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Continued fractions, modular symbols, and non-commutative geometry

Manin, Yuri I. and Marcolli, Matilde (2001) Continued fractions, modular symbols, and non-commutative geometry. . (Submitted)

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Using techniques introduced by D. Mayer, we prove an extension of the classical Gauss–Kuzmin theorem about the distribution of continued fractions, which in particular allows one to take into account some congruence properties of successive convergents. This result has an application to the Mixmaster Universe model in general relativity. We then study some averages involving modular symbols and show that Dirichlet series related to modular forms of weight 2 can be obtained by integrating certain functions on real axis defined in terms of continued fractions. We argue that the quotient PGL(2,Z) \ P^1(R) should be considered as non–commutative modular curve, and show that the modular complex can be seen as a sequence of K0–groups of the related crossed–product C^∗–algebras.

Item Type:Report or Paper (Discussion Paper)
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Additional Information:(Submitted on 1 Feb 2001 (v1), last revised 7 Aug 2001 (this version, v2). We thank Dieter Mayer, Victor Nistor, and Don Zagier for useful conversations. The second author is partially supported by Sofja Kovalevskaya Award.
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Alexander von Humboldt FoundationUNSPECIFIED
Record Number:CaltechAUTHORS:20170713-072319363
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:79052
Deposited By: Ruth Sustaita
Deposited On:13 Jul 2017 16:50
Last Modified:03 Oct 2019 18:15

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