Cusp form
Encyclopedia
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 (see q-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 operator
s 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 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
and called 'Ramanujan's tau function', with the normalization :τ(1) = 1.
In the larger picture of automorphic form
s, 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 representation
s.
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, a branch of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a cusp form is a particular kind of modular form
Modular form
In mathematics, a modular form is a analytic function on the upper half-plane satisfying a certain kind of functional equation and growth condition. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections...
, distinguished in the case of modular forms for the modular group
Modular group
In mathematics, the modular group Γ is a fundamental object of study in number theory, geometry, algebra, and many other areas of advanced mathematics...
by the vanishing in the Fourier series
Fourier series
In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...
expansion (see q-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
Cusp
Cusp may refer to:*Beach cusps, a pointed and regular arc pattern of the shoreline at the beach*Behavioral cusp an important behavior change with far reaching consequences*Cusp catastrophe...
of a modular curve
Modular curve
In number theory and algebraic geometry, a modular curve Y is a Riemann surface, or the corresponding algebraic curve, constructed as a quotient of the complex upper half-plane H by the action of a congruence subgroup Γ of the modular group of integral 2×2 matrices SL...
(in the sense of a point added for compactification
Compactification (mathematics)
In mathematics, compactification is the process or result of making a topological space compact. The methods of compactification are various, but each is a way of controlling points from "going off to infinity" by in some way adding "points at infinity" or preventing such an "escape".-An...
). 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 operator
Hecke operator
In mathematics, in particular in the theory of modular forms, a Hecke operator, studied by , is a certain kind of "averaging" operator that plays a significant role in the structure of vector spaces of modular forms and more general automorphic representations....
s on the space being by scalar multiplication
Scalar multiplication
In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra . In an intuitive geometrical context, scalar multiplication of a real Euclidean vector by a positive real number multiplies the magnitude of the vector without changing its direction...
(Mordell's proof of Ramanujan's identities). Explicitly it is the modular discriminant
- Δ(z, q),
which represents (up to a normalizing constant
Normalizing constant
The concept of a normalizing constant arises in probability theory and a variety of other areas of mathematics.-Definition and examples:In probability theory, a normalizing constant is a constant by which an everywhere non-negative function must be multiplied so the area under its graph is 1, e.g.,...
) the discriminant
Discriminant
In algebra, the discriminant of a polynomial is an expression which gives information about the nature of the polynomial's roots. For example, the discriminant of the quadratic polynomialax^2+bx+c\,is\Delta = \,b^2-4ac....
of the cubic on the right side of the Weierstrass equation of an elliptic curve
Elliptic curve
In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...
; and the 24-th power of the Dedekind eta function
Dedekind eta function
The Dedekind eta function, named after Richard Dedekind, is a function defined on the upper half-plane of complex numbers, where the imaginary part is positive...
. 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 form
Automorphic form
In mathematics, the general notion of automorphic form is the extension to analytic functions, perhaps of several complex variables, of the theory of modular forms...
s, the cusp forms are complementary to Eisenstein series
Eisenstein series
Eisenstein series, named after German mathematician Gotthold Eisenstein, are particular modular forms with infinite series expansions that may be written down directly...
, in a discrete spectrum/continuous spectrum, or discrete series representation/induced representation distinction typical in different parts of spectral theory
Spectral theory
In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. It is a result of studies of linear algebra and the solutions of...
. 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 representation
Cuspidal representation
In number theory, cuspidal representations are certain representations of algebraic groups that occur discretely in L^2 spaces. The term cuspidal is derived, at a certain distance, from the cusp forms of classical modular form theory...
s.