Almost regular quaternary quadratic forms

Bochnak, Jacek; Oh, Byeong-Kweon

Annales de l'Institut Fourier, Volume 58 (2008), p. 1499-1549 / Harvested from Numdam

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Abstract
Résumé

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 $p \leq 37$ . 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.

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 $p \leq 37$ . 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.

Published online : 2008-01-01

MR 2445826 | Zbl 1162.11020

DOI : https://doi.org/10.5802/aif.2391
Classification: 11E12, 11E20
Keywords: Quadratic equations, almost regular quadratic forms

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     author = {Bochnak, Jacek and Oh, Byeong-Kweon},
     title = {Almost regular quaternary quadratic forms},
     journal = {Annales de l'Institut Fourier},
     volume = {58},
     year = {2008},
     pages = {1499-1549},
     doi = {10.5802/aif.2391},
     zbl = {1162.11020},
     mrnumber = {2445826},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_2008__58_5_1499_0}
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Bochnak, Jacek; Oh, Byeong-Kweon. Almost regular quaternary quadratic forms. Annales de l'Institut Fourier, Volume 58 (2008) pp. 1499-1549. doi : 10.5802/aif.2391. http://gdmltest.u-ga.fr/item/AIF_2008__58_5_1499_0/

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