The modular degree, congruence primes, and multiplicity one

  • Amod AgasheEmail author
  • Kenneth A. Ribet
  • William A. Stein


The modular degree and congruence number are two fundamental invariants of an elliptic curve over the rational field. Frey and Müller have asked whether these invariants coincide. We find that the question has a negative answer, and show that in the counterexamples, multiplicity one (defined below) does not hold. At the same time, we prove a theorem about the relation between the two invariants: the modular degree divides the congruence number, and the ratio is divisible only by primes whose squares divide the conductor of the elliptic curve. We discuss the ratio even in the case where the square of a prime does divide the conductor, and we study analogues of the two invariants for modular abelian varieties of arbitrary dimension.

Key words

elliptic curves abelian varieties modular degree congruence primes multiplicity one 



The authors are grateful to M. Baker, F. Calegari, B. Conrad, J. Cremona, G. Frey, H. W. Lenstra, Jr. and B. Noohi for discussions and advice regarding this paper. We would especially like to thank B. Conrad for the material in the appendix and for his suggestions concerning a number of technical facts that are inputs to our arguments. The first author is also grateful to the Max-Planck-Institut für Mathematik for its hospitality during a visit when he partly worked on this paper.


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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Amod Agashe
    • 1
    Email author
  • Kenneth A. Ribet
    • 2
  • William A. Stein
    • 3
  1. 1.Department of MathematicsFlorida State UniversityTallahasseeUSA
  2. 2.Department of MathematicsUC BerkeleyBerkeleyUSA
  3. 3.University of WashingtonSeattleUSA

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