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Concept
Siegel modular form
Summary
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 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 and define
the Siegel upper half-space. Define the symplectic group of level , denoted by as
where is the identity matrix. Finally, let
be a rational representation, where is a finite-dimensional complex vector space.
Given
and
define the notation
Then a holomorphic function
is a Siegel modular form of degree (sometimes called the genus), weight , and level if
for all .
In the case that , we further require that be holomorphic 'at infinity'. This assumption is not necessary for due to the Koecher principle, explained below. Denote the space of weight , degree , and level Siegel modular forms by
Some methods for constructing Siegel modular forms include:
Eisenstein series
Theta functions of lattices (possibly with a pluri-harmonic polynomial)
Saito–Kurokawa lift for degree 2
Ikeda lift
Miyawaki lift
Products of Siegel modular forms.
Official source
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