Heegner Point

Let be the compactification of the quotient of the upper half-plane by the modular group Gamma₀. A Heegner point of discriminant on is the class of a quadratic irrational number satisfying a primitive equation

When is a fundamental discriminant coprime to , the usual Heegner hypothesis that every prime dividing splits in the imaginary quadratic field ensures the existence of such points.

The points are defined over the ring class field of the quadratic order of discriminant ; for fundamental , this is the Hilbert class field. Under a modular parametrization of an elliptic curve, their images and suitable Galois traces give Heegner points on . The Gross-Zagier formula relates their canonical heights to derivatives of Rankin L-series (Gross and Zagier 1986).