Introduction to Modular Forms pp 24-43 | Cite as

# The Petersson Scalar Product

## Abstract

We first define the Riemann surface obtained by taking the quotient of the upper half plane by a subgroup *Г* of *SL*_{2}(**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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