Everything about Cusp Form totally explained
In
number theory, a branch of
mathematics, 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
»
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's 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.
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.
Further Information
Get more info on 'Cusp Form'.
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