Editing Modular form (section)

== Generalizations ==
There are a number of other usages of the term "modular function", apart from this classical one; for example, in the theory of [[Haar measure]]s, it is a function {{math|Δ(''g'')}} determined by the conjugation action.

[[Maass forms]] are [[Analytic function|real-analytic]] [[eigenfunction]]s of the [[Laplacian]] but need not be [[Holomorphic function|holomorphic]]. The holomorphic parts of certain weak Maass wave forms turn out to be essentially Ramanujan's [[mock theta function]]s. Groups which are not subgroups of {{math|SL(2, '''Z''')}} can be considered.

[[Hilbert modular form]]s are functions in ''n'' variables, each a complex number in the upper half-plane, satisfying a modular relation for 2&times;2 matrices with entries in a [[totally real number field]].

[[Siegel modular form]]s are associated to larger [[symplectic group]]s in the same way in which classical modular forms are associated to {{math|SL(2, '''R''')}}; in other words, they are related to [[abelian variety|abelian varieties]] in the same sense that classical modular forms (which are sometimes called ''elliptic modular forms'' to emphasize the point) are related to elliptic curves.

[[Jacobi form]]s are a mixture of modular forms and elliptic functions. Examples of such functions are very classical - the Jacobi theta functions and the Fourier coefficients of Siegel modular forms of genus two - but it is a relatively recent observation that the Jacobi forms have an arithmetic theory very analogous to the usual theory of modular forms.

[[Automorphic form]]s extend the notion of modular forms to general [[Lie group]]s.

Modular integrals of weight {{mvar|k}} are meromorphic functions on the upper half plane of moderate growth at infinity which ''fail to be modular of weight {{mvar|k}}'' by a rational function.

[[Automorphic factor]]s are functions of the form <math>\varepsilon(a,b,c,d) (cz+d)^k</math> which are used to generalise the modularity relation defining modular forms, so that
:<math>f\left(\frac{az+b}{cz+d}\right) = \varepsilon(a,b,c,d) (cz+d)^k f(z).</math>

The function <math>\varepsilon(a,b,c,d)</math> is called the nebentypus of the modular form. Functions such as the [[Dedekind eta function]], a modular form of weight 1/2, may be encompassed by the theory by allowing automorphic factors.