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.

The authors would like to thank the referee for useful comments and suggestions. The work of Y. I. was supported by JSPS KAKENHI Grant Number 21K18577, 24K21512 and 21K13773. The work of T. I. was supported by JSPS KAKENHI Grant Number 21K18577, 24K21512 and 23K20786. The work of T. O. was supported by JSPS KAKENHI Grant Number 18H05233, 20K14295, 21K18577 and 24K21512. The work of T. T. was supported by JSPS KAKENHI Grant Number 21K18577, 24K21512 and 22H01115. The work of Y. U. was supported by JSPS KAKENHI Grant Number 21K18577, 24K21512 and 20K03517. A part of this work was done while the authors were supported by the Sumitomo Foundation FY2018 Grant for Basic Science Research Projects (Grant Number 180044). Most of calculations were done with the aid of the computer algebra systems Magma [2] and Sage [14].