Explicit valuations of division polynomials¶of an elliptic curve
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In this paper, we estimate valuations of division polynomials and compute them explicitely at singular primes. We show that ν?(ψm(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 c1m1 for some constant c1 for a non-torsion point M such that M mod ? is either singular or zero. Furthermore, we show that the common factors of φm(M) and ψm2(M) have valuations at ? asymptotically equal to c2m2 for some constant c2 when M mod ? is singular, which is a generalization of M. Ayad's result.
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