In
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 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
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 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.
Shou-Wu Zhang
Shou-Wu Zhang (; born October 9, 1962) is a Chinese-American mathematician known for his work in number theory and arithmetic geometry. He is currently a Professor of Mathematics at Princeton University.
Biography Early life
Shou-Wu Zhang was b ...
generalized the Gross–Zagier theorem from elliptic curves to the case of modular
abelian varieties (, ).
Birch and Swinnerton-Dyer conjecture
Kolyvagin later used Heegner points to construct
Euler systems, and used this to prove much of 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 an ...
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,
PARI/GP, and
Sage.
References
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*{{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 , 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