Heegner point

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In

Gross–Zagier theorem

The Gross–Zagier theorem (Gross & Zagier 1986) 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). More generally, Gross, Kohnen & Zagier (1987) showed that Heegner points could be used to construct rational points on the curve for each positive integer n, and the heights of these points were the coefficients of a modular form of weight 3/2. Shou-Wu Zhang generalized the Gross–Zagier theorem from elliptic curves to the case of modular abelian varieties (Zhang 2001, 2004, Yuan, Zhang & Zhang 2009).

Birch and Swinnerton-Dyer conjecture

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