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  "body": "\n## Definition\n\nLet $g \\ge 1$ and $k$ be integers, and let $\\mathbb{H}_g$ be the [[Siegel upper half-space|5PTYE97PF6]] of degree $g$.\nLet $\\Gamma_g = Sp(2g, \\Z)$ be the group of integer matrices in the [[symplectic group|X12FS2JAP0]] $Sp(2g, \\R)$.\nA *Siegel modular form* of degree $g$ and weight $k$ is a function $f \\colon \\mathbb{H}_g \\to \\C$ such that:\n\n1. $f$ is [[holomorphic|507C34W0ZT]] in the $g(g+1)/2$ entries $Z_{ij}$, $i \\le j$, of $Z$;\n2. $f\\big((AZ + B)(CZ + D)^{-1}\\big) = \\det(CZ + D)^k f(Z)$ for all $Z \\in \\mathbb{H}_g$ and all $\\begin{pmatrix} A \u0026 B \\\\ C \u0026 D \\end{pmatrix} \\in \\Gamma_g$;\n3. if $g = 1$, $f$ is holomorphic at the cusp: $f$ is bounded on $\\set{Z : \\operatorname{Im} Z \\ge 1}$.\n\n## Remarks\n\nFor $g = 1$ this is the definition of a [[modular form|GWKN5VG78A]] of weight $k$ for $SL_2(\\Z)$.\nFor $g \\ge 2$ the Koecher principle shows that conditions 1 and 2 imply boundedness on every set $\\set{Z : \\operatorname{Im} Z \\ge c I_g}$ with $c \u003e 0$, so no condition at the cusp is needed.\nEvery Siegel modular form has a Fourier expansion $f(Z) = \\sum_T a(T) e^{2\\pi i \\operatorname{tr}(TZ)}$, over half-integral symmetric matrices $T \\ge 0$.\nForms of weight $k$ for subgroups of finite index of $\\Gamma_g$, and vector-valued forms, are defined in the same way.\n\nExample: a constant function $f = c$ is a Siegel modular form of weight $0$, since condition 2 reads $c = \\det(CZ + D)^0 c$.\nExample: let $S$ be a positive definite even unimodular integer matrix of size $m$.\nThe theta series $\\theta_S(Z) = \\sum_{X} e^{\\pi i \\operatorname{tr}(X^T S X Z)}$, over all integer $m \\times g$ matrices $X$, is a Siegel modular form of weight $m/2$.\n\nNon-example: the $j$-invariant $j(z)$ is holomorphic on $\\mathbb{H}_1$ and satisfies condition 2 with $k = 0$.\nBut $j(z) = e^{-2\\pi i z} + 744 + \\cdots$ is unbounded on $\\operatorname{Im} z \\ge 1$, so condition 3 fails.\n",
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