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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.

Contents

Definition and properties

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 Tate (unpublished) defined it globally by observing that the logarithmic height h L associated to a symmetric invertible sheaf L on an abelian variety A is “almost quadratic,” and used this to show that the limit

h ^ L ( P ) = lim N → ∞ h L ( N P ) N 2

exists, defines a quadratic form on the Mordell-Weil group of rational points, and satisfies

h ^ L ( P ) = h L ( P ) + O ( 1 ) ,

where the implied O ( 1 ) constant is independent of P . If L is anti-symmetric, that is [ − 1 ] ∗ L = L , then the analogous limit

h ^ L ( P ) = lim N → ∞ h L ( N P ) N

converges and satisfies h ^ L ( P ) = h L ( P ) + O ( 1 ) , but in this case h ^ L is a linear function on the Mordell-Weil group. For general invertible sheaves, one writes L ⊗ 2 = ( L ⊗ [ − 1 ] ∗ L ) ⊗ ( L ⊗ [ − 1 ] ∗ L − 1 ) as a product of a symmetric sheaf and an anti-symmetric sheaf, and then

h ^ L ( P ) = 1 2 h ^ L ⊗ [ − 1 ] ∗ L ( P ) + 1 2 h ^ L ⊗ [ − 1 ] ∗ L − 1 ( P )

is the unique quadratic function satisfying

h ^ L ( P ) = h L ( P ) + O ( 1 ) and h ^ L ( 0 ) = 0.

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 L in the Néron–Severi group of A . If the abelian variety A 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 A ( K ) . More generally, h ^ L induces a positive definite quadratic form on the real vector space A ( K ) ⊗ R .

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 h ^ 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 A × A ^ , the product of A with its dual.

The elliptic and abelian regulators

The bilinear form associated to the canonical height h ^ on an elliptic curve E is

⟨ P , Q ⟩ = 1 2 ( h ^ ( P + Q ) − h ^ ( P ) − h ^ ( Q ) ) .

The elliptic regulator of E/K is

Reg ⁡ ( E / K ) = det ( ⟨ P i , P j ⟩ ) 1 ≤ i , j ≤ r ,

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

Reg ⁡ ( A / K ) = det ( ⟨ P i , η j ⟩ P ) 1 ≤ i , j ≤ r .

(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

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)      h ^ ( P ) ≥ c ( K ) log ⁡ max { Norm K / Q ⁡ Disc ⁡ ( E / K ) , h ( j ( E ) ) } for all E / K and all nontorsion P ∈ E ( K ) .
  • (Lehmer)     h ^ ( P ) ≥ c ( E / K ) [ K ( P ) : K ] for all nontorsion P ∈ E ( K ¯ ) .

    • In both conjectures, the constants are positive and depend only on the indicated quantities. (A stronger form of Lang's conjecture asserts that c depends only on the degree [ K : Q ] .) 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 h ^ ( P ) ≥ c ( E / K ) / [ K ( P ) : K ] 3 + ϵ due to Masser. When the elliptic curve has complex multiplication, this has been improved to h ^ ( P ) ≥ c ( E / K ) / [ K ( P ) : K ] 1 + ϵ by Laurent. There are analogous conjectures for abelian varieties, with the nontorsion condition replaced by the condition that the multiples of P form a Zariski dense subset of A , and the lower bound in Lang's conjecture replaced by h ^ ( P ) ≥ c ( K ) h ( A / K ) , where h ( A / K ) is the Faltings height of A / K .

    Generalizations

    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 ϕ ∗ L = L ⊗ d for some integer d > 1. The associated canonical height is given by the Tate limit

    h ^ V , ϕ , L ( P ) = lim n → ∞ h V , L ( ϕ ( n ) ( P ) ) d n ,

    where φ(n) = 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(1) = O(d). If V is defined over a number field and L is ample, then the canonical height is non-negative, and

    h ^ V , ϕ , L ( P ) = 0     ⟺     P   i s   p r e p e r i o d i c   f o r   ϕ .

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

    References

    Néron–Tate height Wikipedia
    (Text) CC BY-SA