Modular Forms on Noncongruence Subgroups

In this paper I will describe some results and open problems connected with noncongruence subgroups of PSL 2 ( z ). Most of these have their origins in the fundamental paper of Atkin and Swinnerton-Dyer [1]. Considering how much we know about congruence subgroups and the associated modular forms, it is remarkable how little we can say in the general case (to avoid cumbersome language I shall speak of “congruence modular forms” and “noncongruence modular forms”). The principal difficulty is the absence of a satisfactory theory of Hecke operators. For congruence subgroups, the Hecke operators not only provide a direct interpretation of Fourier coefficients of modular forms in terms of eigenvalues, but also furnish a link with arithmetic, essentially through the representation theory of adèle groups. Moreover, using the action of Hecke operators one can calculate congruence modular forms with relative ease. For a noncongruence subgroup it is possible to define the Hecke algebra using double cosets (as in Ch. 3 of [9]), but it seems difficult to exploit (I take this opportunity to correct the erroneous assertion to the contrary at the beginning of [6]); and there seem to be no good alternative computational devices with which to calculate noncongruence modular forms. This problem is discussed in detail in [1].