Arithmetics of binary quadratic forms, symmetry of their continued fractions and geometry of their de Sitter world

  • V. Arnold
Original Paper


This article concerns the arithmetics of binary quadratic forms with integer coefficients, the De Sitter’s world and the continued fractions.

Given a binary quadratic forms with integer coefficients, the set of values attaint at integer points is always a multiplicative “tri-group”. Sometimes it is a semigroup (in such case the form is said to be perfect). The diagonal forms are specially studied providing sufficient conditions for their perfectness. This led to consider hyperbolic reflection groups and to find that the continued fraction of the square root of a rational number is palindromic.

The relation of these arithmetics with the geometry of the modular group action on the Lobachevski plane (for elliptic forms) and on the relativistic De Sitter’s world (for the hyperbolic forms) is discussed. Finally, several estimates of the growth rate of the number of equivalence classes versus the discriminant of the form are given.


arithmetics quadratic forms De Sitter’s world continued fraction semigroup tri-group 

Mathematical subject classification:

11A55 11E16 20M99 20M99 53A35 


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Copyright information

© Sociedade Brasileira de Matemática 2003

Authors and Affiliations

  • V. Arnold
    • 1
    • 2
  1. 1. Steklov Mathematical InstituteMoscow V-333, GSP-1RUSSIA
  2. 2. CEREMADEUniversite ParisPARIS Cedex 16-eFRANCE

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