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  "body": "\n## Statement\n\nLet $E$ be an [[elliptic curve|427N6S6QQW]] over $\\Q$ of conductor $N$, with modular parametrisation $\\varphi \\colon X_0(N) \\to E$ given by the [[modularity theorem|CS1NVJBZMH]].\nLet $K$ be an imaginary [[quadratic field|7SV4WA92SM]] of discriminant $D$ such that every prime dividing $N$ splits in $K$.\nLet $x$ be a [[Heegner point|2230ZKE4W9]] for $K$; it is defined over the [[Hilbert class field|WWCJ7CR2R3]] $H$ of $K$.\nLet $y_K \\in E(K)$ be the trace of $\\varphi(x)$ from $H$ to $K$.\nLet $L(E/K, s) = L(E, s) \\, L(E^D, s)$, where $E^D$ is the quadratic twist of $E$ by $K$, and let $\\hat{h}$ be the Néron–Tate canonical [[height|F1FF505PWH]] on $E(K)$.\nThen $L(E/K, 1) = 0$, and there is a real constant $c \u003e 0$, depending on $E$ and $K$ and given explicitly by periods and other invariants, such that\n$$L'(E/K, 1) = c \\cdot \\hat{h}(y_K).$$\n\n## Remarks\n\nIn particular, $y_K$ has infinite order if and only if $L'(E/K, 1) \\ne 0$, since $\\hat{h}$ vanishes exactly on torsion points.\nThe Heegner hypothesis forces the sign of the functional equation of $L(E/K, s)$ to be $-1$, which gives $L(E/K, 1) = 0$.\nThe proof computes both sides.\nThe derivative is computed by the Rankin–Selberg method, and the height by local intersection theory of Heegner points on $X_0(N)$.\nGross and Zagier then compare the two as Fourier coefficients of modular forms.\nGross and Zagier proved it in 1986, with some additional conditions on $D$ that later work removed.\nKolyvagin's Euler system argument adds the following consequence.\nIf $L(E, s)$ has a zero of order at most $1$ at $s = 1$, then the rank of $E(\\Q)$ equals that order and the [[Tate–Shafarevich group|BK9WBMF56G]] of $E$ is finite.\n\n## Explanations\n\n### L1 human\n\nThe theorem gives an exact formula linking two very different numbers.\nOne is the slope of the $L$-function of $E$ at the central point, an analytic quantity.\nThe other is the height of a specific point built from complex multiplication, which measures how complicated the point's coordinates are.\nSo a nonzero slope guarantees an explicit point of infinite order.\n\n### L2 human\n\nThis is the strongest known evidence for the [[Birch and Swinnerton-Dyer conjecture|ZWEFBG40H9]], covering analytic rank $0$ and $1$.\nGross and Zagier also used it, with Goldfeld's earlier work, to give an effective lower bound for class numbers of imaginary quadratic fields.\nGeneralisations to Shimura curves and higher-dimensional settings are an active area.\n",
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