On Elliptic Sequences over Commutative Rings
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866914450887409664 |
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| author | Xu, Junyan |
| author_facet | Xu, Junyan |
| contents | We define elliptic sequences over a commutative ring as sequences indexed by the (positive) integers satisfying a 4-parameter, highly symmetric family of homogeneous quartic relations among terms which we call elliptic relations. We classify elliptic sequences over a field into three types, and show that most of them are dilated multiples of standard elliptic divisibility sequences (EDSs) which form countably many 4-dimensional families. In particular, we show standard EDSs are elliptic in a purely algebraic way using intricate implications among elliptic relations, without relying on complex analytic theory of Weierstrass functions. We shall use results presented here to give a purely algebraic treatment of division polynomials in a follow-up paper. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_05280 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On Elliptic Sequences over Commutative Rings Xu, Junyan Number Theory Commutative Algebra 11, 13 We define elliptic sequences over a commutative ring as sequences indexed by the (positive) integers satisfying a 4-parameter, highly symmetric family of homogeneous quartic relations among terms which we call elliptic relations. We classify elliptic sequences over a field into three types, and show that most of them are dilated multiples of standard elliptic divisibility sequences (EDSs) which form countably many 4-dimensional families. In particular, we show standard EDSs are elliptic in a purely algebraic way using intricate implications among elliptic relations, without relying on complex analytic theory of Weierstrass functions. We shall use results presented here to give a purely algebraic treatment of division polynomials in a follow-up paper. |
| title | On Elliptic Sequences over Commutative Rings |
| topic | Number Theory Commutative Algebra 11, 13 |
| url | https://arxiv.org/abs/2604.05280 |