mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a Heegner point is a point on a

modular curve In number theory and algebraic geometry, a modular curve ''Y''(Γ) is a Riemann surface, or the corresponding algebraic curve, constructed as a quotient of the complex upper half-plane H by the action of a congruence subgroup Γ of the modular grou ...

that is the image of a quadratic imaginary point of the

upper half-plane In mathematics, the upper half-plane, \,\mathcal\,, is the set of points in the Cartesian plane with > 0. Complex plane Mathematicians sometimes identify the Cartesian plane with the complex plane, and then the upper half-plane corresponds to t ...

. They were defined by

Bryan Birch Bryan John Birch FRS (born 25 September 1931) is a British mathematician. His name has been given to the Birch and Swinnerton-Dyer conjecture. Biography Bryan John Birch was born in Burton-on-Trent, the son of Arthur Jack and Mary Edith Birch. ...

and named after

Kurt Heegner Kurt Heegner (; 16 December 1893 – 2 February 1965) was a German private scholar from Berlin, who specialized in radio engineering and mathematics. He is famous for his mathematical discoveries in number theory and, in particular, the Stark–He ...

, who used similar ideas to prove Gauss's conjecture on imaginary

quadratic fields In algebraic number theory, a quadratic field is an algebraic number field of degree two over \mathbf, the rational numbers. Every such quadratic field is some \mathbf(\sqrt) where d is a (uniquely defined) square-free integer different from 0 ...

of class number one.

Gross–Zagier theorem

The Gross–Zagier theorem describes the

height Height is measure of vertical distance, either vertical extent (how "tall" something or someone is) or vertical position (how "high" a point is). For example, "The height of that building is 50 m" or "The height of an airplane in-flight is abou ...

of Heegner points in terms of a derivative of the

L-function In mathematics, an ''L''-function is a meromorphic function on the complex plane, associated to one out of several categories of mathematical objects. An ''L''-series is a Dirichlet series, usually convergent on a half-plane, that may give ris ...

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 In arithmetic geometry, the Mordell–Weil group is an abelian group associated to any abelian variety A defined over a number field K, it is an arithmetic invariant of the Abelian variety. It is simply the group of K-points of A, so A(K) is the Mo ...

has rank at least 1). More generally, showed that Heegner points could be used to construct

rational point In number theory and algebraic geometry, a rational point of an algebraic variety is a point whose coordinates belong to a given field. If the field is not mentioned, the field of rational numbers is generally understood. If the field is the field ...

s 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 In mathematics, particularly in algebraic geometry, complex analysis and algebraic 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 function ...

(, ).

Birch and Swinnerton-Dyer conjecture

Kolyvagin later used Heegner points to construct

Euler system In mathematics, an Euler system is a 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 ...

s, and used this to prove much of the Birch–Swinnerton-Dyer conjecture for rank 1 elliptic curves. Brown proved the

Birch–Swinnerton-Dyer conjecture In mathematics, the Birch and Swinnerton-Dyer conjecture (often called the Birch–Swinnerton-Dyer conjecture) describes the set of rational solutions to equations defining an elliptic curve. It is an open problem in the field of number theory a ...

for most rank 1 elliptic curves over global fields of positive characteristic .

Computation

Heegner points can be used to compute very large rational points on rank 1 elliptic curves (see for a survey) that could not be found by naive methods. Implementations of the algorithm are available in

Magma Magma () is the molten or semi-molten natural material from which all igneous rocks are formed. Magma is found beneath the surface of the Earth, and evidence of magmatism has also been discovered on other terrestrial planets and some natural sa ...

PARI/GP PARI/GP is a computer algebra system with the main aim of facilitating number theory computations. Versions 2.1.0 and higher are distributed under the GNU General Public License. It runs on most common operating systems. System overview The ...

, and

Sage Sage or SAGE may refer to: Plants * ''Salvia officinalis'', common sage, a small evergreen subshrub used as a culinary herb ** Lamiaceae, a family of flowering plants commonly known as the mint or deadnettle or sage family ** ''Salvia'', a large ...

References

*. *. * *. *. *. *. *. *. *. *{{Citation, last=Zhang , first=Shou-Wu , editor1-last=Darmon , editor1-first=Henri , editor-link1=Henri Darmon , editor2-last=Zhang , editor2-first=Shou-Wu , title=Heegner points and Rankin L-series , chapter=Gross–Zagier formula for GL(2) II , publisher=

Cambridge University Press Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing hou ...

, series= Mathematical Sciences Research Institute Publications , isbn=978-0-521-83659-3 , mr=2083206 , year=2004 , volume=49 , pages=191–214 , chapter-url=http://www.msri.org/communications/books/Book49 , doi=10.1017/CBO9780511756375. Algebraic number theory Elliptic curves