Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-06T02:43:57.773Z Has data issue: false hasContentIssue false
  • English
  • Français

Modular Parametrizations of Elliptic Curves

Published online by Cambridge University Press:  20 November 2018

D. Zagier
D. Zagier*
Affiliation:
University of MarylandCollege Park Maryland, U.S.A. Max-planck-institut für mathematik, BonnFRG
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Many — conjecturally all — elliptic curves E/ have a "modular parametrization," i.e. for some N there is a map φ from the modular curve X0(N) to E such that the pull-back of a holomorphic differential on E is a modular form (newform) f of weight 2 and level N. We describe an algorithm for computing the degree of φ as a branched covering, discuss the relationship of this degree to the "congruence primes" for f (the primes modulo which there are congruences between f and other newforms), and give estimates for the size of this degree as a function of N.

Keywords

14K0710D1210D23
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 28 , Issue 3 , 01 September 1985 , pp. 372 - 384
Copyright
Copyright © Canadian Mathematical Society 1985

References

1. Birch, B.J. and Kuyk, W., [eds.] Modular Functions of One Variable IV, Springer Lecture Notes 476, Berlin-Heidelberg-New York, 1975, Table I, pp. 81–113.Google Scholar
2. Doi, K. and Ohta, M., On some congruences between cusp forms on Γ0(N), in Modular Functions of One Variable V , Springer Lecture Notes 601, Berlin-Heidelberg-New York, 1977, pp. 91–105.Google Scholar
3. Goldfeld, D., The conjectures of Birch and Swinnerton-Dyer and the class numbers ofquadratic fields, Soc. Math. France, Astérisque 41-42 (1977), pp. 219–277.Google Scholar
4. Gross, B. and Zagier, D., Points de Heegner et dérivées de fonctions L, C.R. Acad. Se. Paris 297 (1983), pp. 85–87.Google Scholar
5. Haberland, K., Perioden von Modulformen einer Variablen und Gruppencohomologie, Math. Nachr. 112 (1983), pp. 245–282.Google Scholar
6. Mazur, B., Modular curves and the Eisenstein ideal, Publ. Math. I.H.E.S. 47 (1977), pp. 33–186.Google Scholar
7. Mazur, B., Rational isogenics of prime degree, Invent. Math. 44 (1978), pp. 129—162 Google Scholar
8. Mazur, B. and Swinnerton-Dyer, H. P. F., Arithmetic of Weil curves, Invent. Math. 25 (1974), pp. 1–61.Google Scholar
9. Oesterlé, J., Nombres de classes des corps quadratiques imaginaires, Séminaire Bourbaki 1983—1984, Exposé 631.Google Scholar
10. Ribet, K., Mod p Hecke operators and congruences between modular forms, Invent. Math. 71 (1983), pp. 193–205.Google Scholar
11. Shimura, G., Introduction to the Arithmetic Theory of Automorphic Functions, Princeton, 1971, Chapter VII. Google Scholar
12. Swinnerton-Dyer, H. P. F. and Birch, B.J., Elliptic curves and modularfunctions, in Modular Functions of One Variable IV , Springer Lecture Notes 476, Berlin-Heidelberg-New York, 1975, pp. 2—32 Google Scholar