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  "body": "\n## Summary\n\nAn automorphic form is a function on a Lie group $G$ that is invariant under a discrete subgroup $\\Gamma$.\nIt must also satisfy strong finiteness and growth conditions.\n[[Modular forms|GWKN5VG78A]] and [[Maass wave forms|3R959YSNYP]] give the basic examples, for the group $\\mathrm{SL}_2(\\R)$ and $\\Gamma = \\mathrm{SL}_2(\\Z)$ or a congruence subgroup.\nAutomorphic forms carry deep arithmetic information, and they are one side of the correspondences predicted by the [[Langlands program|J9T2M084W5]].\n\n## Explanations\n\n### L1 human\n\nA periodic function on the line repeats under the group of integer shifts.\nAn automorphic form is the same idea for a much richer group of symmetries.\nAn example is the group of real $2 \\times 2$ matrices of determinant one, with its subgroup of integer matrices.\nBecause so many symmetries are imposed, only special functions survive, and the space of such functions with given extra conditions is finite-dimensional.\nFor example, the discriminant function $\\Delta(z) = q \\prod_{n \\ge 1} (1 - q^n)^{24}$ with $q = e^{2\\pi i z}$ is a modular form, and so gives an automorphic form on $\\mathrm{SL}_2(\\R)$.\nIts coefficients encode facts about primes, which is why number theorists care about these functions.\n",
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