Automorphic forms for subgroups of the modular group

The theory of theta constants with rational characteristics is developed from the point of view of automorphic functions for the principal congruence subgroups of the modular group PSL(2, ℤ). New identities are derived and particular emphasis is given to the level 3 case where a striking generalization of the classical λ-function is obtained. This paper is dedicated to Prof. J. Thompson in recognition of his many contributions to mathematics and of his being the recipient of the Wolf Prize in Mathematics Research by HMF supported in part by the Paul and Gabriella Rosenbaum Foundation and partially sponsored by the Edmund Landau Center for Research in Mathematical Analysis, supported by the Minerva Foundation (Germany). Research by IK supported in part by NSF Grants DMS 9003361 and 9204092.