Noncommutative geometry of rational elliptic curves

Abstract

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