Euclidean Spaces, (2003), Springer-Verlag, Berlin Zbl1017.11031MR1957723
  14. Y. Mimura, Universal quadratic forms 
  15. O. T. O’Meara, Introduction to Quadratic Forms, (1963), Springer-Verlag Zbl0107.03301
  16. G. Pall, A. Ross, An extension of a problem of Kloosterman, Amer. J. Math. 68 (1946), 59-65 Zbl0060.11002MR14378
  17. J.-P. Serre, A Course in Arithmetic, (1973), Springer-Verlag Zbl0256.12001MR344216
  18. W. Tartakowsky, Die Gesamtheit der Zahlen, die durch eine positive quadratische Form F ( x 1 , ⋯ , x s ) ( s ≥ 4 ) darstellbar sind, Isv. Akad. Nauk SSSR 7 (1929), 111-122, 165-195 Zbl56.0882.04
  19. G. L. Watson, Some problems in the theory of numbers, (1953) 
  20. G. L. Watson, Transformations of a quadratic form which do not increase the class-number, Proc. London Math. Soc. (3) 12 (1962), 577-587 Zbl0107.26901MR142512

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