Noncommutative geometry of rational elliptic curves

We study an interplay between operator algebras and geometry of rational elliptic curves. Namely, let ${\mathcal{O}}_B$ be the Cuntz-Krieger algebra given by square matrix $B=(b-1,1,b-2,1)$, where $b$ is an integer greater or equal to two. It is proved, that there exists a dense self-adjoint sub-algebra of ${\mathcal{O}}_B$, which is isomorphic (modulo an ideal) to a twisted homogeneous coordinate ring of the rational elliptic curve $$\mathcal{E}$({\Bbb Q})={(x,y,z) \in {\Bbb P}^2({\Bbb C}) ~|~ y^2z=x(x-z)(x-{b-2\over b+2}z)}$.