Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-05T22:40:54.445Z Has data issue: false hasContentIssue false

Modular symbols for 1(N) and elliptic curves with everywhere good reduction

Published online by Cambridge University Press:  24 October 2008

J. E. Cremona
Affiliation:
Department of Mathematics, University of Exeter, North Park Road, Exeter EX4 4QE

Abstract

The modular symbols method developed by the author in 4 for the computation of cusp forms for 0(N) and related elliptic curves is here extended to 1(N). Two applications are given: the verification of a conjecture of Stevens 14 on modular curves parametrized by 1(N); and the study of certain elliptic curves with everywhere good reduction over real quadratic fields of prime discriminant, introduced by Shimura and related to Pinch's thesis 10.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1992

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

1Atkin, A. O. L. and Li, W.. Twists of newforms and pseudo-eigenvalues of Hecke operators. Invent. Math. 48 (1978), 221243.CrossRefGoogle Scholar
2Birch, B. J. and Kuyk, W. (editors). Modular Functions of One Variable IV. Lecture Notes in Math. vol. 476 (Springer-Verlag, 1975).CrossRefGoogle Scholar
3Casselman, W.. Abelian varieties with many endomorphisms and a conjecture of Shimura's. Invent. Math. 12 (1971), 225236.CrossRefGoogle Scholar
4Cremona, J. E.. Algorithms for Modular Elliptic Curves (Cambridge University Press, 1992).Google Scholar
5Cremona, J. E.. Abelian varieties with extra twist, cusp forms, and elliptic curves over imaginary quadratic fields. J. London Math. Soc., to appear.Google Scholar
6Deligne, P. and Rapoport, M.. Les schmas des modules de courbes elliptiques. In Modular Functions of One Variable II, Lecture Notes in Math. vol. 349 (Springer-Verlag, 1973), pp. 143316.CrossRefGoogle Scholar
7Lang, S.. Introduction to Modular Forms (Springer-Verlag, 1976).Google Scholar
8Manin, Ju. I.. Parabolic points and zeta-functions of modular curves. Math. USSR-lzv. 6 (1972), 1964.CrossRefGoogle Scholar
9Naganuma, H.. On the coincidence of two Dirichlet series associated with cusp forms of Hecke's ;Neben type and Hilbert modular forms over a real quadratic field. J. Math. Soc. Japan 25 (1973), 54755.CrossRefGoogle Scholar
10Pinch, R. G. E.. Elliptic curves over number fields. D.Phil, thesis, Oxford University (1982).Google Scholar
11Pinch, R. G. E.. Elliptic curves with everywhere good reduction. Preprint.Google Scholar
12Shimura, G.. Introduction to the Arithmetic Theory of Automorphic Functions (Math. Soc. Japan, 1971).Google Scholar
13Shiota, K.. On the explicit models of Shimura's elliptic curves. J. Math. Soc. Japan 38 (1986), 649658.CrossRefGoogle Scholar
14Stevens, G.. Stickelberger elements and modular parametrizations of elliptic curves. Invent. Math. 948 (1989), 75106.CrossRefGoogle Scholar
15Dyer, H. P. F. Swinnerton and Birch, B. J.. Elliptic curves and modular functions. In Modular Functions of One Variable IV, Lecture Notes in Math. vol. 476 (Springer-Verlag, 1975).Google Scholar