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  "body": "\n## Statement\n\nLet $p$ be a prime, and let $\\overline{\\F}_p$ be an algebraic closure of the [[finite field|0ZXZ82FTX2]] $\\F_p$.\nLet $\\bar\\rho \\colon \\Gal(\\overline{\\Q}/\\Q) \\to \\mathrm{GL}_2(\\overline{\\F}_p)$ be a continuous [[Galois representation|JQEYEQJY2G]] that is irreducible and odd, that is, $\\det \\bar\\rho(c) = -1$ for a complex conjugation $c$.\nThen $\\bar\\rho$ is modular, in the following sense.\nThere is a normalised [[cusp form|A50Y012RBP]] $f = \\sum_{n \\ge 1} a_n q^n$ of some weight $k \\ge 2$ and level $N \\ge 1$ that is an eigenform of all [[Hecke operators|S4SV3QZ8B1]].\nLet $\\mathrm{Frob}_\\ell$ denote a [[Frobenius element|H5145REFMW]] at a prime $\\ell$.\nThere is also a ring homomorphism $a \\mapsto \\bar{a}$ from the ring generated by the $a_n$ to $\\overline{\\F}_p$ such that, for every prime $\\ell \\nmid pN$,\n$$\\tr \\bar\\rho(\\mathrm{Frob}_\\ell) = \\bar{a}_\\ell.$$\n\n## Remarks\n\nSerre stated the conjecture in 1987 in a strong form.\nIn it, $N$ is the Artin conductor of $\\bar\\rho$ away from $p$, and $k$ is an explicit weight $k(\\bar\\rho)$ read off from the restriction of $\\bar\\rho$ to inertia at $p$.\nRibet's level-lowering theorem and later work showed that the weak form above implies the strong form.\nKhare and Wintenberger proved the conjecture, with a step by Kisin, in work published in 2009.\nTheir proof is an induction on the prime $p$ and the conductor, using modularity lifting theorems and the existence of compatible families of Galois representations.\nOddness is necessary: for every modular eigenform the attached representation is odd.\nExample: for an elliptic curve $E$ over $\\Q$, the action on the $p$-torsion points $E[p]$ gives an odd $\\bar\\rho$, since $\\det \\bar\\rho$ is the cyclotomic character modulo $p$.\nWhen this $\\bar\\rho$ is irreducible, its modularity also follows from the [[modularity theorem|CS1NVJBZMH]].\n\n## Explanations\n\n### L1 human\n\nSerre's conjecture says that every two-dimensional Galois representation modulo $p$ of the right kind comes from a modular form.\nThe \"right kind\" is just irreducible and odd, two conditions that are easy to check.\nSo a large class of arithmetic objects is matched with analytic ones.\n\n### L2 human\n\nThe strong form gives a route to Fermat's Last Theorem.\nA putative solution would give a representation of level $2$ and weight $2$, and there are no nonzero cusp forms of that level and weight.\nThe theorem is a mod $p$ case of the [[Langlands program|J9T2M084W5]] for $\\mathrm{GL}_2$ over $\\Q$.\nIt also implies that the Artin conjecture holds for odd two-dimensional complex Galois representations of $\\Q$.\n",
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