Let O be√an order of index m in the maximal order of a quadratic number field k = Q( d). Let Od,m be the orthogonal Z-group of the associated norm form qd,m. We describe the structure of the pointed set Hfl1(Z, Od,m), which classifies quadratic forms isomorphic (properly or improperly) to qd,m in the flat topology. Gauss classified quadratic forms of fundamental discriminant and showed that the composition of any binary Z-form of discriminant ∆k with itself belongs to the principal genus. Using cohomological language, we extend these results to forms of certain non-fundamental discriminants.
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Manuscrit reçu le 3 novembre 2016, révisé le 24 juin 2019, accepté le 28 septembre 2019. 2010 Mathematics Subject Classification. 11E41, 11E72, 11E12. Mots-clefs. flat cohomology, quadratic forms, quadratic orders. This work was supported by grant 1246/2014 from the Germany-Israel Foundation. The first author was also supported by a Chateaubriand Fellowship of the Embassy of France in Israel, 2016.
© Société Arithmétique de Bordeaux, 2019, tous droits réservés.
- Fat cohomology
- Quadratic forms
- Quadratic orders