Quantum Elliptic Curves I: Algebraic Case

A complex elliptic curve $E$ can be defined as the quotient of the analytic space $\mathbb{C}^*$ by a discrete action of the cyclic group $q^{\mathbb{Z}}$ for $\vert q\vert \neq 1$. We study the boundary case when $\vert q\vert =1$, which leads to the notion of a quantum elliptic curve and a conjectural equivalence of categories that one might call a noncommutative GAGA.