Modular Form - Generalizations

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 measures, it is a function Δ(g) determined by the conjugation action.

Maass forms are real-analytic eigenfunctions of the Laplacian but need not be holomorphic. The holomorphic parts of certain weak Maass wave forms turn out to be essentially Ramanujan's mock theta functions. Groups which are not subgroups of SL(2,Z) can be considered.

Hilbert modular forms are functions in n variables, each a complex number in the upper half-plane, satisfying a modular relation for 2×2 matrices with entries in a totally real number field.

Siegel modular forms are associated to larger symplectic groups in the same way in which the forms we have discussed are associated to SL(2,R); in other words, they are related to abelian varieties in the same sense that our forms (which are sometimes called elliptic modular forms to emphasize the point) are related to elliptic curves.

Jacobi forms 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 forms extend the notion of modular forms to general Lie groups.