The Hecke-Eisenstein and Klein Forms

Abstract

We have seen in Chapter X, § 3 that the modular form G _k has a q-expansion

$$ {G_k} = 2\zeta (k) + \cdots $$

So the value of the ordinary zeta function appears as the constant term of a modular form (Eisenstein series, as it is called). This phenomenon, first exploited by Klingen [Kl 3] and Siegel [Si 4], has been highly developed by Serre [Se 5] and others. We give here more examples.