Almost regular quaternary quadratic forms  [ Formes quadratiques quaternaires presque régulières ]
Annales de l'Institut Fourier, Tome 58 (2008) no. 5, p. 1499-1549
Nous étudions les formes quadratiques entières quaternaires (c’est-à-dire à quatre variables) qui sont définies positives et presque régulières. Nous montrons en particulier qu’une telle forme n’est p-anisotrope que pour au plus un nombre premier p. De plus, pour un nombre premier p, il existe une forme quadratique quaternaire presque régulière p-anisotrope si et seulement si p37. Nous étudions également les genres contenant une forme quadratique presque régulière p-anisotrope. Nous démontrons plusieurs résultats de finitude concernant les familles de ces genres et établissons des critères effectifs presque réguliers.
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 p37. 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.
DOI : https://doi.org/10.5802/aif.2391
Classification:  11E12,  11E20
Mots clés: équations quadratiques, formes quadratiques presque régulières
@article{AIF_2008__58_5_1499_0,
     author = {Bochnak, Jacek and Oh, Byeong-Kweon},
     title = {Almost regular quaternary quadratic forms},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {58},
     number = {5},
     year = {2008},
     pages = {1499-1549},
     doi = {10.5802/aif.2391},
     mrnumber = {2445826},
     zbl = {1162.11020},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2008__58_5_1499_0}
}
Bochnak, Jacek; Oh, Byeong-Kweon. Almost regular quaternary quadratic forms. Annales de l'Institut Fourier, Tome 58 (2008) no. 5, pp. 1499-1549. doi : 10.5802/aif.2391. http://www.numdam.org/item/AIF_2008__58_5_1499_0/

[1] Andrianov, A. N. Quadratic Forms and Hecke operators, Springer-Verlag, Berlin (1987) | MR 884891 | Zbl 0613.10023

[2] Bhargava, M. On the Conway-Schneeberger fifteen theorem, in ‘Quadratic Forms and Their Applications’ (Dublin), pp.27–37, Amer. Math. Soc., Providence, RI, Contemporary Math., Tome 272 (2000) | Zbl 0987.11027

[3] Bhargava, M.; Hanke, J. Universal quadratic forms and the 290 theorem (to appear in Invent. Math.)

[4] Bochnak, J.; Oh, B.-K. Almost universal quadratic forms: an effective solution of a problem of Ramanujan (to appear in Duke Math.Journal)

[5] Cassels, J. W. S. Rational Quadratic Forms, Academic Press, London (1978) | MR 522835 | Zbl 0395.10029

[6] Chan, W. K.; Earnest, A.; Oh, B.-K. Regularity properties of positive definite integral quadratic forms. Algebraic and arithmetic theory of quadratic forms, pp.59–71, Amer. Math. Soc., Providence, RI, Contemp. Math., Tome 344 (2004) | MR 2058667 | Zbl pre02154478

[7] Chan, W. K.; Oh, B.-K. Finiteness theorems for positive definite n-regular quadratic forms, Trans. Amer. Math. Soc., Tome 355 (2003), pp. 2385-2396 | Article | MR 1973994 | Zbl 1026.11046

[8] Chan, W. K.; Oh, B.-K. Positive ternary quadratic forms with finitely many exceptions, Proc. Amer. Math. Soc., Tome 132 (2004), pp. 1567-1573 | Article | MR 2051115 | Zbl 1129.11310

[9] Gerstein, L. The growth of class numbers of quadratic forms, Amer. J. Math., Tome 94 (1972), pp. 221-236 | Article | MR 319889 | Zbl 0252.10018

[10] Hanke, J. Universal quadratic forms and the 290 theorem (http://www.math.duke.edu/~jonhanke/290/Universal-290.html)

[11] Jagy, W. C.; Kaplansky, I.; Schiemann, A. There are 913 regular ternary forms, Mathematika, Tome 44 (1997), pp. 332-341 | Article | MR 1600553 | Zbl 0923.11060

[12] Kloosterman, H. D. On the representation of numbers in the form ax 2 +by 2 +cz 2 +dt 2 , Acta Math., Tome 49 (1926), pp. 407-464 | Article

[13] Martinet, J. Perfect Lattices in Euclidean Spaces, Springer-Verlag, Berlin (2003) | MR 1957723 | Zbl 1017.11031

[14] Mimura, Y. Universal quadratic forms (http://www.kobepharma-u.ac.jp/~math/notes/note05.html)

[15] O’Meara, O. T. Introduction to Quadratic Forms, Springer-Verlag (1963) | Zbl 0107.03301

[16] Pall, G.; Ross, A. An extension of a problem of Kloosterman, Amer. J. Math., Tome 68 (1946), pp. 59-65 | Article | MR 14378 | Zbl 0060.11002

[17] Serre, J.-P. A Course in Arithmetic, Springer-Verlag (1973) | MR 344216 | Zbl 0256.12001

[18] Tartakowsky, W. Die Gesamtheit der Zahlen, die durch eine positive quadratische Form F(x 1 ,,x s )(s4) darstellbar sind, Isv. Akad. Nauk SSSR, Tome 7 (1929), p. 111-122, 165-195

[19] Watson, G. L. Some problems in the theory of numbers, University of London (1953) (Ph. D. Thesis)

[20] Watson, G. L. Transformations of a quadratic form which do not increase the class-number, Proc. London Math. Soc. (3), Tome 12 (1962), pp. 577-587 | Article | MR 142512 | Zbl 0107.26901