Skip to main content
SpringerLink
Log in

Arithmetic of Modular Curves and Applications

  • Conference paper
Algorithmic Algebra and Number Theory

Abstract

The aim of this article is to describe a computational approach to the study of the arithmetic of modular curves Xo(N)and to give applications of these computations.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. J. Antoniadis, M. Bungert, and G. Frey.Properties of twists of elliptic curves.J. Reine Angew. Math., 405:1–28, 1990.

    MathSciNet  MATH  Google Scholar 

  2. A. Atkin and J. Lehner.Hecke operators on ro{m).Math. Ann. , (185):134–160, 1970.

    Google Scholar 

  3. J. Basmaji. Ein Algorithmus zur Berechnung von Hecke-Operatoren und Anwendung auf modulare Kurven. PhD thesis, Universitat (GH) Essen, 1996.

    Google Scholar 

  4. B. Birch. Hecke actions on classes of ternary quadratic forms. In A. Petho et. al., editor, Computational Number Theory, pages 191–212, Berlin, New York, 1991. deGruyter.

    Chapter  Google Scholar 

  5. H. Carayol. Sur les representations l-adiques associees aux formes modulaires de Hilbert. Ann. Sci. Ec. Norm. Super., IV.Ser. 19, 409–468 (1986).

    MathSciNet  MATH  Google Scholar 

  6. J. Cremona. Algorithms for Modular Elliptic Curves. Cambridge University Press, Cambridge, 1992.

    MATH  Google Scholar 

  7. J.E. Cremona. Computing the degree of modular parametrization of a modular elliptic curve. Math. Comp.,64:1235–1250, 1995.

    Article  MathSciNet  MATH  Google Scholar 

  8. H. Darmon, F. Diamond, and R. Taylor. Fermat’s Last Theorem.Current Development in Mathematics, pages 1–170, Boston, 1995. International Press.

    Google Scholar 

  9. P. Deligne and M. Rapoport. Les schemas de modules de courbes elliptiques. In W. Kuyk and P. Deligne, editors, Modular Functions in One Variable II,number 349 in Lecture Notes in Mathematics, pages 143–316, Berlin, Heidelberg, 1973. Springer-Verlag.

    Google Scholar 

  10. P. Deligne and J.-P. Serre.Formes modulaires de poids 1. Ann. sci. E.N.S.4e ser. , pages 507–530, 1974.

    MathSciNet  Google Scholar 

  11. M. Eichler. The Basis Problem for Modular Forms and Traces of the Heeke Operators. In Modular Functions of one Variable I,volume 320 of Lecture Notes in Mathematics, pages 75–152, Berlin, Heidelberg, 1972. Springer-Verlag.

    Google Scholar 

  12. G.Frey. Links between solutions of A- B= Cand elliptic curves. In Number Theory (Ulm 1987), volume 1380 of Lecture Notes in Mathematics, pages 31–62, 1987.

    Google Scholar 

  13. G. Frey, editor. On Arlin’s Conjecture for Odd 2-dimensional Representations, volume 1583 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, Heidelberg, 1994.

    Book  Google Scholar 

  14. G. Frey. On Ternary Equations of Fermat Type and Relations with Elliptic Curves. In G. Cornell, J.H. Silverman, and G. Stevens, editors, Modular Forms and Fermat’s Last Theorem, pages 527–548, Berlin, Heidelberg, 1997. Springer-Verlag.

    Google Scholar 

  15. K. Haberland. Perioden von Modulformen einer Variablen und Gruppencohomologie, I. Math. Nachr .., 112:245–282, 1983.

    Article  MathSciNet  Google Scholar 

  16. C. Hahne. Die kanonische Höhe auf hyperelliptischen Jacobischen Varietaten undErzeuger von modularen Mordell- Weil Gittern. PhD thesis, Universitat (GH) Essen, 1998. to appear.

    Google Scholar 

  17. H. Hida. Congruences of Cusp Forms and special values of their zeta functions. Invent. math.,63:225–261, 1981.

    Article  MathSciNet  MATH  Google Scholar 

  18. E. Halberstadt and A. Kraus.Sur la comparaison galoisienne des points de torsion des courbes elliptiques. C.R. Acad. Sci., Paris Ser. I, 322(4)313–316, 1996.

    MathSciNet  MATH  Google Scholar 

  19. W. Kampkotter. Explizite Gleichungen fUr Jacobische Varietaten hperelliptischer, Kurven. PhD thesis, Universitat (GH) Essen, 1991.

    Google Scholar 

  20. N.M. Katz and B. Mazur. Arithmetic Moduli of Elliptic curves. Princeton University Press, Princeton, NJ, 1985.

    MATH  Google Scholar 

  21. A. Kraus and J. Oesterle.Sur une question de B. Mazur.Math. Ann.,293:259–275, 1992.

    Article  MathSciNet  MATH  Google Scholar 

  22. E. Kaltofen and B. Saunders. On Wiedemann’s Method of Solving Sparse Linear Systems.InProc. 9th Int. Symp., AAECC-9,,number 539 in Lect. Notes Comput. Sci., pages 29–38, 1991.

    MathSciNet  Google Scholar 

  23. E. Kani and W. Schanz.Modular diagonal quotient surfaces.Mathematische Zeitschrift, 227:337–366, 1998.

    Article  MathSciNet  MATH  Google Scholar 

  24. I. Kiming and X. Wang.Examples of 2-dimensional odd Galois repesentations of A5-typeover Q satisfying the Artin conjecture. In G. Frey, editor, On Arlin’s Conjecture for Odd 2-dimensional Representations,volume 1585 of Lecture Notes , in Mathematics, pages 109–121, Berlin, Heidelberg, 1994. Springer-Verlag.

    Chapter  Google Scholar 

  25. S. Lang. Survey of Diophantine Geometry. Springer-Verlag, Berlin, Heidelberg, 1997.

    MATH  Google Scholar 

  26. Q. Liu. Modeles minimaux des courbes de genre deux. J. Reine Angew. Math., 453:127–164, 1994.

    Google Scholar 

  27. J. Manin. Cyclotomic Fields and Modular Curves. Russian Math. Surveys, 26(6)7–78, 1971.

    Article  MathSciNet  MATH  Google Scholar 

  28. J. Manin. Parabolic Points and Zeta-FUnctions of Modular Curves. Math. USSR

    Google Scholar 

  29. Izvestija, 6(1)19–-64, 1972.

    Google Scholar 

  30. B. Mazur. Modular Curves and the Eisenstein ideal. Publ. Math. IHES, 47:33–186, 1977.

    MathSciNet  MATH  Google Scholar 

  31. B. Mazur. Rational Isogenes of Prime Degree. Invent. math.,44:129-162, 1978.

    Article  MathSciNet  MATH  Google Scholar 

  32. L. Merel. Quelques aspectes arithmetiques et geometriques de la theorie des symbols modulaires. PhD thesis, Universite Paris 6, 1993.

    Google Scholar 

  33. L. Merel. Universal Fourier expansions of modular forms. In G. Frey, editor, On Arlin’s Conjecture for Odd 2-dimensional Representations,volume 1585 of Lecture Notes in Mathematics, pages 59–94, Berlin, Heidelberg, 1994. Springer-Verlag.

    Chapter  Google Scholar 

  34. J.F. Mestre. La methode des graphes. Exemples et applications. In Proceedings of the international conference on class numbers and fundamental units of algebraic number fields, pages 217–242, Katata/ Japan, 1986.

    Google Scholar 

  35. J.F. Mestre. Construction de courbes de genre 2 a partir de leurs modules. In T. Mora and C. Traverso, editors, Effective Methods in Algebraic Geometry,volume 94 of Progress in Mathematics, pages 313–334. Birkhauser Verlag, 1991.

    Google Scholar 

  36. L. Mai and M.R. Murty. The Phragmen-Lindelof theorem and modular elliptic curves.Contemp. Math., 166:335–340, 1994.

    MathSciNet  Google Scholar 

  37. M. Miiller. Arithmetik von Modulkurven. PhD thesis, Universitat (GH) Essen, 1998. to appear.

    Google Scholar 

  38. David Mumford. Abelian Varieties. Oxford University Press, Oxford, 2 edition,1974.

    MATH  Google Scholar 

  39. David Mumford. Curves and Their Jacobians. University of Michigan Press, Ann Arbor, 1975.

    MATH  Google Scholar 

  40. D. Mumford. Tata lectures on Theta I-III. Birkhauser Verlag, Ziirich, 1983.

    Google Scholar 

  41. A. Ogg. On the Weierstrass Points of Xo(N) . Illinois J. Math. ,22(1):31–35,1978.

    MathSciNet  MATH  Google Scholar 

  42. C. Poor. On the hyperelliptic locus. Duke Math. J., 76(3)809–884, 1994.

    Article  MathSciNet  MATH  Google Scholar 

  43. K. Ribet. On mod pHecke operators and congruences between modular forms. Invent. math., 71:193–205, 1983.

    Article  MathSciNet  MATH  Google Scholar 

  44. D. Rohrlich. Modular Curves, Hecke Correspondences and L-Functions. In G. Cornell,J.H. Silverman, and G. Stevens, editors, Modular Forms and Fermat’s Last Theorem, pages 41–100, Berlin, Heidelberg, 1997. Springer-Verlag.

    Chapter  Google Scholar 

  45. R. Schoof. Elliptic curves over finite fields and the computation of square roots mod p. Math. Comp., 43:483–494, 1985.

    MathSciNet  Google Scholar 

  46. J.P. Serre. Modular forms of weight one and Galois representations. In A. Frohlich, editor, Algebraic Number Fields, pages 193–268, New York, 1977. Academic Press.

    Google Scholar 

  47. G. Shimura. Introductionto the Arithmetic Theory of Automorphic Functions.Princeton University Press, Princeton, New Jersey, 1971.

    MATH  Google Scholar 

  48. G. Shimura. On modular forms of half integral weight. Annals of Mathematics,97:440–481, 1973.

    Article  MathSciNet  MATH  Google Scholar 

  49. G. Shimura. On the factors of the jacobian variety of a modular function field. J. Math. Soc. Japan, 25(3)523–544, 1973.

    Article  MathSciNet  MATH  Google Scholar 

  50. J.-L. Waldspurger. Sur les coefficients de Fourier des formes modulaires de poids demi-entiers. Journal de Mathimatiques pureset appliquees, 60:375–484, 1981.

    MathSciNet  MATH  Google Scholar 

  51. X. Wang. The Hecke operators on the cusp forms of ro(N)with nebentype. In G. Frey, editor, On Arlin’s Conjecture for Odd 2-dimensional Representations,volume 1585 of Lecture Notes in Mathematics, pages 59–94, Berlin, Heidelberg, 1994. Springer-Verlag.

    Google Scholar 

  52. X. Wang. 2-dimensional simple factors of Jo(N) . Manuscripta mathematica,87:179–197,1995.

    Article  MathSciNet  MATH  Google Scholar 

  53. H.-J. Weber.Hyperelliptic simple Factors of Jo(N)with dimension at least 3.Experimental Math., 1997. to appear.

    Google Scholar 

  54. A. Weil. Zum Beweis des Torellischen Satzes. Nachr. Akad. Wiss. Göttingen,2:33–53, 1957.

    MathSciNet  Google Scholar 

  55. D. Zagier. Modular Parametrization of Elliptic Curves. Can. Math. Bull.,28(3):372–384, 1985.

    Article  MathSciNet  MATH  Google Scholar 

  56. D. Zagier. Hecke operators and periods of modular forms. In S. Gelbart, R. Howe,and P. Sarnak, editors, Festschrift in honor of 1.1. Piatetski-Shapiro on the occasion of his sixtieth birthday,volume 3 of Isr. Math. Conf. Proc.., pages 321–336. The Weizmann Science Press of Israel, 1990.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Experimentelle Mathematik, Universität GH Essen, Ellernstr. 29, 45326, Essen, Germany

    Gerhard Frey & Michael Müller

Authors
  1. Gerhard Frey
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Michael Müller
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Interdisziplinäres Zentrum für Wissenschaftliches Rechnen, der Universität Heidelberg, Im Neuenheimer Feld 368, D-69120, Heidelberg, Germany

    B. Heinrich Matzat

  2. Fachbereich Mathematik, Universität Kaiserslautern, Postfach 3049, D-67653, Kaiserslautern, Germany

    Gert-Martin Greuel

  3. Lehrstuhl D für Mathematik, RWTH Aachen, Templergraben 64, D-52062, Aachen, Germany

    Gerhard Hiss

Rights and permissions

Reprints and permissions

Copyright information

© 1999 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Frey, G., Müller, M. (1999). Arithmetic of Modular Curves and Applications. In: Matzat, B.H., Greuel, GM., Hiss, G. (eds) Algorithmic Algebra and Number Theory. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-59932-3_2

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-59932-3_2

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-64670-9

  • Online ISBN: 978-3-642-59932-3

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation