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On complex and noncommutative tori

Author: Igor Nikolaev
Journal: Proc. Amer. Math. Soc. 134 (2006), 973-981
MSC (2000): Primary 14H52, 46L85
Published electronically: September 28, 2005
MathSciNet review: 2196027
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Abstract: The ``noncommutative geometry'' of complex algebraic curves is studied. As a first step, we clarify a morphism between elliptic curves, or complex tori, and $C^*$-algebras $T_{\theta}=\{u,v~\vert~vu=e^{2\pi i\theta}uv\}$, or noncommutative tori. The main result says that under the morphism, isomorphic elliptic curves map to the Morita equivalent noncommutative tori. Our approach is based on the rigidity of the length spectra of Riemann surfaces.

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Additional Information

Igor Nikolaev
Affiliation: Department of Mathematics, University of Calgary, 2500 University Drive N.W., Calgary, Canada T2N 1N4

Keywords: Elliptic curve, noncommutative torus
Received by editor(s): February 25, 2003
Received by editor(s) in revised form: November 2, 2004
Published electronically: September 28, 2005
Communicated by: Michael Stillman
Article copyright: © Copyright 2005 American Mathematical Society

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