Almost regular quaternary quadratic forms

Bochnak, Jacek; Oh, Byeong-Kweon

Annales de l'Institut Fourier, Tome 58 (2008) no. 5, pp. 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 $p\le 37$. 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.

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.

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