Arithmetics of binary quadratic
forms,symmetry of their continued fractions and geometry of
their de Sitter world |
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Authors: | V Arnold |
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Institution: | (1) Steklov Mathematical Institute, Gubkina str. 8, Moscow V-333, GSP-1, 117966, RUSSIA;(2) Present address: CEREMADE, Universite Paris, 9—Dauphine Place du Marechal de Lattre de Tassigny, 75775 PARIS Cedex 16-e, FRANCE |
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Abstract: | 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.Partially supported by RFBR, grant 02-01-00655. |
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Keywords: | arithmetics quadratic forms De Sitter s world" target="_blank">gif" alt="rsquo" align="BASELINE" BORDER="0">s world continued fraction semigroup tri-group |
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