Explicit valuations of division polynomials of an elliptic curve (Q1275221)

The authors estimate valuations of division polynomials of elliptic curves defined over a number field \(K\). It is shown that at a prime \(\mathfrak{p}\) one has that \(v_{\mathfrak{p}}(\psi_{m}(M))\) is asymptotically equal to \(v_{\mathfrak{p}}(m)\) for a non-torsion point \(M\) such that \(M \)mod \(\mathfrak{p}\) is non-zero and non-singular. Moreover, \(v_{\mathfrak{p}}(\psi_{m}(M))\) is asymptotically equal to \(c_{1}m^{2}\) for some constant \(c_{1}\) and for some non-torsion point \(M\) such that \(M \)mod \(\mathfrak{p}\) is either singular or zero. Furthermore they show that the common factors of \(\phi_{m}(M)\) and \(\psi_{m}(M)^{2}\) have valuations at \(\mathfrak{p}\) asymptotically equal to \(c_{2}m^{2}\), for some constant \(c_{2}\) when \(M \)mod \(\mathfrak{p}\) is singular. This allows a generalization of previous results by \textit{M. Ayad} on the computations of \(S\)-integral points on a rank \(1\) elliptic curve [Manuscr. Math. 76, 305-324 (1992; Zbl 0773.14014)] and also on finding periods of elliptic series [Ann. Inst. Fourier 43, 585-618 (1993; Zbl 0781.11007)].