On Franke's theorem in the simplest case

Abstract

For level one spherical automorphic forms on the upper half-plane, we prove directly that every automorphic form is a sum of a cusp form and a linear combination of Laurent coefficients of the standard Eisenstein series. This is the simplest instance of Franke's general theorem, which asserts that automorphic forms on a reductive group are spanned by Laurent coefficients of Eisenstein series induced from cuspidal automorphic forms on Levi subgroups. Unlike Franke's general argument, ours does not invoke Langlands' construction of the discrete automorphic spectrum from cuspidal Eisenstein series. It rests instead on basic analytic properties of automorphic forms and Green's identity.