Harmonic analysis in weighted L2-spaces

In this article, we prove a conjecture of Borel, which states that the inclusion of the space of automorphic forms into the space of all C-infinity-functions induces an isomorphism on the cohomology with coefficients in a finite-dimensional representation. We also prove that every automorphic form is a sum of derivatives of Eisenstein series and apply these results to the generalisation of the Manin-Drinfeld theorem and to the trace formula for Hecke operators.