Heegner point

In mathematics, a Heegner point is a point on a modular elliptic 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 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, together with Kohnen, Gross and Zagier 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 Gross-Zagier theorem from elliptic curve to the case of abelian variety. Brown proved the Birch-Swinnerton-Dyer conjecture for most rank 1 elliptic curves over global fields of positive characteristic.

References

*Heegner, Kurt [http://www.springerlink.com/openurl.asp?genre=article&id=doi:10.1007/BF01174749 "Diophantische Analysis und Modulfunktionen."] Math. Z. 56, (1952). 227--253.
* [http://assets.cambridge.org/052183/659X/excerpt/052183659X_excerpt.pdf Heegner points: the beginnings] by B. Birch, in "Heegner Points and Rankin L-Series" (Mathematical Sciences Research Institute Publications) by Henri Darmon (Editor), Shou-wu Zhang (Editor), ISBN 0-521-83659-X
*Gross, Benedict H.; Zagier, Don B. [http://www.springerlink.com/openurl.asp?genre=article&id=doi:10.1007/BF01388809 "Heegner points and derivatives of L-series."] Invent. Math. 84 (1986), no. 2, 225-320.
*Gross, B.; Kohnen, W.; Zagier, D. [http://springerlink.metapress.com/openurl.asp?genre=article&id=doi:10.1007/BF01458081 "Heegner points and derivatives of L-series. II."] Math. Ann. 278 (1987), no. 1-4, 497-562.
* Brown, M.L.; Heegner modules and elliptic curves. Springer Verlag Lecture Notes In Mathematics No. 1849, Springer Verlag 2004 (517pp).