On the Expansions of Certain Modular Forms of Positive Dimension

Abstract

1. A definition of a modular form of positive dimension has been given in a paper by Rademacher and the author.2 In that paper we found the Fourier expansions of those forms F(r) which belong to the full modular group and which have only polar singularities at the parabolic point T == i when measured in the uniformizing variable x e2wriT. In the present paper we shall also restrict ourselves to functions which belong to the full group and which have only polar singularities at -r == io, but in the definition of a modular form we shall omit the restriction that F(T) be analytic in the upper half-plane and shall merely assume that F(7-) has, as singularities in the fundamental region,3 at most a finite number of poles and possibly a polar singularity at ioo. This problem of determining expansions of modular forms having poles in the upper half-plane was partially considered by Hardy and Ramanujan.4 However they considered only forms o-t positive inltegral dimension which have no singularities at the parabolic points. The generalization to forms of real positive dimension presents no difficulties. We have only to introduce the roots of uiiity E(a, b, c, d) and e21iaL in the transformation formulas (1. 11) and (1. 12) below, and to carry them through the analysis. However in order to take care of forms having singularities at ioo we have to evaluate certain integrals which Hardy and Ramanujan were able to eliminate by means of simple estimates. This part of the work is contained in sections 3, 4, 5, and 6. In this paper we shall consider most of the integrals in the r-plane rather than in the x-plane, where x e2w1r. The original path owr of section 2 is taken in the i-plane so that we may avoid the poles of Fl(T) which are easier to treat there than in the x-plane. After this point much of the work could