The Gross-Zagier formula relates the Néron-Tate height of a Heegner point to the central derivative of a Rankin L-function. Combined with Kolyvagin's theory of Euler systems, it establishes the weak Birch and Swinnerton-Dyer conjecture for elliptic curves over Q of analytic rank one.

We begin by outlining the ideas behind Gross and Zagier's earlier factorization formula for differences of singular moduli, which can be viewed as the local height computation in the special case N=1. For general N, Gross's algebraic approach computes non-archimedean local height pairings, while Zagier's analytic approach computes archimedean local height pairings. On the L-function side, Rankin's method expresses the central derivative as the Petersson inner product with an explicit cusp form. The Gross-Zagier formula then follows by comparing this cusp form with the generating series of height pairings, up to an oldform.

We conclude with a brief overview of recent developments in the field.