Modular properties of elliptic algebras

TL;DR

This paper demonstrates the modular invariance of Feigin and Odesskii's elliptic algebras, allowing their consistent definition over elliptic curves and points, independent of the choice of complex structure.

Contribution

The authors prove the modular invariance property of elliptic algebras $Q_{n,k}( ext{eta}| au)$, enabling their well-defined construction over elliptic curves and points.

Findings

Elliptic algebras are invariant under SL(2,Z) transformations.

Allows unambiguous definition of $Q_{n,k}(E,\xi)$ for elliptic curves and points.

Supports the use of these algebras in geometric and algebraic contexts.

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.