Modular properties of elliptic algebras

Abstract

Fix a pair of relatively prime integers $n>k\ge 1$, and a point $(\eta\,|\,\tau)\in\mathbb{C}\times\mathbb{H}$, where $\mathbb{H}$ denotes the upper-half complex plane, and let ${{a\;\,b}\choose{c\,\;d}}\in\mathrm{SL}(2,\mathbb{Z})$. We show that Feigin and Odesskii's elliptic algebras $Q_{n,k}(\eta\,|\,\tau)$ have the property $Q_{n,k}\big(\frac{\eta}{c\tau+d}\,\big\vert\,\frac{a\tau+b}{c\tau+d}\big)\cong Q_{n,k}(\eta\,|\,\tau)$. As a consequence, given a pair $(E,\xi)$ consisting of a complex elliptic curve $E$ and a point $\xi\in E$, one may unambiguously define $Q_{n,k}(E,\xi):=Q_{n,k}(\eta\,|\,\tau)$ where $\tau\in\mathbb{H}$ is any point such that $\mathbb{C}/\mathbb{Z}+\mathbb{Z}\tau\cong E$ and $\eta\in\mathbb{C}$ is any point whose image in $E$ is $\xi$. This justifies Feigin and Odesskii's notation $Q_{n,k}(E,\xi)$ for their algebras.