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| In [[mathematics]], '''Siegel modular forms''' are a major type of [[automorphic form]]. These stand in relation to the conventional ''elliptic'' [[modular form]]s as [[abelian varieties]] do in relation to [[elliptic curve]]s; 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 group]]s.
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| The modular forms of the theory are [[holomorphic function]]s on the set of [[symmetric matrix|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 function]]s of [[several complex variables]].
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| Siegel modular forms were first investigated by [[Carl Ludwig Siegel]] in the 1930s for the purpose of studying [[quadratic form]]s 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]].
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| ==Definition==
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| ===Preliminaries===
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| Let <math>g, N \in \mathbb{N}</math> and define
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| :<math>\mathcal{H}_g=\left\{\tau \in M_{g \times g}(\mathbb{C}) \ \big| \ \tau^{\top}=\tau, \textrm{Im}(\tau) \text{ positive definite} \right\},</math> the [[Siegel upper half-space]]. Define the [[symplectic group]] of level <math>N</math>, denoted by
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| :<math>\Gamma_g(N),</math>
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| as
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| :<math>\Gamma_g(N)=\left\{ \gamma \in GL_{2g}(\mathbb{Z}) \ \big| \ \gamma^{\top} \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\},</math>
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| where <math>I_g</math> is the <math>g \times g</math> [[identity matrix]]. Finally, let
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| :<math>\rho:\textrm{GL}(g,\mathbb{C}) \rightarrow \textrm{GL}(V)</math>
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| be a [[rational representation]], where <math>V</math> is a finite-dimensional complex [[vector space]].
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| ===Siegel modular form===
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| Given
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| :<math>\gamma=\begin{pmatrix} A & B \\ C & D \end{pmatrix}</math> | |
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| and
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| :<math>\gamma \in \Gamma_g(N),</math>
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| define the notation
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| :<math>(f\big|\gamma)(\tau)=(\rho(C\tau+D))^{-1}f(\gamma\tau).</math>
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| Then a [[holomorphic]] function
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| :<math>f:\mathcal{H}_g \rightarrow V</math>
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| is a ''Siegel modular form'' of degree <math>g</math>, weight <math>\rho</math>, and level <math>N</math> if
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| :<math>(f\big|\gamma)=f.</math>
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| In the case that <math>g=1</math>, we further require that <math>f</math> be holomorphic 'at infinity'. This assumption is not necessary for <math>g>1</math> due to the Koecher principle, explained below. Denote the space of weight <math>\rho</math>, degree <math>g</math>, and level <math>N</math> Siegel modular forms by
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| :<math>M_{\rho}(\Gamma_g(N)).</math>
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| ==Koecher principle==
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| The theorem known as the ''Koecher principle'' states that if <math>f</math> is a Siegel modular form of weight <math>\rho</math>, level 1, and degree <math>g>1</math>, then <math>f</math> is bounded on subsets of <math>\mathcal{H}_g</math> of the form
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| :<math>\left\{\tau \in \mathcal{H}_g \ | \textrm{Im}(\tau) > \epsilon I_g \right\},</math>
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| where <math>\epsilon>0</math>. Corollary to this theorem is the fact that Siegel modular forms of degree <math>g>1</math> have [[Fourier expansion]]s and are thus holomorphic at infinity.<ref>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 form]]s was apparently known earlier, after Fritz Gotzky, ''Uber eine zahlentheoretische Anwendung von Modulfunktionen zweier Veranderlicher'', Math. Ann. 100 (1928), pp. 411-37</ref>
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| ==References==
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| *Helmut Klingen. ''Introductory Lectures on Siegel Modular Forms'', Cambridge University Press (May 21, 2003), ISBN 0-521-35052-2
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| ==Notes==
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| <references/>
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| ==External links==
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| *[http://www.citebase.org/fulltext?format=application/pdf&identifier=oai:arXiv.org:math/0605346 Gerard van der Geer, lecture notes on Siegel modular forms (PDF)]
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| [[Category:Modular forms]]
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