Quantum Elliptic Curves I: Algebraic Case

TL;DR

This paper explores the algebraic structure of quantum elliptic curves, extending classical elliptic curve theory to a noncommutative setting when the parameter q lies on the unit circle.

Contribution

It introduces the concept of quantum elliptic curves at the boundary case where |q|=1 and proposes a conjectural noncommutative GAGA equivalence of categories.

Findings

Defined quantum elliptic curves for |q|=1

Proposed a conjectural noncommutative GAGA equivalence

Extended classical elliptic curve theory to a noncommutative setting

Abstract

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.