Singular modular forms and theta-series

For large weights Eisenstein and Poincaré series were an appropriate method of constructing modular forms. But for small weights these series diverge and so we have to look for another method. In the one-variable case theta-functions are at least as important as Eisenstein or Poincaré series. Although theta-series in several variables appeared early in Siegel's analytic theory of quadratic forms, rather little attention was paid to them during an extended period afterwards. Only much later theta-functions did gain more respect by the impressive work of A.N. Andrianov, M. Eichler, E. Freitag and J.-I. Igusa, amongst others, and nowadays their outstanding importance is acknowledged without any doubt. The significance of theta-series is rooted in number theory. There are no difficulties concerned with convergence or non-vanishing theorems; on the other hand the transformation formula is difficult and in general only automorphic forms for subgroups of the modular group arise. Finally, theta-functions do not fit into Petersson's metrization theory. In this section we deal with theta-series only to the extent necessary for the theory of singular forms, and we restrict ourselves to just a few remarks about recent developments at the end of the section.

Singular forms are introduced as counterparts to cusp forms by the following

Definition 1

A modular form f of degree n and weight k is called singular if its Fourier coefficients a(t) vanish for all half-integral positive t.