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arXiv · 2310.01013

Periods modulo $p$ of integer sequences associated with division polynomials of genus $2$ curves

Yasuhiro Ishitsuka·Tetsushi Ito·Tatsuya Ohshita·Takashi Taniguchi·Yukihiro Uchida
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Abstract

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.

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Yasuhiro Ishitsuka, Tetsushi Ito, Tatsuya Ohshita, Takashi Taniguchi, Yukihiro Uchida. 2023-10-02. Periods modulo $p$ of integer sequences associated with division polynomials of genus $2$ curves. https://arxiv.org/abs/2310.01013

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