# Cusp form

In number theory, a branch of mathematics, a **cusp form** is a particular kind of modular form with a zero constant coefficient in the Fourier series expansion.

## Introduction

A cusp form is distinguished in the case of modular forms for the modular group by the vanishing in the Fourier series expansion (see *q*-expansion)

of the constant coefficient *a _{0}*. 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.

## Dimension

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 *a _{1}* = 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

- τ(
*n*)

and called 'Ramanujan's tau function', with the normalization :τ(1) = 1.

## Related concepts

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.

## References

- 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