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
This Fourier expansion exists as a consequence of the presence in the modular group's action on the upper half-plane via 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
Dimension
The dimensions of spaces of cusp forms are, in principle, computable via the Riemann–Roch theorem. For example, the Ramanujan tau 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
which represents (up to a
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.
Consider a standard parabolic subgroup of some reductive group (over , the adele ring), an automorphic form on is called cuspidal if for all parabolic subgroups such that we have , where is the standard minimal parabolic subgroup. The notation for is defined as .
References
- ISBN 0-387-90040-3
- ISBN 0-691-08092-5
- ISBN 0-691-08156-5
- ISBN 978-0521418935