Siegel modular form

Siegel's modular forms are generalizations of modular forms in several complex variables and are examples of automorphic forms .

They are defined on Siegel's half-space , the space of complex symmetrical matrices with a positive definite imaginary part. Siegel's modular forms are holomorphic functions on Siegel's half-space that meet an automorphism condition. ${\displaystyle {\mathcal {H}}_{g}}$ ${\displaystyle g\times g}$

They are related to Abelian varieties in a similar way as elliptical modular forms to elliptical curves. They were originally introduced by Carl Ludwig Siegel in 1935 as part of his analytical theory of quadratic forms and are used in number theory.

There are Siegel modular forms, which are constructed in the same way as Eisenstein for modular forms, and those that are theta functions of square forms. The theory was based as closely as possible on that of the elliptical module forms.

definition

Be

${\displaystyle {\begin{aligned}\Gamma _{g}&=Sp_{2g}(\mathbb {Z} )=\left\{\gamma \in GL_{2g}(\mathbb {Z} )\ {\big |}\ \gamma ^{\mathrm {T} }{\begin{pmatrix}0&I_{g}\\-I_{g}&0\end{pmatrix}}\gamma ={\begin{pmatrix}0&I_{g}\\-I_{g}&0\end{pmatrix}}\right\}\\&=\left\{{\begin{pmatrix}A&B\\C&D\end{pmatrix}}\in GL_{2g}(\mathbb {Z} ):A^{T}C=C^{T}A\,,B^{T}D=D^{T}B\,,A^{T}D-C^{T}B=I_{g}\,\right\}\end{aligned}}}$

the group of symplectic matrices with values ​​in whole numbers (Siegel module group). Where is the unit matrix. Examples are the matrices with a symmetrical matrix . ${\displaystyle I_{g}}$ ${\displaystyle g\times g}$ ${\displaystyle {\begin{pmatrix}I_{g}&S\\0&I_{g}\end{pmatrix}}}$ ${\displaystyle S=S^{T}}$

The group operates over the Siegel hemisphere

${\displaystyle Z\to (AZ+B){(CZ+D)}^{-1}}$.

A Siegel modular form is a holomorphic in the Siegel half-space function with ${\displaystyle f}$

${\displaystyle f((AZ+B){(CZ+D)}^{-1})={(\det \,(CZ+D))}^{k}f(Z)}$.

${\displaystyle g}$is the degree (sometimes gender), the weight. ${\displaystyle k}$

In addition, it is required that the modular form is limited in the Siegel half-space (for this follows from the so-called Koecher principle). ${\displaystyle g>1}$

The following applies:

${\displaystyle f(Z+S)=f(Z)}$for all integral symmetric matrices ${\displaystyle g\times g}$ ${\displaystyle S=S^{T}}$

${\displaystyle f(UZU^{T})={(\det \,U)}^{k}f(Z)}$ for all ${\displaystyle U\in GL_{g}(\mathbb {Z} )}$

${\displaystyle f(-Z^{-1})={(\det \,Z)}^{k}f(Z)}$

This provides the transformation behavior among the generators of the Siegel module group . ${\displaystyle Sp_{2g}(\mathbb {Z} )}$

It can be shown that Siegel modular forms have a Fourier expansion.

${\displaystyle f(Z)=\sum _{T}a(T)e^{\pi iSp(TZ)}}$

with symmetric ( ) positive semidefinite matrices T (short:) . ${\displaystyle T=T^{T}}$ ${\displaystyle T\geq 0}$

In arithmetic applications, instead of the symplectic group , a congruence subgroup is used (with a natural number , the level): ${\displaystyle \Gamma _{g}=Sp_{2g}(\mathbb {Z} )}$ ${\displaystyle \Gamma _{g}(N)}$ ${\displaystyle N}$

${\displaystyle \Gamma _{g}(N)=\left\{\gamma \in \Gamma _{g}:\gamma \equiv I_{2g}\mod N\right\},}$

Note: There is also an extended definition in which the Siegel module form is vector-valued (the Siegel module form defined above is then called scalar-valued).

A rational representation is used for the definition of the weight

${\displaystyle \rho :{\textrm {GL}}_{g}(\mathbb {C} )\rightarrow {\textrm {GL}}(V)}$

is used in a complex vector space . With the definition ${\displaystyle V}$

${\displaystyle (f{\big |}\gamma )(Z)=(\rho (CZ+D))^{-1}f(\gamma Z).}$

is the holomorphic function

${\displaystyle f:{\mathcal {H}}_{g}\rightarrow V}$

a Siegel modular form of degree , if ${\displaystyle g}$

${\displaystyle (f{\big |}\gamma )=f}$

for everyone . ${\displaystyle \gamma \in \Gamma _{g}}$

literature

• Eberhard Freitag: Siegel modular forms, Springer 1983
• Eberhard Freitag: Siegelsche Module Functions, Annual Report DMV, Volume 79, 1977, pp. 79-86, pdf
• Helmut Klingen: Introductory Lectures on Siegel Modular Forms, Cambridge University Press 1990

Web links

• Winfried Kohnen, A short course on Siegel modular forms, pdf