In number theory, the Néron–Tate height (or canonical height) is a quadratic form on the Mordell–Weil group of rational points of an abelian variety defined over a global field.
It is named after André Néron and John Tate.
Tate (unpublished) defined it globally by observing that the logarithmic height
is “almost quadratic,” and used this to show that the limit exists, defines a quadratic form on the Mordell–Weil group of rational points, and satisfies where the implied
, then the analogous limit converges and satisfies
is a linear function on the Mordell-Weil group.
For general invertible sheaves, one writes
as a product of a symmetric sheaf and an anti-symmetric sheaf, and then is the unique quadratic function satisfying The Néron–Tate height depends on the choice of an invertible sheaf on the abelian variety, although the associated bilinear form depends only on the image of
is defined over a number field K and the invertible sheaf is symmetric and ample, then the Néron–Tate height is positive definite in the sense that it vanishes only on torsion elements of the Mordell–Weil group
induces a positive definite quadratic form on the real vector space
On an elliptic curve, the Néron–Severi group is of rank one and has a unique ample generator, so this generator is often used to define the Néron–Tate height, which is denoted
On abelian varieties of higher dimension, there need not be a particular choice of smallest ample line bundle to be used in defining the Néron–Tate height, and the height used in the statement of the Birch–Swinnerton-Dyer conjecture is the Néron–Tate height associated to the Poincaré line bundle on
The bilinear form associated to the canonical height
on an elliptic curve E is The elliptic regulator of E/K is where P1,...,Pr is a basis for the Mordell–Weil group E(K) modulo torsion (cf.
The elliptic regulator does not depend on the choice of basis.
Then the abelian regulator of A/K is defined by choosing a basis Q1,...,Qr for the Mordell–Weil group A(K) modulo torsion and a basis η1,...,ηr for the Mordell–Weil group B(K) modulo torsion and setting (The definitions of elliptic and abelian regulator are not entirely consistent, since if A is an elliptic curve, then the latter is 2r times the former.)
The elliptic and abelian regulators appear in the Birch–Swinnerton-Dyer conjecture.
There are two fundamental conjectures that give lower bounds for the Néron–Tate height.
In the first, the field K is fixed and the elliptic curve E/K and point P ∈ E(K) vary, while in the second, the elliptic Lehmer conjecture, the curve E/K is fixed while the field of definition of the point P varies.
In both conjectures, the constants are positive and depend only on the indicated quantities.
(A stronger form of Lang's conjecture asserts that
[3][5] The best general result on Lehmer's conjecture is the weaker estimate
[6] When the elliptic curve has complex multiplication, this has been improved to
form a Zariski dense subset of
, and the lower bound in Lang's conjecture replaced by
A polarized algebraic dynamical system is a triple
consisting of a (smooth projective) algebraic variety
The associated canonical height is given by the Tate limit[8] where
yields a canonical height associated to the line bundle relation
General references for the theory of canonical heights