Siegel modular form

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 (with some extra level structure) should be and are constructed as quotients of the Siegel upper half-space rather than the upper half-plane by discrete groups.

Siegel modular forms are holomorphic functions on the set of symmetric n × n matrices with positive definite imaginary part; the forms must satisfy an automorphy condition. Siegel modular forms can be thought of as multivariable modular forms, i.e. as special functions of several complex variables.

Siegel modular forms were first investigated by Carl Ludwig Siegel (1939) for the purpose of studying quadratic forms analytically. These primarily arise in various branches of number theory, such as arithmetic geometry and elliptic cohomology. Siegel modular forms have also been used in some areas of physics, such as conformal field theory and black hole thermodynamics in string theory.

Let ${\displaystyle g,N\in \mathbb {N} }$ and define

the Siegel upper half-space. Define the symplectic group of level ${\displaystyle N}$, denoted by ${\displaystyle \Gamma _{g}(N),}$ as

where ${\displaystyle I_{g}}$ is the ${\displaystyle g\times g}$ identity matrix. Finally, let

be a rational representation, where ${\displaystyle V}$ is a finite-dimensional complex vector space.

Given