Noncommutative geometry of rational elliptic curves

TL;DR

This paper explores the connection between operator algebras and the geometry of rational elliptic curves, establishing an isomorphism with a twisted coordinate ring for certain Cuntz-Krieger algebras.

Contribution

It demonstrates a novel link between Cuntz-Krieger algebras and the algebraic geometry of rational elliptic curves, providing a new perspective on their interplay.

Findings

Existence of a dense self-adjoint sub-algebra isomorphic to a twisted coordinate ring

Establishment of a connection between operator algebras and elliptic curve geometry

Identification of a specific algebraic structure related to 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 $\mathcal{E}({\Bbb Q})=\{(x,y,z) \in {\Bbb P}^2({\Bbb C}) ~|~ y^2z=x(x-z)(x-{b-2\over b+2}z)\}$.