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Modular forms and the Shimura-Taniyama Conjecture

  • Ze-Li Dou
  • Qiao Zhang

Abstract

The concept of modular form are based on very natural considerations. In this chapter we recount some rudiments of the theory of modular forms without assuming any previous knowledge of the subject on the reader’s part. The number theoretic interest of the subject becomes apparent when we describe the Hecke operators on the spaces of modular forms and the L-functions attached to eigenforms. The connection between elliptic curves and modular forms of weight 2 is briefly described towards the end in order to state the celebrated Shimura-Taniyama Conjecture, which is now a theorem of A. Wiles, et al. See [Wi95] and related articles.

Keywords

Meromorphic Function Modular Form Elliptic Curve Elliptic Curf Elliptic Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [Bo97]
    Borel, A. Automorphic forms on SL2(ℝ). Cambridge University Press, 1997.Google Scholar
  2. [Bu97]
    Bump, D. Automorphic forms and representations. Cambridge University Press, 1997.Google Scholar
  3. [Ge75]
    Gelbart, S. Automorphic forms on adele groups. Princeton University Press, 1975.Google Scholar
  4. [Gu62]
    Gunning, R. Lectures on modular forms (notes by A. Brumer). Princeton University Press, 1962.Google Scholar
  5. [Iw02]
    Iwaniec, H. Spectral methods of automorphic forms, 2nd ed. American Mathematical Society, 2002.Google Scholar
  6. [Kn92]
    Knapp, A. Elliptic curves. Princeton University Press, 1992.Google Scholar
  7. [Mi89]
    Miyake, T. Modular forms. Trans. Y. Maeda. Springer-Verlag, 1989.Google Scholar
  8. [Se73]
    Serre, J.-P. A course in arithmetic. (Translation of Cours d’arithmétic.) Springer-Verlag, 1973.Google Scholar
  9. [Sh71]
    Shimura, G. Introduction to the arithmetic theory of automorphic functions. Iwanami Shoten, Princeton University Press, 1971.zbMATHGoogle Scholar
  10. [Wi95]
    Wiles, A. “Modular elliptic curves and Fermat’s last theorem.” Ann. of Math. (2) 141 (1995): 443–551.MathSciNetCrossRefGoogle Scholar

Copyright information

© Science Press Beijing and Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Ze-Li Dou
    • 1
  • Qiao Zhang
    • 1
  1. 1.Department of MathematicsTexas Christian UniversityFort WorthUSA

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