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

TL;DR

This paper investigates the periodicity modulo primes of sequences derived from division polynomials of genus 2 curves, extending known results from elliptic curves to higher genus cases.

Contribution

It establishes the periodic nature of these sequences modulo almost all primes and relates their periods to the order of points on the Jacobian, generalizing Ward's elliptic curve results.

Findings

Sequences are periodic modulo all but finitely many primes p.

The period relates explicitly to the order of points on the Jacobian.

Generalizes elliptic divisibility sequence properties to genus 2 curves.

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.