The Gross-Zagier formula relates the first derivative at the central point 
of a Rankin L-series to the canonical
height of a Heegner point. For a normalized
weight-2 modular newform 
, a suitable character 
of the class group of an
imaginary quadratic field, and the corresponding
Heegner point 
,
it has the schematic form
 |
where 
is an explicit positive factor depending on the normalizations and 
is the canonical height.
Consequently, a nonzero derivative implies that 
has infinite order. Together with Kolyvagin's work, the
Gross-Zagier formula shows that a modular elliptic
curve of analytic rank one has Mordell-Weil rank one and a finite Tate-Shafarevich
group (Gross and Zagier 1986, Kolyvagin 1989).
-Series."
Invent. Math. 84, 225-320, 1986.