Heegner point
Encyclopedia
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, 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 system
s, 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.
Elliptic curve
In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...
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
Kurt Heegner
Kurt Heegner was a German private scholar from Berlin, who specialized inradio engineering and mathematics...
, who used similar ideas to prove Gauss's conjecture on imaginary quadratic fields of class number one.
The Gross–Zagier theorem describes the height
Néron–Tate height
In number theory, the Néron–Tate height is a quadratic form on the Mordell-Weil group of rational points of an abelian variety defined over a global field...
of Heegner points in terms of a derivative of the L-function
L-function
The theory of L-functions has become a very substantial, and still largely conjectural, part of contemporary analytic number theory. In it, broad generalisations of the Riemann zeta function and the L-series for a Dirichlet character are constructed, and their general properties, in most cases...
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, 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 system
Euler system
In mathematics, an Euler system is collection of compatible elements of Galois cohomology groups indexed by fields. They were introduced by in his work on Heegner points on modular elliptic curves, which was motivated by his earlier paper and the work of...
s, 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
Abelian variety
In mathematics, particularly in algebraic geometry, complex analysis and number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions...
. Brown proved the Birch–Swinnerton-Dyer conjecture for most rank 1 elliptic curves over global fields of positive characteristic.