Hostname: page-component-6766d58669-vgfm9 Total loading time: 0 Render date: 2026-05-20T20:43:58.260Z Has data issue: false hasContentIssue false
  • English
  • Français

Integral Springer Theorem for Quaternionic Forms

Published online by Cambridge University Press:  11 January 2016

Luis Arenas-Carmona
Luis Arenas-Carmona*
Affiliation:
Universidad de Chile, Facultad de Ciencias, Casilla 653, Santiago, Chile, learenas@uchile.cl
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

J. S. Hsia has conjectured an arithmetical version of Springer Theorem, which states that no two spinor genera in the same genus of integral quadratic forms become identified over an odd degree extension. In this paper we prove by examples that the corresponding result for quaternionic skew-hermitian forms does not hold in full generality. We prove that it does hold for unimodular skew-hermitian lattices under all extensions and for lattices whose discriminant is relatively prime to 2 under Galois extensions.

Keywords

11E4111E0811E12

Information

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 187 , September 2007 , pp. 157 - 174
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2007

References

[1] Arenas-Carmona, L. E., Applications of spinor class fields: embeddings of orders and quaternionic lattices, Ann. Inst. Fourier, 53 (2003), 2021–2038.Google Scholar
[2] Arenas-Carmona, L. E., Spinor norm for local skew-hermitian forms, Contemporary Math., 344 (2004), 19–29.Google Scholar
[3] Beli, C. N., Integral spinor norm groups over dyadic local fields, J. Number Th., 102 (2003), 125–182.CrossRefGoogle Scholar
[4] Benhamand, J. W. and Hsia, J. S., On exceptional spinor representations, Nagoya Math. J., 87 (1982), 247–260.Google Scholar
[5] Böoge, S., Spinorgeschlechter schiefhermitescher Formen, Arch. Math., XXI (1970), 172–184.CrossRefGoogle Scholar
[6] Brown, K. S., Cohomology of groups, Springer-Verlag, New York, 1994.Google Scholar
[7] Earnest, A. G. and Hsia, J. S., Spinor norms of local integral rotations II, Pacific J. Math., 61 (1975), 71–86; also ibid. 115 (1984), 493–494.Google Scholar
[8] Earnest, A. G. and Hsia, J. S., Spinor genera under field extensions II: 2 unramified in the bottom field, Am. Journal of Math., 100 (1978), 523–538.Google Scholar
[9] Estes, D. R. and Hsia, J. S., Spinor genera under field extensions IV: Spinor class fields, Japanese J. Math., 16 (1990), 341–350.Google Scholar
[10] Hsia, J. S., Spinor norms of local integral rotations I, Pacific J. of Math., 57 (1975), 199–206.Google Scholar
[11] Hsia, J. S., Arithmetic of indefinite quadratic forms, Contemporary Math., 249 (1999), 1–15.CrossRefGoogle Scholar
[12] Kneser, M., Klassenzahlen indefiniter quadratischen Formen in drei oder mehr Veränderlichen, Arch. Math., VII (1956), 323–332.Google Scholar
[13] Kneser, M., Lectures on Galois cohomology of classical groups, TATA Institute of Fundamental Research, Bombay, 1969.Google Scholar
[14] O’meara, O. T., Introduction to quadratic forms, Academic Press, New York, 1963.Google Scholar
[15] Platonov, V. P., Bondarenko, A. A. and Rapinchuk, A. S., Class numbers and groups of algebraic groups, Math. USSR Izv., 14 (1980), 547–569.Google Scholar
[16] Platonov, V. P. and Rapinchuk, A. S., Algebraic groups and number theory, Academic Press, Boston, 1994.Google Scholar
[17] Scharlau, W., Quadratic and Hermitian forms, Springer Verlag, Berlin, 1985. Universidad de Chile Facultad de Ciencias Casilla 653, Santiago Chile learenas@uchile.cl Google Scholar