Modular Forms for SL2(ℤ)

  • Anton Deitmar
Part of the Universitext book series (UTX)


The notion of a modular form is introduced. The dimension of the space of modular forms is determined and Fourier-expansions of Eisenstein series are determined leading to the first number theoretical applications. The Fourier-coefficients of a modular form are fed into a Dirichlet series thus forming the associated L-function, which is shown to extend to an entire function. Hecke operators acting on modular forms are introduced and it is shown that L-functions of Hecke eigenforms admit Euler products. Finally, non-holomorphic Eisenstein series and Maaß-wave forms, the real-analytic counterparts of modular forms, and there L-functions, are introduced.


Modular Form Half Plane Fourier Expansion Fundamental Domain Eisenstein Series 
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  1. [AS64]
    Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards Applied Mathematics Series, vol. 55. For sale by the Superintendent of Documents, US Government Printing Office, Washington (1964) zbMATHGoogle Scholar
  2. [BCdS+03]
    Bump, D., Cogdell, J.W., de Shalit, E., Gaitsgory, D., Kowalski, E., Kudla, S.S.: An Introduction to the Langlands Program. Birkhäuser Boston, Boston (2003). Lectures presented at the Hebrew University of Jerusalem, Jerusalem, March 12–16, 2001. Edited by Joseph Bernstein and Stephen Gelbart Google Scholar
  3. [Con78]
    Conway, J.B.: Functions of One Complex Variable, 2nd edn. Graduate Texts in Mathematics, vol. 11. Springer, New York (1978) CrossRefGoogle Scholar
  4. [Dei05]
    Deitmar, A.: A First Course in Harmonic Analysis, 2nd edn. Universitext. Springer, New York (2005) zbMATHGoogle Scholar
  5. [HM98]
    Harris, J., Morrison, I.: Moduli of Curves. Graduate Texts in Mathematics, vol. 187. Springer, New York (1998) zbMATHGoogle Scholar
  6. [HH80]
    Hartley, B., Hawkes, T.O.: Rings, Modules and Linear Algebra. Chapman & Hall, London (1980) Google Scholar
  7. [Haz01]
    Hazewinkel, M. (ed.): Encyclopaedia of Mathematics. Supplement, vol. III. Kluwer Academic, Dordrecht (2001) Google Scholar
  8. [Iwa02]
    Iwaniec, H.: Spectral Methods of Automorphic Forms, 2nd edn. Graduate Studies in Mathematics, vol. 53. American Mathematical Society, Providence (2002) zbMATHGoogle Scholar
  9. [KM85]
    Katz, N.M., Mazur, B.: Arithmetic Moduli of Elliptic Curves. Annals of Mathematics Studies, vol. 108. Princeton University Press, Princeton (1985) zbMATHGoogle Scholar
  10. [Lee03]
    Lee, J.M.: Introduction to Smooth Manifolds. Graduate Texts in Mathematics, vol. 218. Springer, New York (2003) Google Scholar
  11. [Rob10]
    Roberts, C.E. Jr.: Ordinary Differential Equations. Applications, Models, and Computing. Textbooks in Mathematics. CRC Press, Boca Raton (2010). With 1 CD-ROM (Windows) zbMATHGoogle Scholar
  12. [Rud87]
    Rudin, W.: Real and Complex Analysis, 3rd edn. McGraw-Hill, New York (1987) zbMATHGoogle Scholar
  13. [SS03]
    Stein, E.M., Shakarchi, R.: Complex Analysis. Princeton Lectures in Analysis, II. Princeton University Press, Princeton (2003) zbMATHGoogle Scholar
  14. [SW71]
    Stein, E.M., Weiss, G.: Introduction to Fourier Analysis on Euclidean Spaces. Princeton Mathematical Series, vol. 32. Princeton University Press, Princeton (1971) zbMATHGoogle Scholar
  15. [Wil95]
    Wiles, A.: Modular elliptic curves and Fermat’s last theorem. Ann. Math. (2) 141(3), 443–551 (1995). doi: 10.2307/2118559 MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag London 2013

Authors and Affiliations

  • Anton Deitmar
    • 1
  1. 1.Inst. MathematikUniversität TübingenTübingenGermany

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