Siegel modular form

In mathematics, Siegel modular forms are holomorphic functions on the set of symmetric nxn matrices with positive definite imaginary part which satisfy an automorphy condition. A Siegel modular form can be thought of as a multivariable modular form. They were first investigated by Carl Ludwig Siegel in the 1930s for the purpose of studying quadratic forms analytically. They 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.

Definition

Preliminaries

Let $g,N\in \mathbb {N}$ and define ${\mathcal {H}}_{g}=\left\{\gamma =\in M_{gxg}(\mathbb {C} )\ {\big |}\ \gamma ^{T}=\gamma ,{\textrm {Im}}(\gamma )>0\right\}$, the Siegel upper half space. Define the symplectic group of level $N$, $\Gamma _{g}(N)$ by, $\Gamma _{g}(N)=\left\{\gamma \in GL_{2g}(\mathbb {Z} )\ {\big |}\ \gamma ^{T}{\begin{pmatrix}0&I_{g}\\-I_{g}&0\end{pmatrix}}\gamma ={\begin{pmatrix}0&I_{g}\\-I_{g}&0\end{pmatrix}}\right\}$, where $I_{g}$ is the $gxg$ 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 $(f{\big |}\gamma )(\tau )=(\rho (C\tau +D))^{-1}f(\gamma \tau )$. 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 be holomorphic at infinity. This assumption is not necessary for $g>1$ due to the Kocher principle. We denote the space of of weight $\rho$, degree $g$, and level $N$ Siegel modular forms by $M_{\rho }(\Gamma _{g}(N))$.

Kocher Principle

The theorem known as the Kocher principle states that if $f$ is a Seigel modular form of weight $\rho$, level 1, and degree $g>1$, then $f$ is bounded on subsets of ${\mathcal {H}}_{g}$ of the form $\left\{\tau \in {\mathcal {H}}_{g}\ |{\textrm {Im}}(\tau )>\epsilon I_{g}\right\}$. Corollary to this theorem is the fact that Siegel modular forms of degree $g>1$ have Fourier expansions and are thus holomorphic at infinity.

Reference

• Klingen, Helmut. Introductory Lectures on Siegel Modular Forms, Cambridge University Press (May 21, 2003), ISBN 0-521-35052-2.