Variational property of cusp forms

Abstract

summed over all reduced rational fractions p/q (including 1/0). The construction involved perturbations. A modification of t,he construction could be used to define Eisenstein series corresponding to subgroups of the modular group. Further investigation, however, reveals that perturbation constructions will yield only Eisenstein series, (meaning that the cusp forms are not so constructible). Although the last result appears negative it leads to another result whose statement constitutes the main theorem (see ?2 below). The statement is independent of perturbation theory altogether and it enables us to detect the presence of a cusp form by the so-called period-polynomials of a modular form. It also is possible to construct a formalism which is in itself of interest. In fact a comparable formalism has been discovered by another method by Bol (see [5]), but apparently has never been applied to the present purpose. The method of variation of boundary has been used in the past not only for minimal problems but also for establishing identities among modules of Riemann surfaces (see [74 p. 316]). According to a private communication, M. Schiffer has, in unpublished work, used the method of interior variations (see [7, p. 283 ]) to define and establish the dimension of theta-functions with poles defined on the covering surface of a punctured plane. The present work by contrast might be said, roughly speaking, to use perturbations around the boundary (or cusp) points of the covering surface, leading to Eisenstein series (without poles), or, as is even more important, leading only to these Eisenstein series. As is well known, the Eisenstein series have rather elementary Fourier coefficients [2] which depend, for instance, on divisor functions and it is therefore of value to be able to say that a modular function is linearly expressible in terms of Eisenstein series alone. 2. Main theorem. We consider a congruence sugbroup 9 of linear fractional transformations, T, S, * . *,