Periods modulo $p$ of integer sequences associated with division polynomials of genus $2$ curves
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arXiv
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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| Acceso en línea: | |
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| _version_ | 1866917378683568128 |
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| author | Ishitsuka, Yasuhiro Ito, Tetsushi Ohshita, Tatsuya Taniguchi, Takashi Uchida, Yukihiro |
| author_facet | Ishitsuka, Yasuhiro Ito, Tetsushi Ohshita, Tatsuya Taniguchi, Takashi Uchida, Yukihiro |
| contents | We study an integer sequence associated with Cantor's division polynomials of a genus 2 curve having an integral point. We show that the reduction modulo $p$ of such a sequence is periodic for all but finitely many primes $p$, and describe the relation between the period of the reduction modulo $p$ of the sequence and the order of the integral point on the reduction modulo $p$ in the Jacobian variety explicitly. This generalizes Ward's results on elliptic divisibility sequences associated with division polynomials of elliptic curves. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_01013 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Periods modulo $p$ of integer sequences associated with division polynomials of genus $2$ curves Ishitsuka, Yasuhiro Ito, Tetsushi Ohshita, Tatsuya Taniguchi, Takashi Uchida, Yukihiro Number Theory Algebraic Geometry 11G30 (Primary) 11B50, 14H25 (Secondary) We study an integer sequence associated with Cantor's division polynomials of a genus 2 curve having an integral point. We show that the reduction modulo $p$ of such a sequence is periodic for all but finitely many primes $p$, and describe the relation between the period of the reduction modulo $p$ of the sequence and the order of the integral point on the reduction modulo $p$ in the Jacobian variety explicitly. This generalizes Ward's results on elliptic divisibility sequences associated with division polynomials of elliptic curves. |
| title | Periods modulo $p$ of integer sequences associated with division polynomials of genus $2$ curves |
| topic | Number Theory Algebraic Geometry 11G30 (Primary) 11B50, 14H25 (Secondary) |
| url | https://arxiv.org/abs/2310.01013 |