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  "body": "\n## Statement\n\nThe following is the global reciprocity conjecture for $\\mathrm{GL}_n$, one central part of the program.\nLet $F$ be an [[algebraic number field|MWBDFS2KJ5]], let $G_F$ be its absolute Galois group, let $n \\ge 1$, and let $\\ell$ be a prime.\nFix a field isomorphism $\\iota \\colon \\overline{\\Q}_\\ell \\to \\C$.\nLet $\\rho \\colon G_F \\to \\mathrm{GL}_n(\\overline{\\Q}_\\ell)$ be an irreducible [[Galois representation|JQEYEQJY2G]].\nAssume that $\\rho$ is unramified at all but finitely many [[places|BT5V3TSNY6]] of $F$ and de Rham at the places above $\\ell$.\nThen there is a cuspidal automorphic representation $\\pi$ of $\\mathrm{GL}_n(\\mathbb{A}_F)$, where $\\mathbb{A}_F$ is the [[adele ring|SEH68BA1QQ]] of $F$, with the following property.\nFor all but finitely many finite places $v$ of $F$, both $\\rho$ and $\\pi_v$ are unramified at $v$.\nLet $\\mathrm{Frob}_v$ be a [[Frobenius element|H5145REFMW]] at $v$.\nThen the Satake parameters of $\\pi_v$ are, up to the standard normalising twist, the images under $\\iota$ of the eigenvalues of $\\rho(\\mathrm{Frob}_v)$.\n\n## Remarks\n\nThe Langlands program is a web of conjectures proposed by Robert Langlands from 1967 on.\nIt relates Galois representations, [[automorphic forms|3C73HND6XA]] and representations of reductive groups over local and global fields.\nBesides reciprocity, its main parts are the principle of functoriality, which predicts transfers of automorphic representations between different reductive groups, and the local Langlands correspondence.\nFor $n = 1$, reciprocity is a reformulation of [[class field theory|BEKCEYHCHX]] in terms of [[Hecke characters|R65XKHN4D1]].\nThe [[modularity theorem|CS1NVJBZMH]] for elliptic curves over $\\Q$ and [[Serre's modularity conjecture|RWXXNZNQYJ]] are proved cases for $n = 2$ and $F = \\Q$.\nThe local Langlands correspondence for $\\mathrm{GL}_n$ over a $p$-adic field was proved by Harris and Taylor and, independently, by Henniart.\nIn general, both reciprocity and functoriality remain open.\n\n## Explanations\n\n### L1 human\n\nThe Langlands program predicts a dictionary between two worlds.\nOn one side are symmetries of the solutions of polynomial equations, recorded by Galois groups and their representations.\nOn the other side are highly symmetric functions from analysis, the automorphic forms.\nThe dictionary says that the counts of solutions modulo primes, on the first side, match the coefficients of functions on the second side.\nProved pieces of this dictionary, such as the modularity of elliptic curves, led to the proof of Fermat's Last Theorem.\n",
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