Two-periodic elliptic helices: classification and geometry

TL;DR

This paper classifies and explores two-periodic elliptic helices as noncommutative analogues of line bundles on elliptic curves, extending classical double cover theory to a noncommutative setting with new examples and classifications.

Contribution

It introduces a classification of two-periodic elliptic helices, generalizing double cover theory to noncommutative geometry and providing explicit examples with unique and multiple classes.

Findings

Existence of unique numerical class of elliptic helices for certain parameters.

Existence of multiple distinct classes of elliptic helices for other parameters.

Contrast with classical (commutative) case where classes are typically unique.

Abstract

Let $k$ denote an algebraically closed field of characteristic zero and let $X$ denote a smooth elliptic curve over $k$. In this paper, motivated by work in \cite{CN}, we think of two-periodic elliptic helices as noncommutative analogues of degree two line bundles over $X$. We classify and study two-periodic elliptic helices in order to generalize the theory of double covers of $\mathbb{P}^{1}$ by $X$ to the noncommutative setting. This leads to the following problem: given an integer $d>2$ and a real number $\theta \in \mathbb{Q}+\mathbb{Q}\sqrt{d^2-4}$, classify elliptic helices inducing double covers of $\mathbb{P}^{1}_{d}$ by ${\sf C}^{\theta}$, where $\mathbb{P}^{1}_{d}$ is Piontkovski's noncommutative projective line and ${\sf C}^{\theta}$ is Polischuk's noncommutative elliptic curve. We find examples of $d$ and $\theta$ such that there is essentially one numerical class of elliptic helices and examples of $d$ and $\theta$ such that there are several distinct numerical classes of elliptic helices, in contrast to the commutative situation.