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. A modular function is a function that, like a modular form, is invariant with respect to the modular group...

In mathematics, Siegel modular forms are a major type of automorphic form. These stand in relation to the conventional elliptic modular forms as abelian varieties do in relation to elliptic curves; the complex manifolds constructed as in the theory are basic models for what a moduli space for abelian varieties (with some extra level structure) should be, as quotients of the Siegel upper half-space rather than the upper half-plane by discrete groups. The modular forms of the theory are holomorphic functions on the set of symmetric n × n matrices with positive definite imaginary part; the forms must...