Explicit valuations of division polynomials of an elliptic curve

Abstract

In this paper, we estimate valuations of division polynomials and compute them explicitely at singular primes. We show that nu(p) (psi(m)(M)) is asymptotically equal to nu(p)(m) for a non-torsion point M such that M mod p is non-zero and non-singular, and it is asymptotically equal to c(1)m(2) for some constant c(1) for a non-torsion point M such that M mod p is either singular or zero. Furthermore, we show that the common factors of phi(m) (M) and psi(m)(2)(M) have valuations at p asymptotically equal to c(2)m(2) for some constant. c(2) when M mod p is singular, which is a generalization of M. Ayad's result.