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  "body": "\n## Statement\n\nLet $E$ be an [[elliptic curve|427N6S6QQW]] over $\\Q$ with conductor $N$.\nFor a prime $p \\nmid N$, let $a_p = p + 1 - \\#\\tilde{E}(\\F_p)$, where $\\tilde{E}$ is the reduction of $E$ modulo $p$.\nFor $p \\mid N$, let $a_p$ be $1$, $-1$ or $0$ when the reduction of $E$ at $p$ is split multiplicative, non-split multiplicative or additive.\nLet\n$$L(E, s) = \\prod_{p \\nmid N} \\left(1 - a_p p^{-s} + p^{1 - 2s}\\right)^{-1} \\prod_{p \\mid N} \\left(1 - a_p p^{-s}\\right)^{-1},$$\nwhich converges for $\\mathrm{Re}(s) \u003e 3/2$ and extends to an entire function of $s$ by the [[modularity theorem|CS1NVJBZMH]].\nLet $r$ be the rank of the finitely generated abelian group $E(\\Q)$ of [[rational points|P7V1BQJPSS]].\nThen $L(E, s)$ has a zero of order exactly $r$ at $s = 1$.\n\n## Remarks\n\nThe refined form of the conjecture predicts the leading coefficient.\n$$\\lim_{s \\to 1} \\frac{L(E, s)}{(s - 1)^r} = \\frac{\\Omega_E \\cdot \\mathrm{Reg}_E \\cdot \\#\\text{Ш}(E/\\Q) \\cdot \\prod_{p \\mid N} c_p}{(\\# E(\\Q)_{\\mathrm{tors}})^2},$$\nwhere $\\Omega_E$ is the real period, $\\mathrm{Reg}_E$ the regulator of the canonical height pairing, $\\text{Ш}(E/\\Q)$ the [[Tate–Shafarevich group|BK9WBMF56G]] (conjectured to be finite), and $c_p$ the Tamagawa numbers.\nStatus: the conjecture is open; it is one of the Clay Millennium Prize Problems.\nKnown results: if the order of vanishing of $L(E, s)$ at $s = 1$ is $0$ or $1$, then the rank equals it and $\\text{Ш}(E/\\Q)$ is finite.\nThis follows from the [[Gross–Zagier theorem|BX558DTV2D]] together with the work of Kolyvagin.\nCoates and Wiles had earlier proved that if $E$ has [[complex multiplication|D3HEJTVY7G]] and $L(E, 1) \\ne 0$, then $E(\\Q)$ is finite.\nBirch and Swinnerton-Dyer formulated the conjecture in the 1960s after computer experiments on the products $\\prod_{p \\le x} \\#\\tilde{E}(\\F_p)/p$.\n\n## Explanations\n\n### L1 human\n\nThe rank measures how many independent rational points an elliptic curve has, but it is hard to compute.\nThe $L$-function packs together the numbers of points of the curve modulo every prime.\nThe conjecture says that the rank can be read off from the $L$-function: it is the order of its zero at $s = 1$.\nHeuristically, a curve with many rational points tends to have many points modulo $p$, which pushes the Euler product toward zero at $s = 1$.\n",
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      "first_stated": "1965",
      "posed_by": "Bryan Birch and Peter Swinnerton-Dyer"
    },
    "kind": "conjecture",
    "msc": [
      "11G40",
      "11G05",
      "14G10"
    ],
    "name": "Birch and Swinnerton-Dyer conjecture",
    "same_as": {
      "wikidata": "Q527863",
      "wikipedia": "Birch and Swinnerton-Dyer conjecture"
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  "link": "https://math2.tamnd.com/node/ZWEFBG40H9/",
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  "msc": [
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    "14G10"
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  "name": "Birch and Swinnerton-Dyer conjecture",
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/ZWE/ZWEFBG40H9.md",
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  "url": "https://math2.tamnd.com/wiki/birch-and-swinnerton-dyer-conjecture/",
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