Siegel modular form

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 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 in the 1930s 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.

Contents

Definition

Preliminaries

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

\mathcal{H}_g=\left\{\tau \in M_{g \times g}(\mathbb{C}) \ \big| \ \tau^{T}=\tau, \textrm{Im}(\tau) \text{ positive definite} \right\},

the Siegel upper half-space. Define the symplectic group of level N, denoted by

\Gamma_g(N),

as

\Gamma_g(N)=\left\{ \gamma \in GL_{2g}(\mathbb{Z}) \ \big| \ \gamma^{T} \begin{pmatrix} 0 & I_g \\ -I_g & 0 \end{pmatrix} \gamma= \begin{pmatrix} 0 & I_g \\ -I_g & 0 \end{pmatrix} , \ \gamma \equiv I_{2g}\mod N\right\},

where I_g is the g \times g identity matrix. Finally, let

\rho:\textrm{GL}(g,\mathbb{C}) \rightarrow \textrm{GL}(V)

be a rational representation, where V is a finite-dimensional complex vector space.

Siegel modular form

Given

\gamma=\begin{pmatrix} A & B \\ C & D \end{pmatrix}

and

\gamma \in \Gamma_g(N),

define the notation

(f\big|\gamma)(\tau)=(\rho(C\tau%2BD))^{-1}f(\gamma\tau).

Then a holomorphic function

f:\mathcal{H}_g \rightarrow V

is a Siegel modular form of degree g, weight \rho, and level N if

(f\big|\gamma)=f.

In the case that g=1, we further require that f be holomorphic 'at infinity'. This assumption is not necessary for g>1 due to the Koecher principle, explained below. Denote the space of weight \rho, degree g, and level N Siegel modular forms by

M_{\rho}(\Gamma_g(N)).

Koecher principle

The theorem known as the Koecher principle states that if f is a Siegel modular form of weight \rho, level 1, and degree g>1, then f is bounded on subsets of \mathcal{H}_g of the form

\left\{\tau \in \mathcal{H}_g \ | \textrm{Im}(\tau) > \epsilon I_g \right\},

where \epsilon>0. Corollary to this theorem is the fact that Siegel modular forms of degree g>1 have Fourier expansions and are thus holomorphic at infinity.[1]

References

Notes

  1. ^ This was proved by Max Koecher, Zur Theorie der Modulformen n-ten Grades I, Mathematische. Zeitschrift 59 (1954), 455–466. A corresponding principle for Hilbert modular forms was apparently known earlier, after Fritz Gotzky, Uber eine zahlentheoretische Anwendung von Modulfunktionen zweier Veranderlicher, Math. Ann. 100 (1928), pp. 411-37

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