# Heegner point

In mathematics, a **Heegner point** is a point on a modular curve that is the image of a quadratic imaginary point of the upper half-plane. They were defined by Bryan Birch and named after Kurt Heegner, who used similar ideas to prove Gauss's conjecture on imaginary quadratic fields of class number one.

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.

Kolyvagin later used Heegner points to construct Euler systems, and used this to prove much of the Birch–Swinnerton-Dyer conjecture for rank 1 elliptic curves. Shouwu Zhang generalized the Gross–Zagier theorem from elliptic curves to the case of modular abelian varieties. Brown proved the Birch–Swinnerton-Dyer conjecture for most rank 1 elliptic curves over global fields of positive characteristic. (Brown 1994)

Heegner points can be used to compute very large rational points on rank 1 elliptic curves (see (Watkins 2006) for a survey) that could not be found by naive methods. Implementation of the algorithm is available in Magma and PARI/GP

## References

- Birch, B., "Heegner points: the beginnings", in Darmon, Henri; Zhang, Shou-wu,
*Heegner Points and Rankin L-Series*(PDF), Mathematical Sciences Research Institute Publications,**49**, Cambridge University Press, ISBN 0-521-83659-X, MR 2083207. - Brown, M. L. (2004),
*Heegner modules and elliptic curves*, Lecture Notes In Mathematics,**1849**, Springer-Verlag, ISBN 3-540-22290-1, MR 2082815. - Darmon, Henri; Zhang, Shou-Wu, eds. (2004),
*Heegner points and Rankin L-series*, Mathematical Sciences Research Institute Publications,**49**, Cambridge University Press, ISBN 978-0-521-83659-3, MR 2083206 - Gross, Benedict H.; Zagier, Don B. (1986), "Heegner points and derivatives of L-series",
*Inventiones Mathematicae*,**84**(2): 225–320, doi:10.1007/BF01388809, MR 0833192. - Gross, B.; Kohnen, W.; Zagier, D. (1987), "Heegner points and derivatives of L-series. II",
*Mathematische Annalen*,**278**(1–4): 497–562, doi:10.1007/BF01458081, MR 0909238. - Heegner, Kurt (1952), "Diophantische Analysis und Modulfunktionen",
*Mathematische Zeitschrift*,**56**(3): 227–253, doi:10.1007/BF01174749, MR 0053135. - Watkins, Mark (2006),
*Some remarks on Heegner point computations*, arXiv:math/0506325v2. - Brown, Mark (1994), "On a conjecture of Tate for elliptic surfaces over finite fields",
*Proc. London Math. Soc.*,**69**(3): 489–514, doi:10.1112/plms/s3-69.3.489.