Noncommutative geometry of rational elliptic curves

Abstract

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 $\mathcal{E}(\mathbb{Q})=\{(x,y,z)\in {\mathbb{P}}^2(\mathbb{C})|y^{2}z=x(x-z)(x-\frac{b-2}{b+2}z)\}$.

to appear Annals of Functional Analysis

Cited by in corpus (2)

• Noncommutative geometry of elliptic surfaces
• Elliptic curves and continued fractions