A Heegner point of discriminant 
on 
is the class of a quadratic surd 
satisfying a primitive equation
 |
where 
,
, 
, and 
. Here 
denotes the compactification
of the quotient of the upper half-plane by the
modular group Gamma0. 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).
-Series."
Invent. Math. 84, 225-320, 1986.