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  "body": "\n## Definition\n\nLet $N \\ge 1$ be an integer.\nLet $X_0(N)$ be the modular curve of level $N$.\nIts non-cuspidal points over $\\C$ correspond to isomorphism classes of pairs $(E, C)$.\nHere $E$ is an [[elliptic curve|427N6S6QQW]] over $\\C$ and $C \\subset E$ is a cyclic subgroup of order $N$.\nLet $K$ be an imaginary [[quadratic field|7SV4WA92SM]] with ring of integers $\\mathcal{O}_K$.\nA *Heegner point* of $X_0(N)$ for $K$ is a point given by a pair $(E, C)$ such that $\\End(E) \\cong \\End(E/C) \\cong \\mathcal{O}_K$.\nSo both $E$ and $E/C$ have [[complex multiplication|D3HEJTVY7G]] by $\\mathcal{O}_K$.\n\nLet $E$ be an elliptic curve over $\\Q$ of conductor $N$ with a modular parametrisation $\\varphi \\colon X_0(N) \\to E$.\nThe images $\\varphi(x)$ of Heegner points $x$ are the Heegner points of $E$.\n\n## Remarks\n\nHeegner points for $K$ exist exactly when $\\mathcal{O}_K$ has an ideal $\\mathfrak{n}$ with $\\mathcal{O}_K/\\mathfrak{n} \\cong \\Z/N\\Z$; then $(\\C/\\mathcal{O}_K, \\mathfrak{n}^{-1}/\\mathcal{O}_K)$ is one.\nThis holds, for example, under the Heegner hypothesis that every prime dividing $N$ splits in $K$.\nBy the theory of complex multiplication, Heegner points are defined over the [[Hilbert class field|WWCJ7CR2R3]] $H$ of $K$, and the trace $y_K = \\sum_{\\sigma \\in \\Gal(H/K)} \\varphi(x)^\\sigma$ lies in $E(K)$.\n\nExample: for $N = 1$, $X_0(1)$ is the $j$-line and the condition is only that $E$ has CM by $\\mathcal{O}_K$.\nFor $K = \\Q(i)$, the curve $\\C/\\Z[i]$ has endomorphism ring $\\Z[i] = \\mathcal{O}_K$, so the point $j = 1728$ is a Heegner point for $K$.\n\nNon-example: the curve $\\C/(\\Z + 2i\\Z)$ has endomorphism ring $\\Z[2i]$, an order of conductor $2$ in $\\Q(i)$.\nThis is not the maximal order $\\Z[i]$, so its point $j(2i)$ of $X_0(1)$ is not a Heegner point for $\\Q(i)$ in this sense.\n\nSome authors also allow pairs with CM by a non-maximal order of $K$, and speak of Heegner points of conductor $c$.\n",
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