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  "body": "\n## Statement\n\nLet $E$ be an [[elliptic curve|427N6S6QQW]] over $\\Q$ with conductor $N$, a positive integer whose prime factors are the primes where $E$ has bad reduction.\nFor each prime $p \\nmid N$, let $\\tilde{E}$ be the reduction modulo $p$ of a minimal Weierstrass model of $E$.\nIt is a curve over the [[finite field|0ZXZ82FTX2]] $\\F_p$; let $a_p = p + 1 - \\# \\tilde{E}(\\F_p)$.\nThen there is a [[cusp form|A50Y012RBP]] $f(\\tau) = \\sum_{n \\ge 1} b_n q^n$, with $q = e^{2\\pi i \\tau}$, of weight $2$ for the [[congruence subgroup|1H6R856P3C]] $\\Gamma_0(N)$, such that:\n\n1. $f$ is a normalised newform: $b_1 = 1$ and $f$ is an eigenform of all [[Hecke operators|S4SV3QZ8B1]] that does not come from a lower level;\n2. $b_n \\in \\Z$ for all $n$;\n3. $b_p = a_p$ for every prime $p \\nmid N$.\n\n## Remarks\n\nEquivalently, the $L$-function of $E$ equals the $L$-function of $f$, or there is a nonconstant morphism $X_0(N) \\to E$ defined over $\\Q$ from the modular curve $X_0(N)$.\nWiles, with a key step completed jointly with Taylor, proved the theorem for semistable curves in 1995; Breuil, Conrad, Diamond and Taylor completed the general case in 2001.\nThe proof shows that the $\\ell$-adic [[Galois representation|JQEYEQJY2G]] of $E$ is modular.\nIt lifts the modularity of the reduction modulo $3$, known by the Langlands–Tunnell theorem, with a modularity lifting theorem.\nA switch to the prime $5$ handles the cases where the reduction modulo $3$ is not irreducible.\nExample: $y^2 + y = x^3 - x^2$ has conductor $11$, and it corresponds to $f = q \\prod_{n \\ge 1} (1 - q^n)^2 (1 - q^{11n})^2$.\nThe semistable case, combined with the earlier work of Frey, Serre and Ribet, implies Fermat's Last Theorem.\n\n## Explanations\n\n### L1 human\n\nCounting the points of an elliptic curve modulo each prime produces a list of numbers.\nThe theorem says that this list always appears as the coefficients of a very symmetric function, a modular form.\nIt was known as the Taniyama–Shimura–Weil conjecture before its proof.\n\n### L2 human\n\nModularity gives the analytic continuation and functional equation of $L(E, s)$, which are needed even to state the [[Birch and Swinnerton-Dyer conjecture|ZWEFBG40H9]].\nThe parametrisation $X_0(N) \\to E$ lets one construct [[Heegner points|2230ZKE4W9]] on $E$, the input of the [[Gross–Zagier theorem|BX558DTV2D]].\nIt is a case of the reciprocity conjectures of the [[Langlands program|J9T2M084W5]].\n",
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