Relations between cusp forms on congruence and noncongruence groups

Abstract

We construct a quadratic relation between cusp forms of weight two on four genus 1 subgroups of SL2 (Z). Two of the subgroups are congruence and two are noncongruence. We then generalize this to subgroups of F(N) of index 2. Let A be a finite-index subgroup of SL2(Z), and let XA be the associated Riemann surface (*/A. If K is a number field and C/K is a projective nonsingular curve, we say that C/K is a model for XA if there exists a morphism : C * JPl defined over K and a complex analytic isomorphism p: XA 0(QC) such that p(oo) is rational over K and XA P 0 (C) (1) 1 jjc * /SL2 (Z) i PI1 (C) commutes. (Here, the left hand arrow is the natural projection map.) It is known that for any A one can find a C/K as above, and, in fact, Belyi's theorem [B] tells us that the converse is true as well. In light of this, one may wish to study the usual arithmetic objects associated with congruence groups (i.e., Hecke operators, Fourier coefficients, and so forth) for arbitrary subgroups of finite index in SL2(Z). While it is true that on a general noncongruence subgroup, Hecke operators and Fourier coefficients are not related, there are nevertheless results concerning both. (See [Be] for Hecke operators on noncongruence subgroups and [S] for Fourier coefficients. From [Be], we see that Hecke operators on noncongruence subgroups give essentially no new information, but the Atkin-Swinnerton-Dyer congruences of [S] are highly nontrivial.) What has not been previously studied, however, is the relationship between cusp forms on related congruence and noncongruence groups. Since cusp forms on congruence groups are to some extent understood, any explicit information we obtain may be viewed as shedding light on the noncongruence forms. In particular, an actual polynomial relationship between forms would lead to an infinite number of relations between Fourier coefficients. One may view this as a Received by the editors November 16, 1998. 1991 Mathematics Subject Classification. Primary lIF1I; Secondary 1F30.