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

Luis Arenas-Carmona[1]

  • [1] Universidad de Chile, Facultad de Ciencias, Casilla 653, Santiago (Chili)

Annales de l'Institut Fourier (2003)

  • Volume: 53, Issue: 7, page 2021-2038
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Arenas-Carmona, Luis. "Applications of spinor class fields: embeddings of orders and quaternionic lattices." Annales de l'Institut Fourier 53.7 (2003): 2021-2038. <http://eudml.org/doc/116092>.

@article{Arenas2003,
abstract = {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.},
affiliation = {Universidad de Chile, Facultad de Ciencias, Casilla 653, Santiago (Chili)},
author = {Arenas-Carmona, Luis},
journal = {Annales de l'Institut Fourier},
keywords = {spinor norm; spinor genus; class fields; skew-Hermitian forms; maximal orders; central simple algebras},
language = {eng},
number = {7},
pages = {2021-2038},
publisher = {Association des Annales de l'Institut Fourier},
title = {Applications of spinor class fields: embeddings of orders and quaternionic lattices},
url = {http://eudml.org/doc/116092},
volume = {53},
year = {2003},
}

TY - JOUR
AU - Arenas-Carmona, Luis
TI - Applications of spinor class fields: embeddings of orders and quaternionic lattices
JO - Annales de l'Institut Fourier
PY - 2003
PB - Association des Annales de l'Institut Fourier
VL - 53
IS - 7
SP - 2021
EP - 2038
AB - 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.
LA - eng
KW - spinor norm; spinor genus; class fields; skew-Hermitian forms; maximal orders; central simple algebras
UR - http://eudml.org/doc/116092
ER -

References

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  1. L.E. Arenas-Carmona, Spinor genera under field extensions for skew-Hermitian forms and cohomology, (2000) 
  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 (2002), AMS Zbl1152.11325
  3. J.W. Benham, J.S. Hsia, On exceptional spinor representations, Nagoya Math. J 87 (1982), 247-260 Zbl0455.10013MR676594
  4. S. Böge, Spinorgeschlechter schiefhermitescher Formen, Arch. Math. XXI (1970), 172-184 Zbl0362.10020MR277476
  5. J. Brzezinski, Spinor class groups of orders, J. Algebra 84 (1983), 468-481 Zbl0529.16002MR723403
  6. C. Chevalley, L'arithmétique dans les algèbres de matrices, Exposés Mathématiques 323 (1936), Hermann, Paris Zbl0014.29006
  7. T. Chinburg, E. Friedman, An embedding Theorem for quaternion algebras, J. London Math. Soc 60 (1999), 33-44 Zbl0940.11053MR1721813
  8. D.R. Estes, J.S. Hsia, Spinor genera under field extensions IV: Spinor Class Fields, Japanese J. Math 16 (1990), 341-350 Zbl0725.11019MR1091167
  9. J.S. Hsia, Representations by spinor genera, Pacific J. of Math 63 (1976), 147-152 Zbl0328.10018MR424685
  10. J.S. Hsia, Y.Y. Shao, F. Xu, Representations of indefinite quadratic forms, J. reine angew. Math 494 (1998), 129-140 Zbl0883.11016MR1604472
  11. J.S. Hsia, Arithmetic of indefinite quadratic forms, Contemporary Math 249 (1999), 1-15 Zbl0992.11032MR1732345
  12. M. Kneser, Strong approximation, Algebraic groups and discrete subgroups 9 (1966), 187-197, Amer. Math. Soc Zbl0201.37904
  13. M. Kneser, Lectures on Galois cohomology of classical groups, (1969) Zbl0246.14008
  14. S. Lang, Algebraic Number Theory, (1994), Springer Verlag, New York Zbl0811.11001MR1282723
  15. O.T. O, ' Meara, Introduction to quadratic forms, (1963), Academic Press, New York MR152507
  16. V.P. Platonov, A.A. Bondarenko, A. S. Rapinchuk, Class numbers and groups of algebraic groups, Math. USSR Izv 14 (1980), 547-569 Zbl0464.20031
  17. V.P. Platonov, A.S. Rapinchuk, Algebraic groups and number theory, (1994), Academic Press, Boston Zbl0841.20046MR1278263
  18. L.H. Rowen, Ring Theory, (1988), Academic Press, San Diego Zbl0651.16001MR1095047
  19. W. Scharlau, Quadratic and Hermitian forms, (1985), Springer Verlag, Berlin Zbl0584.10010MR770063
  20. R. Schulze-Pillot, private letter to E. Friedman, (2000) 
  21. J.-P. Serre, Cohomologie Galoisienne, (1997), Springer Verlag, Berlin Zbl0812.12002

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