Addendum to “p-adic properties of division polynomials and elliptic divisibility sequences”

Abstract

After the (online) publication of [4], the author became aware of a significant overlap of results, although not of methodology, with a beautiful paper of Mohamed Ayad [2]. (See also [1].) In this addendum we briefly compare the results of [2] and [4], using the notation from the latter. Thus Ayad’s elliptic divisibility sequence
m = dm−1 m(M) is denoted instead by Fm(P ). We fix an odd prime p such that P ∈ E(Q) is nonsingular modulo p, and we write r for the rank of apparition of the sequence modulo p, or equivalently, for the order of P modulo p. Assuming that r ≥ 3, Ayad proves [2, Theoreme D] that the elliptic divisibility sequence is periodic modulo p for all μ ≥ 1. This may be compared with the author’s similar result [4, Theorem 8] over number fields and for r ≥ 2, but under the assumption that the elliptic curve E has good ordinary reduction. Ayad also gives a fairly explicit formula for the period πμ, which allows him to prove that πμ = min{1, pμ−ν}π1 for the constant ν = ordp(Fr(P )). This relation between the periods modulo various powers of p is Ayad’s main objective in [2] (“notre objectif principal dans ce travail”), and as an application he obtains nontrivial results for S-integer points on rank one subgroups of E(Q). The author’s purpose in [4] was rather different. The main goal was to prove the existence and algebraicity of the p-adic limit of certain subsequences of Fm(P ). More precisely, Theorem 7 of [4] (over Q) says that if E has good ordinary reduction, then there is a power q = p so that for every m ≥ 1, the limit