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) :\sigma a_n q^n 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 :z\mapsto z+1. 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 &rarr; 0 is the limit in the upper half-plane as the imaginary part of z &rarr; &infin;. 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 &tau;(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 :&delta;(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 :&tau;(n) and called 'ramanujan's tau function', with the normalization :&tau;(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 science+business me
Modular_forms