The Gross–Zagier theorem  describes the height of  Heegner points in terms of a derivative of the L-function of the elliptic curve at the point s = 1. In particular if the elliptic curve has (analytic) rank 1, then the Heegner points can be used to construct a rational point on the curve of infinite order (so the Mordell–Weil group has rank at least 1).
The Gross–Zagier theorem connects the height of Heegner points on an elliptic curve to the derivative of its L-function evaluated at s=1.  This connection is particularly relevant when the elliptic curve has analytic rank 1, as it allows for the construction of a rational point with infinite order on the curve, which implies that the Mordell–Weil group has a rank of at least 1.