The Petersson Scalar Product

  • Serge Lang
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 222)


We first define the Riemann surface obtained by taking the quotient of the upper half plane by a subgroup Г of SL2(Z), of finite index, and we show how to complete it to a compact Riemann surface X Г . We then define modular forms and cusp forms for such subgroups. In a sense, these generalize the notion of differential form of the first kind on the Riemann surface defined above. Just as one can define a scalar product for differentials of the first kind on X Г , one can extend the definition of this product to arbitrary cusp forms. The Hecke operators act essentially as a trace mapping, from one level to another. They act as Hermitian operators with respect to this scalar product.


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Copyright information

© Springer-Verlag Berlin Heidelberg 1976

Authors and Affiliations

  • Serge Lang
    • 1
  1. 1.Department of MathematicsYale UniversityNew HavenUSA

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