Néron defined the Néron–Tate height as a sum of local heights. Although the global Néron–Tate height is quadratic, the constituent local heights are not quite quadratic. Tate defined it globally by observing that the logarithmic height associated to a symmetric invertible sheaf on an abelian variety 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 constant is independent of. If is anti-symmetric, that is, then the analogous limit converges and satisfies, but in this case 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 in the Néron–Severi group of. If the abelian variety is defined over a number fieldK 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. More generally, 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 without reference to a particular line bundle. 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 product of with its dual.
The elliptic and abelian regulators
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 modulo torsion . The elliptic regulator does not depend on the choice of basis. More generally, let A/K be an abelian variety, let B ≅ Pic0 be the dual abelian variety to A, and let P be the Poincaré line bundle on A × B. Then the abelian regulator of A/K is defined by choosing a basis Q1,…,Qr for the Mordell-Weil group A modulo torsion and a basis η1,…,ηr for the Mordell-Weil group B modulo torsion and setting The elliptic and abelian regulators appear in the Birch–Swinnerton-Dyer conjecture.
Lower bounds for the Néron–Tate height
There are two fundamental conjectures that give lower bounds for the Néron–Tate height. In the first, the fieldK is fixed and the elliptic curve E/K and point P ∈ E vary, while in the second, the elliptic Lehmer conjecture, the curveE/K is fixed while the field of definition of the point P varies.
for all and all nontorsion
for all nontorsion
In both conjectures, the constants are positive and depend only on the indicated quantities. It is known that the abc conjecture implies Lang's conjecture, and that the analogue of Lang's conjecture over one dimensional characteristic 0 function fields is unconditionally true. The best general result on Lehmer's conjecture is the weaker estimate due to Masser. When the elliptic curve has complex multiplication, this has been improved to by Laurent. There are analogous conjectures for abelian varieties, with the nontorsion condition replaced by the condition that the multiples of form a Zariski dense subset of, and the lower bound in Lang's conjecture replaced by, where is the Faltings height of.
Generalizations
A polarized algebraic dynamical system is a triple consisting of a algebraic varietyV, a self-morphism φ : V → V, and a line bundle L on V with the property that for some integer d > 1. The associated canonical height is given by the Tate limit where φ = φ o φ o … o φ is the n-fold iteration of φ. For example, any morphism φ : PN → PN of degree d > 1 yields a canonical height associated to the line bundle relation φ*O = O. If V is defined over a number field and L is ample, then the canonical height is non-negative, and , φ2, φ3
