Preview: Reduction Theory

  • Winfried Scharlau
  • Hans Opolka
Part of the Undergraduate Texts in Mathematics book series (UTM)


The main emphasis of this book has been on the theory of quadratic forms, and we have given special attention to reduction theory. The main question of reduction theory can be formulated in the following way. Let us consider the real-valued quadratic forms in n variables. We look for inequalities for the coefficients such that every form is integrally equivalent to one and only one reduced form, i.e., to a form which satisfies all these inequalities. (From now on, without again stating this explicitly, we will confine ourselves to positive forms.)


Fundamental Domain Semidirect Product Reduction Theory Positive Form Poisson Summation Formula 
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  1. H. Minkowski: Gesammelte Abhandlungen. (Particularly: Diskontinuitätsbereich für arithmetische Äquivalenz, Band 2, S. 53–100.)Google Scholar
  2. C. L. Siegel: Gesammelte Abhandlungen,4 vols., Springer-Verlag, Berlin, Heidelberg, New York, 1966, 1979. (Particularly: The Volume of the Fundamental Domain for Some Infinite Groups, Vol. 1, 459–468; A Mean Value Theorem in the Geometry of Numbers, Vol. 3, 39–46.)Google Scholar
  3. A. Weil: Collected Papers, 3 Vols., Springer-Verlag, 1979, Berlin, Heidelberg, New York. (Particularly: Sur quelques résultats de Siegel,Vol. 1, 339–357.)Google Scholar
  4. G. P. L. Dirichlet: Über die Reduktion der positiven quadratischen Formen mit drei unbestimmten ganzen Zahlen. Werke II.Google Scholar

Copyright information

© Springer Science+Business Media New York 1985

Authors and Affiliations

  • Winfried Scharlau
    • 1
  • Hans Opolka
    • 1
  1. 1.Mathematisches InstitutUniversität MünsterMünsterWest Germany

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