The Sharp Composition of Automorphic Distributions

Abstract

In this section, we begin our proof of the main formula (5.38) (or (5.62). First, there are two reasons why we do not follow the lines of the heuristic approach in Section5, neither of which has to do with the difficulty of the approach. The first one is that, from the very start, the original definition as a series (3.1) or (5.32) of 3v is only valid if Re v > 1 while, from the point of view of the spectral analysis of automorphic distributions, the case when y is pure imaginary is more important: the method below will take us directly to this case. Still, let us observe that it is precisely because our heuristic section was based on the Definition (5.32) of at that it allowed us to get some true understanding of the role played by the Dirichlet-Hecke operators G(s) in the formula. The second, and related, reason is that a proof based on (5.32) would depend on a definition of Eisenstein series which could not generalize to more general automorphic distributions. On the contrary, our present proof is based on the Fourier series expansion (3.25), which generalizes to the case of cusp-distributions (4.4).