Periods modulo $p$ of integer sequences associated with division polynomials of genus $2$ curves
Number Theory
2026-04-03 v2 Algebraic Geometry
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.
Keywords
Cite
@article{arxiv.2310.01013,
title = {Periods modulo $p$ of integer sequences associated with division polynomials of genus $2$ curves},
author = {Yasuhiro Ishitsuka and Tetsushi Ito and Tatsuya Ohshita and Takashi Taniguchi and Yukihiro Uchida},
journal= {arXiv preprint arXiv:2310.01013},
year = {2026}
}
Comments
18 pages; to appear in New York Journal of Mathematics