Three-periodic helices on elliptic curves and their associated regular algebras

TL;DR

This paper studies three-periodic elliptic helices on smooth elliptic curves, characterizing their endomorphism algebras, growth properties, and connections to elliptic algebras, including new constructions with exponential growth.

Contribution

It generalizes previous results on endomorphism algebras of elliptic helices, linking noetherianity, polynomial growth, and Markov triples, and constructs new families with exponential growth.

Findings

Endomorphism algebra is a quotient of the quadratic cover by degree three normal elements.

Endomorphism algebra is noetherian iff it has polynomial growth, with ranks forming a Markov triple.

The associated noncommutative algebra is noetherian, GK-three, and ${ m Proj}$-equivalent to an elliptic algebra.

Abstract

Let $k$ denote an algebraically closed field of characteristic zero and let $X$ denote a smooth elliptic curve over $k$. Given a three-periodic elliptic helix $\underline{\mathcal{E}}$ of vector bundles over $X$ with endomorphism $\mathbb{Z}$-algebra $\operatorname{End} \underline{\mathcal{E}}$ and quadratic cover $\mathbb{S}^{nc}(\underline{\mathcal{E}})$, we prove that $\operatorname{End} \underline{\mathcal{E}}$ is the quotient of $\mathbb{S}^{nc}(\underline{\mathcal{E}})$ by a degree three family of normal elements, generalizing a result of the authors to the case in which $\operatorname{dim }(\operatorname{End} \underline{\mathcal{E}})_{i, i+1}$ isn't a constant function of $i$. We then show that $\operatorname{End} \underline{\mathcal{E}}$ is noetherian if and only if it has polynomial growth, and in this case, the ranks of any three consecutive bundles in the helix are a Markov triple. Furthermore, in this case $\mathbb{S}^{nc}(\underline{\mathcal{E}})$ is a noetherian GK-three $\mathbb{Z}$-algebra which is ${\sf Proj }$-equivalent to an elliptic algebra. We conclude the paper by constructing several new families of elliptic helices with exponential growth.