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  "body": "\n## Summary\n\nA Heegner point is a special point on the modular curve $X_0(N)$.\nIt corresponds to a pair of elliptic curves with [[complex multiplication|D3HEJTVY7G]] by the ring of integers of one imaginary [[quadratic field|7SV4WA92SM]] $K$.\nSuch points are defined over the [[Hilbert class field|WWCJ7CR2R3]] of $K$.\nThrough the [[modularity theorem|CS1NVJBZMH]], they give points on [[elliptic curves|427N6S6QQW]] over $\\Q$.\nThe [[Gross–Zagier theorem|BX558DTV2D]] relates their heights to derivatives of $L$-functions.\n\n## Explanations\n\n### L1 human\n\nRational points of infinite order on an elliptic curve are hard to find, and there is in general no formula for them.\nHeegner points are the main general construction that produces such points.\nThe idea is to start from very symmetric elliptic curves, those with complex multiplication, whose coordinates are algebraic numbers with known fields of definition.\nMapping these special points to a given elliptic curve and summing over their conjugates gives a point on that curve defined over the quadratic field $K$.\nHeegner used such points in his solution of the class number one problem.\n",
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