, Volume 97, Issue 3, pp 319-328

Explicit valuations of division polynomials¶of an elliptic curve

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Abstract:

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