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  "body": "\n## Summary\n\nA Galois representation is a continuous [[representation|WYGRXCG2RN]] of the absolute [[Galois group|0XTY86BZKY]] of a field on a finite-dimensional vector space or free module.\nThe field is usually a number field or a local field.\nTypical coefficients are the complex numbers, finite fields and $\\ell$-adic numbers.\nGalois representations arise from the torsion points of [[elliptic curves|427N6S6QQW]], from the cohomology of varieties, and from [[modular forms|GWKN5VG78A]].\nMatching them with automorphic objects is the content of the [[Langlands program|J9T2M084W5]].\n\n## Explanations\n\n### L1 human\n\nThe absolute Galois group of $\\Q$ records all the symmetries of all algebraic numbers at once; it is far too large to study directly.\nA Galois representation looks at this group through matrices: each symmetry becomes an invertible matrix, compatibly with composition.\nFor example, the symmetries act on the roots of unity of order $\\ell^k$ for all $k$, and this gives the cyclotomic character, a $1 \\times 1$ representation.\nThe torsion points of an elliptic curve give a $2 \\times 2$ representation, which encodes how many points the curve has modulo each prime.\n",
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