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 provides a way to understand the relationship between Heegner points and the L-function of an elliptic curve.  Specifically, it shows how these points can be used to generate rational points on the curve with infinite order, which implies that the Mordell–Weil group has a rank of at least 1.