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

Yasuhiro Ishitsuka, Tetsushi Ito, Tatsuya Ohshita, Takashi Taniguchi, Yukihiro Uchida

TL;DR

This work extends Ward’s theory of elliptic divisibility sequences to genus $2$ curves by studying the reduction modulo primes of the Cantor division-polynomial sequence $\{c_n\}$ with respect to an integral point $P$ on a hyperelliptic curve $C$ of genus $2$. Central to the method are Cantor’s division polynomials expressed through the hyperelliptic sigma function, yielding Somos-type bilinear recurrences that control the behavior of $c_n$ and enable periodicity results modulo almost all primes $p$. The authors prove that for odd primes $p$ not dividing the discriminant or certain initial terms, the sequence $\{c_n\}$ is periodic modulo $p$ and that the period satisfies $\mathrm{Per}_p(\bm{c}) = d\,\operatorname{ord}_p(D_P)$, where $d$ divides $p-1$ and is determined by explicit quantities $\alpha_p,\beta_p$ in $\mathbb{F}_p$. This yields an explicit analogue of Ward’s divisibility-period relation in the genus $2$ setting and provides an effective upper bound $\mathrm{Per}_p(\bm{c}) \le (p-1)(1+\sqrt{p})^4$, with further refinement available via the stated parameters.

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.

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

TL;DR

This work extends Ward’s theory of elliptic divisibility sequences to genus curves by studying the reduction modulo primes of the Cantor division-polynomial sequence with respect to an integral point on a hyperelliptic curve of genus . Central to the method are Cantor’s division polynomials expressed through the hyperelliptic sigma function, yielding Somos-type bilinear recurrences that control the behavior of and enable periodicity results modulo almost all primes . The authors prove that for odd primes not dividing the discriminant or certain initial terms, the sequence is periodic modulo and that the period satisfies , where divides and is determined by explicit quantities in . This yields an explicit analogue of Ward’s divisibility-period relation in the genus setting and provides an effective upper bound , with further refinement available via the stated parameters.

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 of such a sequence is periodic for all but finitely many primes , and describe the relation between the period of the reduction modulo of the sequence and the order of the integral point on the reduction modulo in the Jacobian variety explicitly. This generalizes Ward's results on elliptic divisibility sequences associated with division polynomials of elliptic curves.
Paper Structure (6 sections, 15 theorems, 69 equations)

This paper contains 6 sections, 15 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Cantor's division polynomials
  3. Periodicity of the values of Cantor's division polynomials
  4. Proof of the main theorems
  5. Proof of Theorem \ref{['thm:analytic_psi']}
  6. Numerical calculation of periods and orders

Key Result

Theorem 1.1

Let $\bm{c} \coloneqq \{ c_n \}_{n\in\mathbb{Z}} \coloneqq \{ \psi_n(x_P) \}_{n\in\mathbb{Z}}$ be the integer sequence associated with the division polynomials of a hyperelliptic curve $C$ and its integral point $P$ on $C \backslash \{ \infty \}$ defined as above. Assume that $c_3 c_4 c_5 c_6 c_7 (c

Theorems & Definitions (45)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • ...and 35 more