Vol. 78, No. 1, 1978

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The Petersson inner product and the residue of an Euler product

Carlos Moreno

Vol. 78 (1978), No. 1, 149–155

1. Introduction. Implicit in the work of Rankin [6] and explicit in the work of Petersson [5] is a formula connecting the Petersson inner product of two holomorphic modular forms with a residue of the Dirichlet series formed with the products of the Fourier coefficients of the two modular forms. In view of the modern group theoretic interpretation of the eigenfunctions of the Hecke operators as unitary representations of an adèle group, it appears that the ideas of Rankin and Petersson may have wider applicability; for example they may relate to multiplicity-one problems in the theory of automorphic representations. The purpose of this note is to extend these ideas to real analytic modular forms.

Mathematical Subject Classification
Primary: 10D12, 10D12
Received: 15 February 1978
Published: 1 September 1978
Carlos Moreno
CUNY, Bernard M Baruch Coll
United States