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On the Modular Function and Its Importance for Arithmetic

  • Paula B. Cohen
Conference paper
Part of the Lecture Notes in Physics book series (LNP, volume 550)

Abstract

The modular function
$$ j(\tau ) = \exp ( - 2i\pi \tau ) + 744 + \sum\limits_{n = 1}^\infty {a_n \exp (2i\pi n\tau )} , a_n \in Z, $$
automorphic with respect to the action of SL(2,Z) on the Poincaré upper half plane of those τ ∈ C with positive imaginary part, is very important for the theory of elliptic curves and of modular forms. Indeed, the values of j parametrise the isomorphism classes over C of elliptic curves. In this lecture, we give an introduction to the modular function, and explain in particular a celebrated result of Th. Schneider (1937) which says that the j function takes an algebraic value at an algebraic point τ if and only if τ is imaginary quadratic, that is the associated class of elliptic curves has complex multiplication. We also discuss some more recent results.

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References

  1. 1.
    H. Cohen, Elliptic Curves, in From Number Theory to Physics, M. Waldschmidt, P. Moussa, J.-M. Luck, C. Itzykson, eds., Springer-Verlag, New York Berlin Heidelberg Tokyo Hong Kong Barcelona Budapest (1989).Google Scholar
  2. 2.
    P. B. Cohen, Humbert surfaces and transcendence properties of automorphic functions, Rocky Mountain J. Math. 26, No.3 (1996), pp. 987–1001.zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    P. Cohen, J. Wolfart, Modular embeddings for some non-arithmetic Fuchsian groups, Acta Arith. LVI, (1990), pp. 93–110.MathSciNetGoogle Scholar
  4. 4.
    N. Koblitz, Introduction to Elliptic Curves and Modular Forms, Springer-Verlag, New York Berlin Heidelberg Tokyo (1984).zbMATHGoogle Scholar
  5. 5.
    Th. Schneider, Arithmetische Untersuchungen elliptischer Integrale, Math. Ann 113, (1937), pp. 1–13.CrossRefMathSciNetGoogle Scholar
  6. 6.
    K. Takeuchi, A characterisation of arithmetic Fuchsian groups, J. Math.Soc. Japan 27, No.4 (1975), pp. 600–612.zbMATHMathSciNetGoogle Scholar
  7. 7.
    K. Takeuchi, Arithmetic triangle groups, J. Math. Soc. Japan 29, (1977), pp. 91–106.zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Paula B. Cohen
    • 1
  1. 1.CNRS, UMR 8524, UFR de MathématiquesUniversité des Sciences et Technologies de LilleVilleneuve d’AscqFrance

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