• Multiplicative arithmetic of finite quadratic forms over Dedekind rings

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    • Keywords

       

      Quadratic forms; multiplicative properties; rings of automorphs; Siegel theorem

    • Abstract

       

      Letq(X) be a quadratic form in an even numberm of variables with coefficients in a Dedekind ringK. Let us assume that the setsR(q,a) = {NKm;q(N) = a} of representations of elementsa ofK by the formq are finite. Then certain multiplicative relations are obtained by elementary means between the setsR(q,a) andR(q,ab), whereb is a product of prime elementsρ ofK with finite coefficientsK/ρK. The relations imply similar multiplicative relations between the numbers of elements of the setsR(q,a), which formerly could be obtained only in some special cases like the case whenK = ℤ is the ring of rational integers and only by means of the theory of Hecke operators on the spaces of theta-series. As an application, an almost elementary proof of the Siegel theorem on the mean number of representations of integers by integral positive quadratic forms of determinant 1 is given.

    • Author Affiliations

       

      Anatoli Andrianov1 2

      1. Sonderforschungsbereich 170, “Geometrie und Analysis”, Mathematisches Institut der Universität, Bunsenstr. 3-5, Göttingen - D-3400, Germany
      2. St. Petersburg Branch of the Steklov Mathematical Institute, Fontanka 27, St. Petersburg - 191011, Russia
    • Dates

       
  • Proceedings – Mathematical Sciences | News

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