On traces of the Brandt-Eichler matrices

Juliusz Brzezinski

Journal de théorie des nombres de Bordeaux (1998)

  • Volume: 10, Issue: 2, page 273-285
  • ISSN: 1246-7405

Abstract

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We compute the numbers of locally principal ideals with given norm in a class of definite quaternion orders and the traces of the Brandt-Eichler matrices corresponding to these orders. As an application, we compute the numbers of representations of algebraic integers by the norm forms of definite quaternion orders with class number one as well as we obtain class number relations for some CM-fields.

How to cite

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Brzezinski, Juliusz. "On traces of the Brandt-Eichler matrices." Journal de théorie des nombres de Bordeaux 10.2 (1998): 273-285. <http://eudml.org/doc/248171>.

@article{Brzezinski1998,
abstract = {We compute the numbers of locally principal ideals with given norm in a class of definite quaternion orders and the traces of the Brandt-Eichler matrices corresponding to these orders. As an application, we compute the numbers of representations of algebraic integers by the norm forms of definite quaternion orders with class number one as well as we obtain class number relations for some CM-fields.},
author = {Brzezinski, Juliusz},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {integral quadratic forms; definite quaternion orders; Brandt-Eichler matrices; representations of algebraic integers by norm forms; class number one; class number relations; CM-fields},
language = {eng},
number = {2},
pages = {273-285},
publisher = {Université Bordeaux I},
title = {On traces of the Brandt-Eichler matrices},
url = {http://eudml.org/doc/248171},
volume = {10},
year = {1998},
}

TY - JOUR
AU - Brzezinski, Juliusz
TI - On traces of the Brandt-Eichler matrices
JO - Journal de théorie des nombres de Bordeaux
PY - 1998
PB - Université Bordeaux I
VL - 10
IS - 2
SP - 273
EP - 285
AB - We compute the numbers of locally principal ideals with given norm in a class of definite quaternion orders and the traces of the Brandt-Eichler matrices corresponding to these orders. As an application, we compute the numbers of representations of algebraic integers by the norm forms of definite quaternion orders with class number one as well as we obtain class number relations for some CM-fields.
LA - eng
KW - integral quadratic forms; definite quaternion orders; Brandt-Eichler matrices; representations of algebraic integers by norm forms; class number one; class number relations; CM-fields
UR - http://eudml.org/doc/248171
ER -

References

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  3. [3] J. Brzezinski, Definite quaternion orders of class number one. J. Théor. Nombres Bordeaux7 (1995), 93-96. Zbl0843.11048MR1413568
  4. [4] C.W. Curtis, I. Reiner, Methods of representation theory. Vol. I. With applications to finite groups and orders. Pure and Applied Mathematics. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1981. MR632548
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  10. [10] F. Götzky, Über eine zahlentheoretische Anwendung von Modulfunktionen zweier Veränderlicher. Math. Ann.100 (1928), 411-437. Zbl54.0407.01MR1512493JFM54.0407.01
  11. [11] H. Hasse, Über die Klassenzahl abelscher Zahlkörper. Akademie-Verlag, Berlin1952. Zbl0046.26003MR49239
  12. [12] A. Hurwitz, Über die Zahlentheorie der Quaternionen. Nachr. k. Gesell. Wiss. Göttingen, Math.- Phys. Kl. (1886), 313-340. JFM27.0162.01
  13. [13] I. Kaplansky, Submodules of quaternion algebras. Proc. London Math. Soc.(3) 19 (1969), 219-232. Zbl0212.39102MR240142
  14. [14] I. Pays, Arbres, orders maximaux et formes quadratiques entières. in Number Theory: Séminaire de théorie des nombres de Paris (1992 -93), ed. David Sinnou, Cambridge University Press1995. Zbl0826.11016MR1345181
  15. [15] J.V. Uspensky, On Gierster's classnumber relations. Amer. J. Math.50 (1928), 93-122. Zbl54.0178.06MR1506656JFM54.0178.06
  16. [16] M.-F. Vignéras, Arithmétique des Algebres de Quaternions, Lect. Notes in Math.800, Springer-Verlag, Berlin-Heidelberg -New York1980. Zbl0422.12008MR580949

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