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.

A cusp form is distinguished in the case of modular forms for the modular group by the vanishing of the constant coefficient *a*_{0} in the Fourier series expansion (see *q*-expansion)

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 imaginary part of *z* → ∞. Taking the quotient by the modular group, 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 Ramanujan tau function *τ*(*n*) arises as the sequence of Fourier coefficients of the cusp form of weight 12 for the modular group, with *a*_{1} = 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 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 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.

The **modularity theorem** states that elliptic curves over the field of rational numbers are related to modular forms. Andrew Wiles proved the modularity theorem for semistable elliptic curves, which was enough to imply Fermat's Last Theorem. Later, a series of papers by Wiles's former students Brian Conrad, Fred Diamond and Richard Taylor, culminating in a joint paper with Christophe Breuil, extended Wiles's techniques to prove the full modularity theorem in 2001.

In mathematics, a **modular form** is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a 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 with number theory. Modular forms appear in other areas, such as algebraic topology, sphere packing, and string theory.

In mathematics, the **Dedekind eta function**, named after Richard Dedekind, is a modular form of weight 1/2 and is a function defined on the upper half-plane of complex numbers, where the imaginary part is positive. It also occurs in bosonic string theory.

In mathematics, **Weierstrass's elliptic functions** are elliptic functions that take a particularly simple form. They are named for Karl Weierstrass. This class of functions are also referred to as **p-functions** and they are usually denoted by the symbol ℘. They play an important role in theory of elliptic functions. A ℘-function together with its derivative can be used to parameterize elliptic curves and they generate the field of elliptic functions with respect to a given period lattice.

In mathematics, Felix Klein's **j-invariant** or **j function**, regarded as a function of a complex variable τ, is a modular function of weight zero for SL(2, **Z**) defined on the upper half-plane of complex numbers. It is the unique such function which is holomorphic away from a simple pole at the cusp such that

In mathematics, in particular in the theory of modular forms, a **Hecke operator**, studied by Hecke (1937), 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.

In number theory and algebraic geometry, a **modular curve***Y*(Γ) 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(2, **Z**). The term modular curve can also be used to refer to the **compactified modular curves***X*(Γ) which are compactifications obtained by adding finitely many points to this quotient. The points of a modular curve parametrize isomorphism classes of elliptic curves, together with some additional structure depending on the group Γ. This interpretation allows one to give a purely algebraic definition of modular curves, without reference to complex numbers, and, moreover, prove that modular curves are defined either over the field **Q** of rational numbers or a cyclotomic field **Q**(ζ_{n}). The latter fact and its generalizations are of fundamental importance in number theory.

In mathematics, the **Ramanujan conjecture**, due to Srinivasa Ramanujan (1916, p.176), states that Ramanujan's tau function given by the Fourier coefficients *τ*(*n*) of the cusp form Δ(*z*) of weight 12

The **Ramanujan tau function**, studied by Ramanujan (1916), is the function defined by the following identity:

**Eisenstein series**, named after German mathematician Gotthold Eisenstein, are particular modular forms with infinite series expansions that may be written down directly. Originally defined for the modular group, Eisenstein series can be generalized in the theory of automorphic forms.

A **modular elliptic curve** is an elliptic curve *E* that admits a parametrisation *X*_{0}(*N*) → *E* by a modular curve. This is not the same as a modular curve that happens to be an elliptic curve, something that could be called an elliptic modular curve. The modularity theorem, also known as the Taniyama–Shimura conjecture, asserts that every elliptic curve defined over the rational numbers is modular.

In mathematics, a **mock modular form** is the holomorphic part of a harmonic weak Maass form, and a **mock theta function** is essentially a mock modular form of weight 1/2. The first examples of mock theta functions were described by Srinivasa Ramanujan in his last 1920 letter to G. H. Hardy and in his lost notebook. Sander Zwegers discovered that adding certain non-holomorphic functions to them turns them into harmonic weak Maass forms.

In mathematics, **Siegel modular forms** are a major type of automorphic form. These generalize conventional *elliptic* modular forms which are closely related to elliptic curves. The complex manifolds constructed in the theory of Siegel modular forms are Siegel modular varieties, which are basic models for what a moduli space for abelian varieties should be and are constructed as quotients of the Siegel upper half-space rather than the upper half-plane by discrete groups.

In mathematics, a **Jacobi form** is an automorphic form on the Jacobi group, which is the semidirect product of the symplectic group Sp(n;R) and the Heisenberg group . The theory was first systematically studied by Eichler & Zagier (1985).

In mathematics, the **Rankin–Selberg method**, introduced by (Rankin 1939) and Selberg (1940), also known as the theory of integral representations of *L*-functions, is a technique for directly constructing and analytically continuing several important examples of automorphic *L*-functions. Some authors reserve the term for a special type of integral representation, namely those that involve an Eisenstein series. It has been one of the most powerful techniques for studying the Langlands program.

In mathematics, the **Langlands–Shahidi method** provides the means to define automorphic L-functions in many cases that arise with connected reductive groups over a number field. This includes Rankin–Selberg products for cuspidal automorphic representations of general linear groups. The method develops the theory of the **local coefficient**, which links to the global theory via Eisenstein series. The resulting *L*-functions satisfy a number of analytic properties, including an important functional equation.

In mathematics, **almost holomorphic modular forms**, also called **nearly holomorphic modular forms**, are a generalization of modular forms that are polynomials in 1/Im(τ) with coefficients that are holomorphic functions of τ. A **quasimodular form** is the holomorphic part of an almost holomorphic modular form. An almost holomorphic modular form is determined by its holomorphic part, so the operation of taking the holomorphic part gives an isomorphism between the spaces of almost holomorphic modular forms and quasimodular forms. The archetypal examples of quasimodular forms are the Eisenstein series E_{2}(τ) (the holomorphic part of the almost holomorphic modular form E_{2}(τ) – 3/πIm(τ)), and derivatives of modular forms.

In mathematics, a **weakly holomorphic modular form** is similar to a holomorphic modular form, except that it is allowed to have poles at cusps. Examples include modular functions and modular forms.

In mathematics, a **weak Maass form** is a smooth function on the upper half plane, transforming like a modular form under the action of the modular group, being an eigenfunction of the corresponding hyperbolic Laplace operator, and having at most linear exponential growth at the cusps. If the eigenvalue of under the Laplacian is zero, then is called a **harmonic weak Maass form**, or briefly a **harmonic Maass form**.

In mathematics, modular forms are particular complex analytic functions on the upper half-plane of interest in complex analysis and number theory. When reduced modulo a prime *p*, there is an analogous theory to the classical theory of complex modular forms and the *p*-adic theory of modular forms.

- Serre, Jean-Pierre,
*A Course in Arithmetic*, Graduate Texts in Mathematics, No. 7, Springer-Verlag, 1978. ISBN 0-387-90040-3 - Shimura, Goro,
*An Introduction to the Arithmetic Theory of Automorphic Functions*, Princeton University Press, 1994. ISBN 0-691-08092-5 - Gelbart, Stephen,
*Automorphic Forms on Adele Groups*, Annals of Mathematics Studies, No. 83, Princeton University Press, 1975. ISBN 0-691-08156-5

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