Applications of spinor class fields: embeddings of orders and quaternionic lattices

Arenas-Carmona, Luis

Annales de l'Institut Fourier, Tome 53 (2003) no. 7, pp. 2021-2038.

Nous étendons la théorie des corps de classe spinoriels et corps de classes spinoriels relatifs à l'étude des problèmes de représentation des réseaux par rapport à plusieurs groupes algébriques linéaires classiques sur les corps de nombres. Nous appliquons cette théorie pour étudier l'ensemble des classes d'isomorphismes d'ordres maximaux dans une algèbre centrale simple qui contiennent un sous-ordre abélien donné. Nous étudions aussi les isométries injectives d'un réseau anti-hermitien quaternionique dans un autre.

We extend the theory of spinor class fields and relative spinor class fields to study representation problems in several classical linear algebraic groups over number fields. We apply this theory to study the set of isomorphism classes of maximal orders of central simple algebras containing a given maximal Abelian suborder. We also study isometric embeddings of one skew-Hermitian Quaternionic lattice into another.

Classification : 11R52, 11E41, 11R56, 11R37, 16G30.
Keywords: spinor norm, spinor genus, class fields, skew-Hermitian forms, maximal orders, central simple algebras
Mot clés : norme spinorielle, genre spinoriel, champs de classe, formes anti-hermitiennes, ordres maximaux, algèbres simples centrales

[1] L.E. Arenas-Carmona Spinor genera under field extensions for skew-Hermitian forms and cohomology (2000) (Ph. D. thesis, Ohio-State University)

[2] L.E. Arenas-Carmona Spinor norm for local skew-Hermitian forms, Proceedings of the International Conference on the Arithmetic and Algebra of Quadratic Forms, Talca (Contemporary Math. (to appear)) (2002) | Zbl

[3] J.W. Benham; J.S. Hsia On exceptional spinor representations, Nagoya Math. J, Volume 87 (1982), pp. 247-260 | MR | Zbl

[4] S. Böge Spinorgeschlechter schiefhermitescher Formen, Arch. Math., Volume XXI (1970), pp. 172-184 | DOI | MR | Zbl

[5] J. Brzezinski Spinor class groups of orders, J. Algebra, Volume 84 (1983), pp. 468-481 | DOI | MR | Zbl

[6] C. Chevalley L'arithmétique dans les algèbres de matrices, Exposés Mathématiques (Actualités scientifiques et industrielles), Volume 323 (1936) | Zbl

[7] T. Chinburg; E. Friedman An embedding Theorem for quaternion algebras, J. London Math. Soc, Volume 60 (1999), pp. 33-44 | DOI | MR | Zbl

[8] D.R. Estes; J.S. Hsia Spinor genera under field extensions IV: Spinor Class Fields, Japanese J. Math, Volume 16 (1990), pp. 341-350 | MR | Zbl

[9] J.S. Hsia Representations by spinor genera, Pacific J. of Math, Volume 63 (1976), pp. 147-152 | MR | Zbl

[10] J.S. Hsia; Y.Y. Shao; F. Xu Representations of indefinite quadratic forms, J. reine angew. Math, Volume 494 (1998), pp. 129-140 | DOI | MR | Zbl

[11] J.S. Hsia Arithmetic of indefinite quadratic forms, Contemporary Math, Volume 249 (1999), pp. 1-15 | MR | Zbl

[12] M. Kneser Strong approximation, Algebraic groups and discrete subgroups (Proc. Symp. Pure Math.), Volume 9 (1966), pp. 187-197 | Zbl

[13] M. Kneser Lectures on Galois cohomology of classical groups (Tata Institute of Fundamental Research, Bombay) (1969) | Zbl

[14] S. Lang Algebraic Number Theory, Springer Verlag, New York, 1994 | MR | Zbl

[15] O.T. O; ' Meara Introduction to quadratic forms, Academic Press, New York, 1963 | MR

[16] V.P. Platonov; A.A. Bondarenko; A. S. Rapinchuk Class numbers and groups of algebraic groups, Math. USSR Izv, Volume 14 (1980), pp. 547-569 | DOI | Zbl

[17] V.P. Platonov; A.S. Rapinchuk Algebraic groups and number theory, Academic Press, Boston, 1994 | MR | Zbl

[18] L.H. Rowen Ring Theory, Academic Press, San Diego, 1988 | MR | Zbl

[19] W. Scharlau Quadratic and Hermitian forms, Springer Verlag, Berlin, 1985 | MR | Zbl

[20] R. Schulze-Pillot private letter to E. Friedman (2000)

[21] J.-P. Serre Cohomologie Galoisienne, Springer Verlag, Berlin, 1997 | Zbl