of the constant coefficient a0. This Fourier expansion exists as a consequence of the presence in the modular group's action on the upper half-plane of the transformation
For other groups, there may be some translation through several units, in which case the Fourier expansion is in terms of a different parameter. In all cases, though, the limit as q → 0 is the limit in the upper half-plane as the imaginary part of z → ∞. Taking the quotient by the modular group, say, this limit corresponds to a cusp of a modular curve (in the sense of a point added for compactification). So, the definition amounts to saying that a cusp form is a modular form that vanishes at a cusp. In the case of other groups, there may be several cusps, and the definition becomes a modular form vanishing at all cusps. This may involve several expansions.
The dimensions of spaces of cusp forms are in principle computable, via the Riemann-Roch theorem. For example, the famous Ramanujan function τ(n) arises as the sequence of Fourier coefficients of the cusp form of weight 12 for the modular group, with a1 = 1. The space of such forms has dimension 1, which means this definition is possible; and that accounts for the action of Hecke operators on the space being by scalar multiplication (Mordell's proof of Ramanujan's identities). Explicitly it is the modular discriminant
- Δ(z, q),
which represents (up to a normalizing constant) the discriminant of the cubic on the right side of the Weierstrass equation of an elliptic curve; and the 24-th power of the Dedekind eta function. The Fourier coefficients here are written
and called 'Ramanujan's tau function', with the normalization :τ(1) = 1.
In the larger picture of automorphic forms, the cusp forms are complementary to Eisenstein series, in a discrete spectrum/continuous spectrum, or discrete series representation/induced representation distinction typical in different parts of spectral theory. That is, Eisenstein series can be 'designed' to take on given values at cusps. There is a large general theory, depending though on the quite intricate theory of parabolic subgroups, and corresponding cuspidal representations.
- Serre, Jean-Pierre, A Course in Arithmetic, Graduate Texts in Mathematics, No. 7, Springer-Verlag, 1978. ISBN 0-387-90040-3
- Shimura, Goro, An Introduction to the Arithmetic Theory of Automorphic Functions, Princeton University Press, 1994. ISBN 0-691-08092-5
- Gelbart, Stephen, Automorphic Forms on Adele Groups, Annals of Mathematics Studies, No. 83, Princeton University Press, 1975. ISBN 0-691-08156-5