manuscripta mathematica

, Volume 97, Issue 3, pp 319–328

Explicit valuations of division polynomials¶of an elliptic curve


  • J. Cheon
    • Electronics and Telecommunications Research Institute, Taejon 305-350, Republic of Korea. e-mail:
  • S. Hahn
    • Department of Mathematics, Korea Advanced Institute of Science and Technology, Taejon 305-701, Republic of Korea. e-mail:

DOI: 10.1007/s002290050104

Cite this article as:
Cheon, J. & Hahn, S. manuscripta math. (1998) 97: 319. doi:10.1007/s002290050104


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 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.

Mathematics Subject Classification (1991):11G05, 11G20, 14H52

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© Springer-Verlag Berlin Heidelberg 1998