Cusp form

In number theory, a cusp form is a particular kind of modular form, distinguished in the case of modular forms for the modular group by the vanishing in the Fourier series expansion

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~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:Sigma a_n q^n

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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

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:zmapsto z+1.

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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.

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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

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:Δ(z, q),

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which represents (up to a normalising constant) the discriminant of the cubic on the RHS of the Weierstrass equation of an elliptic curve; and the 24-th power of the Dedekind eta function. The Fourier coefficients here are written

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:τ(n)

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

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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.

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