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.