Néron–Tate height

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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.

Definition and properties[edit]

Néron defined the Néron–Tate height as a sum of local heights.[1] Although the global Néron–Tate height is quadratic, the constituent local heights are not quite quadratic. Tate (unpublished) defined it globally by observing that the logarithmic height associated to a symmetric invertible sheaf on an abelian variety is “almost quadratic,” and used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 .[2] 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 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 . 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. (However, the height that naturally appears in the statement of the Birch–Swinnerton-Dyer conjecture is twice this height.) 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[edit]

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. Gram determinant). The elliptic regulator does not depend on the choice of basis.

More generally, let A/K be an abelian variety, let B ≅ Pic0(A) 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(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.

Lower bounds for the Néron–Tate height[edit]

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.

  • (Lang)[3]      for all and all nontorsion
  • (Lehmer)[4]     for all nontorsion

In both conjectures, the constants are positive and depend only on the indicated quantities. (A stronger form of Lang's conjecture asserts that depends only on the degree .) 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.[3][5] The best general result on Lehmer's conjecture is the weaker estimate due to Masser.[6] When the elliptic curve has complex multiplication, this has been improved to by Laurent.[7] 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[edit]

A polarized algebraic dynamical system is a triple (V,φ,L) consisting of a (smooth projective) algebraic variety V, 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[8]

where φ(n) = φ o φ o … o φ is the n-fold iteration of φ. For example, any morphism φ : PNPN of degree d > 1 yields a canonical height associated to the line bundle relation φ*O(1) = O(d). If V is defined over a number field and L is ample, then the canonical height is non-negative, and

(P is preperiodic if its forward orbit P, φ(P), φ2(P), φ3(P),… contains only finitely many distinct points.)

References[edit]

  1. ^ Néron, André (1965). "Quasi-fonctions et hauteurs sur les variétés abéliennes". Ann. of Math. (in French). 82: 249–331. doi:10.2307/1970644. MR 0179173. 
  2. ^ Lang (1997) p.72
  3. ^ a b Lang (1997) pp.73–74
  4. ^ Lang (1997) pp.243
  5. ^ Hindry, Marc; Silverman, Joseph H. (1988). "The canonical height and integral points on elliptic curves". Invent. Math. 93 (2): 419–450. doi:10.1007/bf01394340. MR 0948108. Zbl 0657.14018. 
  6. ^ Masser, David W. (1989). "Counting points of small height on elliptic curves". Bull. Soc. Math. France. 117 (2): 247–265. MR 1015810. 
  7. ^ Laurent, Michel (1983). "Minoration de la hauteur de Néron-Tate" [Lower bounds of the Nerón-Tate height]. In Bertin, Marie-José. Séminaire de théorie des nombres, Paris 1981–82 [Seminar on number theory, Paris 1981–82]. Progress in Mathematics (in French). Birkhäuser. pp. 137–151. ISBN 0-8176-3155-0. MR 0729165. 
  8. ^ Call, Gregory S.; Silverman, Joseph H. (1993). "Canonical heights on varieties with morphisms". Compositio Mathematica. 89 (2): 163–205. MR 1255693. 

General references for the theory of canonical heights

External links[edit]

"Canonical height on an elliptic curve". PlanetMath.