Factorization of Elliptic Divisibility Sequences via Modular Curves

Source abstract

In this paper, we give a structural factorization of elliptic divisibility sequences in terms of finitely many modular functions on the modular curve $X_1(N)$. More precisely, we express the functions obtained by pulling back division polynomials along the universal torsion section as products of fixed modular factors. Then we show that this factorization is compatible with the natural action of $(\mathbb{Z}/N\mathbb{Z})^\times$ on the modular curve $X_1(N)$ and so descends to the modular curve $X_0(N)$ via the norm map. As applications, we obtain a perfect power criterion, local and global growth estimates for the associated elliptic divisibility sequences.