Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2604.21900 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911618882863104 |
|---|---|
| author | Chan, Daniel Nyman, Adam |
| author_facet | Chan, Daniel Nyman, Adam |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_21900 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Three-periodic helices on elliptic curves and their associated regular algebras Chan, Daniel Nyman, Adam Rings and Algebras Algebraic Geometry 14A22 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. |
| title | Three-periodic helices on elliptic curves and their associated regular algebras |
| topic | Rings and Algebras Algebraic Geometry 14A22 |
| url | https://arxiv.org/abs/2604.21900 |