nLab Siegel modular form


Siegel modular forms are one generalization of modular forms to functions in more than one complex variable (another such generalization are Hilbert modular forms).

Where ordinary modular forms are locally functions on the moduli stack of elliptic curves over the complex numbers, so Siegel modular forms are locally functions on something like the moduli stack of higher-dimensional complex abelian varieties.


Let g\mathcal{H}_{g} be the Siegel upper half-space of degree gg and let Sp 2g()\Sp_{2g}(\mathbb{Z}) the subgroup of elements in the symplectic group Sp 2g()\Sp_{2g}(\mathbb{R}) with integral entries. Let VV be a finite-dimensional vector space over \mathbb{C} and fix a representation

ρ:GL g()GL(V).\rho:\GL_{g}(\mathbb{C})\to \GL(V).

A Siegel modular form of weight ρ\rho is a holomorphic function f: gVf:\mathcal{H}_{g}\to V such that


for all τ=(A B C D)Sp 2g()\tau=\begin{pmatrix}A & B\\C & D\end{pmatrix}\in\Sp_{2g}(\mathbb{Z}) and all τ g\tau\in\mathcal{H}_{g}. If g=1g=1, we additionally require that ff is holomorphic at \infty. A classical Siegel modular form of weight kk is the special case when the representation ρ\rho is given by taking kk-th powers of the determinant. Further specializing to g=1g=1, this gives classical (also known as elliptic) modular forms.


Gerard van der Greer, Siegel Modular Forms, arxiv:0605346

Last revised on November 23, 2022 at 04:47:21. See the history of this page for a list of all contributions to it.