Ramification in the division fields of an elliptic curve



We study the ramification in the division fields of an elliptic curve. Applications on the class number of quadratic fields and on elliptic curves with prime power conductor are given.


  1. [1]
    B. J. Birch andW. Kuyk (eds.),Modular functions of one variable. IV. Lecture Notes in Mathematics476, Springer-Verlag, Berlin, 1975.Google Scholar
  2. [2]
    A. Brumer andK. Kramer, The rank of elliptic curves.Duke Math. J. 44 (1977), no. 4, 715–743.MATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    B. Conrad, The flat deformation functor. In:Modular forms and Fermat’s last theorem (Boston, MA, 1995), Springer, New York, 1997, pp. 373–420.Google Scholar
  4. [4]
    J. E. Cremona,Algorithms for modular elliptic curves. 2nd ed., Cambridge University Press, Cambridge, 1997.MATHGoogle Scholar
  5. [5]
    B. Edixhoven, A. de Groot, andJ. Top, Elliptic curves over the rationals with bad reduction at only one prime.Math. Comp. 54 (1990), no. 189, 413–419.MATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    T. Hadano, On the conductor of an elliptic curve with a rational point of order 2.Nagoya Math. J. 53 (1974), 199–210.MathSciNetGoogle Scholar
  7. [7]
    Y. Kishi andK. Miyake, Parametrization of the quadratic fields whose class numbers are divisible by three.J. Number Theory 80 (2000), no. 2, 209–217.MATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    A. Kraus, Détermination du poids et due conducteur associés aux représentations des points de p-torsion d’une courbe elliptique.Dissertationes Math. (Rozprawy Mat.) 364 (1997), 39.MathSciNetGoogle Scholar
  9. [9]
    P. Llorente andE. Nart, Effective determination of the decomposition of the rational primes in a cubic field.Proc. Amer. Math. Soc. 87 (1983), no. 4, 579–585.MATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    A. P. OGG, Abelian curves of 2-power conductor.Proc. Cambridge Philos. Soc. 62 (1966), 143–148.MATHMathSciNetCrossRefGoogle Scholar
  11. [11]
    I. Papadopoulos, Sur la classification de Néron des courbes elliptiques en caractéristique résiduelle 2 et 3.J. Number Theory 44 (1993), no. 2, 119–152.MATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    F. Ramaroson,Elliptic curves with conductor a square of a prime. Ph.D. thesis, The Johns Hopkins University, Baltimore, Maryland, 1980.Google Scholar
  13. [13]
    H. Schwerdtfeger,Introduction to group theory. Noordhoff International Publishing, Leyden, 1976.MATHGoogle Scholar
  14. [14]
    J.-P. Serre, Propriétés galoisiennes des points d’ordre fini des courbes elliptiques.Invent. Math. 15 (1972), no. 4, 259–331.MATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    —, Sur les représentations modulaires de degré 2 de Gal(Q/Q).Duke Math. J. 54 (1987), no. 1, 179–230.MATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    J.-P. Serre andJ. Tate, Good reduction of abelian varieties.Ann. of Math. (2)88 (1968), 492–517.CrossRefMathSciNetGoogle Scholar
  17. [17]
    J. H. Silverman,The arithmetic of elliptic curves. Springer-Verlag, New York, 1986.MATHGoogle Scholar
  18. [18]
    —,Advanced topics in the arithmetic of elliptic curves. Springer-Verlag, New York, 1994.MATHGoogle Scholar

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© Mathematische Seminar 2003

Authors and Affiliations

  1. 1.Department of MathematicsThe University of Electro-Communications, ChofuTokyoJapan

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