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  "body": "\n## Summary\n\nA Siegel modular form is a holomorphic function on the [[Siegel upper half-space|5PTYE97PF6]].\nUnder the Siegel modular group $Sp(2g, \\Z)$ it transforms by a power of a determinant factor.\nFor $g = 1$ these are the classical [[modular forms|GWKN5VG78A]].\nSiegel modular forms are sections of line bundles on the moduli space of principally polarized abelian varieties.\nTheir Fourier coefficients often encode arithmetic data, such as numbers of representations of one quadratic form by another.\n\n## Explanations\n\n### L1 human\n\nA classical modular form is a function on the upper half-plane that changes in a prescribed way under the substitutions $z \\mapsto (az + b)/(cz + d)$.\nA Siegel modular form does the same in several variables: its argument is a symmetric matrix with positive definite imaginary part, and the substitutions are given by symplectic integer matrices.\nTheta series of quadratic forms are typical examples.\nAs in one variable, the space of forms of a fixed weight is finite-dimensional.\n",
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