---
tag: NHS23SW780
kind: definition
name: Heegner point
msc: [11G05, 11G15, 11G18]
defines: 2230ZKE4W9
uses: [427N6S6QQW, D3HEJTVY7G, 7SV4WA92SM, WWCJ7CR2R3]
sources:
  - url: https://en.wikipedia.org/wiki/Heegner_point
---

## Definition

Let $N \ge 1$ be an integer.
Let $X_0(N)$ be the modular curve of level $N$.
Its non-cuspidal points over $\C$ correspond to isomorphism classes of pairs $(E, C)$.
Here $E$ is an [[elliptic curve|427N6S6QQW]] over $\C$ and $C \subset E$ is a cyclic subgroup of order $N$.
Let $K$ be an imaginary [[quadratic field|7SV4WA92SM]] with ring of integers $\mathcal{O}_K$.
A *Heegner point* of $X_0(N)$ for $K$ is a point given by a pair $(E, C)$ such that $\End(E) \cong \End(E/C) \cong \mathcal{O}_K$.
So both $E$ and $E/C$ have [[complex multiplication|D3HEJTVY7G]] by $\mathcal{O}_K$.

Let $E$ be an elliptic curve over $\Q$ of conductor $N$ with a modular parametrisation $\varphi \colon X_0(N) \to E$.
The images $\varphi(x)$ of Heegner points $x$ are the Heegner points of $E$.

## Remarks

Heegner 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.
This holds, for example, under the Heegner hypothesis that every prime dividing $N$ splits in $K$.
By 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 $E(K)$.

Example: for $N = 1$, $X_0(1)$ is the $j$-line and the condition is only that $E$ has CM by $\mathcal{O}_K$.
For $K = \Q(i)$, the curve $\C/\Z[i]$ has endomorphism ring $\Z[i] = \mathcal{O}_K$, so the point $j = 1728$ is a Heegner point for $K$.

Non-example: the curve $\C/(\Z + 2i\Z)$ has endomorphism ring $\Z[2i]$, an order of conductor $2$ in $\Q(i)$.
This 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.

Some authors also allow pairs with CM by a non-maximal order of $K$, and speak of Heegner points of conductor $c$.