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 $\left(\genfrac{}{}{0ex}{}{ab}{cd}\right)\in \mathrm{SL}(2,\mathbb{Z})$. We show that Feigin and Odesskii's elliptic algebras $Q_{n,k}(\eta |\tau )$ have the property $Q_{n,k}\left(\frac{\eta }{c\tau +d}\right|\frac{a\tau +b}{c\tau +d})\cong Q_{n,k}(\eta |\tau )$. 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.

17 pages + references; numerous minor changes, in both notation and conventions