Three-periodic helices on elliptic curves and their associated regular algebras
Daniel Chan, Adam Nyman

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 denote an algebraically closed field of characteristic zero and let denote a smooth elliptic curve over . Given a three-periodic elliptic helix of vector bundles over with endomorphism -algebra and quadratic cover , we prove that is the quotient of by a degree three family of normal elements, generalizing a result of the authors to the case in which isn't a constant function of . We then show that 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…
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