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 by the ring of integers of one imaginary quadratic field $K$. Such points are defined over the Hilbert class field of $K$. Through the modularity theorem, they give points on elliptic curves over $\Q$. The Gross–Zagier theorem relates their heights to derivatives of $L$-functions.