Nagoya Mathematical Journal

Integral Springer Theorem for quaternionic forms

Luis Arenas-Carmona

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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.

Article information

Nagoya Math. J., Volume 187 (2007), 157-174.

First available in Project Euclid: 4 September 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11E41: Class numbers of quadratic and Hermitian forms 11E08: Quadratic forms over local rings and fields 11E12: Quadratic forms over global rings and fields


Arenas-Carmona, Luis. Integral Springer Theorem for quaternionic forms. Nagoya Math. J. 187 (2007), 157--174.

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