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The modular degree, congruence primes, and multiplicity one

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Number Theory, Analysis and Geometry

Abstract

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.

Agashe was partially supported by National Science Foundation Grant No. 0603668. Stein was partially supported by National Science Foundation Grant No. 0653968.

To Serge Lang, our friend and teacher, someone who always knew a fact from a hole in the ground

Mathematics Subject Classification (2010): 11J81

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Acknowledgments

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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Authors and Affiliations

  1. Department of Mathematics, Florida State University, Tallahassee, Florida, 32312, USA

    Amod Agashe

  2. Department of Mathematics, UC Berkeley, 3840, 970 Evans Hall, Berkeley, CA, 94720-3840, USA

    Kenneth A. Ribet

  3. University of Washington, Office: 423 Padelford, Seattle, WA, 98195-4350, USA

    William A. Stein

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  1. Amod Agashe
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  2. Kenneth A. Ribet
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  3. William A. Stein
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Correspondence to Amod Agashe .

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Editors and Affiliations

  1. , Department of Mathematics, Columbia University, Broadway 2990, New York, 10027, New York, USA

    Dorian Goldfeld

  2. , Department of Mathematics, City College of New York, New York, 10031, New York, USA

    Jay Jorgenson

  3. , Department of Mathematics, Yale University, Hillhouse Ave. 10, New Haven, 06520, Connecticut, USA

    Peter Jones

  4. , Department of Mathematics, California Institute of Technology, E. California Blvd. 1200, Pasadena, 91125, California, USA

    Dinakar Ramakrishnan

  5. , Department of Mathematics, University of California at Berkeley, Evans Hall 925, Berkeley, 94720-3840, California, USA

    Kenneth Ribet

  6. , Department of Mathematics, Harvard University, Science Center/One Oxford Street, Cambridge, 02138, Massachusetts, USA

    John Tate

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Agashe, A., Ribet, K.A., Stein, W.A. (2012). The modular degree, congruence primes, and multiplicity one. In: Goldfeld, D., Jorgenson, J., Jones, P., Ramakrishnan, D., Ribet, K., Tate, J. (eds) Number Theory, Analysis and Geometry. Springer, Boston, MA. https://doi.org/10.1007/978-1-4614-1260-1_2

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