---
tag: 2230ZKE4W9
kind: concept
name: Heegner point
msc: [11G05, 11G15, 11G18]
sources:
  - url: https://en.wikipedia.org/wiki/Heegner_point
same_as:
  wikidata: Q5697927
  wikipedia: Heegner point
---

## Summary

A Heegner point is a special point on the modular curve $X_0(N)$.
It corresponds to a pair of elliptic curves with [[complex multiplication|D3HEJTVY7G]] by the ring of integers of one imaginary [[quadratic field|7SV4WA92SM]] $K$.
Such points are defined over the [[Hilbert class field|WWCJ7CR2R3]] of $K$.
Through the [[modularity theorem|CS1NVJBZMH]], they give points on [[elliptic curves|427N6S6QQW]] over $\Q$.
The [[Gross–Zagier theorem|BX558DTV2D]] relates their heights to derivatives of $L$-functions.

## Explanations

### L1 human

Rational points of infinite order on an elliptic curve are hard to find, and there is in general no formula for them.
Heegner points are the main general construction that produces such points.
The idea is to start from very symmetric elliptic curves, those with complex multiplication, whose coordinates are algebraic numbers with known fields of definition.
Mapping these special points to a given elliptic curve and summing over their conjugates gives a point on that curve defined over the quadratic field $K$.
Heegner used such points in his solution of the class number one problem.