Quantum Elliptic Curves I: Algebraic Case

Abstract

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

27 pages