On complex and noncommutative tori

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.