Representation fields for commutative orders

Luis Arenas-Carmona[1]

  • [1] Universidad de Chile Departamento de matematicas Facultad de ciencia Casilla 653 Santiago (Chile)

Annales de l’institut Fourier (2012)

  • Volume: 62, Issue: 2, page 807-819
  • ISSN: 0373-0956

Abstract

top
A representation field for a non-maximal order ℌ in a central simple algebra is a subfield of the spinor class field of maximal orders which determines the set of spinor genera of maximal orders containing a copy of ℌ . Not every non-maximal order has a representation field. In this work we prove that every commutative order has a representation field and give a formula for it. The main result is proved for central simple algebras over arbitrary global fields.

How to cite

top

Arenas-Carmona, Luis. "Representation fields for commutative orders." Annales de l’institut Fourier 62.2 (2012): 807-819. <http://eudml.org/doc/251030>.

@article{Arenas2012,
abstract = {A representation field for a non-maximal order $\mathfrak\{H\}$ in a central simple algebra is a subfield of the spinor class field of maximal orders which determines the set of spinor genera of maximal orders containing a copy of $\mathfrak\{H\}$. Not every non-maximal order has a representation field. In this work we prove that every commutative order has a representation field and give a formula for it. The main result is proved for central simple algebras over arbitrary global fields.},
affiliation = {Universidad de Chile Departamento de matematicas Facultad de ciencia Casilla 653 Santiago (Chile)},
author = {Arenas-Carmona, Luis},
journal = {Annales de l’institut Fourier},
keywords = {maximal orders; central simple algebras; spinor genera; spinor class fields},
language = {eng},
number = {2},
pages = {807-819},
publisher = {Association des Annales de l’institut Fourier},
title = {Representation fields for commutative orders},
url = {http://eudml.org/doc/251030},
volume = {62},
year = {2012},
}

TY - JOUR
AU - Arenas-Carmona, Luis
TI - Representation fields for commutative orders
JO - Annales de l’institut Fourier
PY - 2012
PB - Association des Annales de l’institut Fourier
VL - 62
IS - 2
SP - 807
EP - 819
AB - A representation field for a non-maximal order $\mathfrak{H}$ in a central simple algebra is a subfield of the spinor class field of maximal orders which determines the set of spinor genera of maximal orders containing a copy of $\mathfrak{H}$. Not every non-maximal order has a representation field. In this work we prove that every commutative order has a representation field and give a formula for it. The main result is proved for central simple algebras over arbitrary global fields.
LA - eng
KW - maximal orders; central simple algebras; spinor genera; spinor class fields
UR - http://eudml.org/doc/251030
ER -

References

top
  1. Luis Arenas-Carmona, Spinor class fields for sheaves of lattices Zbl1060.11018
  2. Luis Arenas-Carmona, Applications of spinor class fields: embeddings of orders and quaternionic lattices, Ann. Inst. Fourier (Grenoble) 53 (2003), 2021-2038 Zbl1060.11018MR2044166
  3. Luis Arenas-Carmona, Relative spinor class fields: a counterexample, Arch. Math. (Basel) 91 (2008), 486-491 Zbl1158.11048MR2465867
  4. C. Chevalley, L’arithmétique sur les algèbres de matrices, (1936), Herman, Paris Zbl0014.29006
  5. T. Chinburg, E. Friedman, An embedding theorem for quaternion algebras, J. London Math. Soc. 60 (1999), 33-44 Zbl0940.11053MR1721813
  6. B. Linowitz, T. R. Shemanske, Embedding orders into central simple algebras Zbl1272.11126
  7. O. T. O’Meara, Introduction to quadratic forms, (1963), Academic press, New York Zbl0107.03301MR152507
  8. I. Reiner, Maximal orders, (1975), Academic press, London Zbl0305.16001MR1972204
  9. T. R. Shemanske, Split orders and convex polytopes in buildings, J. Number Theory 130 (2010), 101-115 Zbl1267.11116MR2569844
  10. A. Weil, Basic Number Theory, (1973), Springer Verlag, Berlin Zbl0267.12001

NotesEmbed ?

top

You must be logged in to post comments.