Néron–Tate height

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

Néron defined the Néron–Tate height as a sum of local heights.^[1] Tate (unpublished) defined it globally by observing that the logarithmic height h_L associated to an invertible sheaf L on an abelian variety A is “almost quadratic,” and used this to show that the limit

$\hat h_L(P) = \lim_{N\rightarrow\infty}\frac{h_L(NP)}{N^2}$

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

The Néron–Tate height depends on the choice of an invertible sheaf (or an element of the Néron-Severi group) on the abelian variety. If the abelian variety A is defined over a number field K and the invertible sheaf is 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, $\hat h_L$ induces a positive definite quadratic form on the real vector space $A(K)\otimes\mathbb{R}$.

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 $\hat h$ 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.

The elliptic and abelian regulators

The bilinear form associated to the canonical height $\hat h$ on an elliptic curve E is

$\langle P,Q\rangle = \frac{1}{2} \bigl( \hat h(P+Q) - \hat h(P) - \hat h(Q) \bigr) .$

The elliptic regulator of E/K is

$\operatorname{Reg}(E/K) = \det\bigl( \langle P_i,P_j\rangle \bigr)_{1\le i,j\le r},$

where P₁,…,P_r 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,…,Q_r for the Mordell-Weil group A(K) modulo torsion and a basis η₁,…,η_r for the Mordell-Weil group B(K) modulo torsion and setting

$\operatorname{Reg}(A/K) = \det\bigl( \langle P_i,\eta_j\rangle_{P} \bigr)_{1\le i,j\le r}.$

(The definitions of elliptic and abelian regulator are not entirely consistent, since if A is an elliptic curve, then the latter is 2^r 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 curve E/K is fixed while the field of definition of the point P varies.

${\rm {Conjecture}~(Lang)}\quad \hat h(P) \ge c(K) \log(\operatorname{Norm}_{K/\mathbb{Q}}\operatorname{Disc}(E/K))\quad{\rm for~all }~E/K~{ \rm and~all~nontorsion }~P\in E(K).$

${\rm {Conjecture}~(Lehmer)}\quad \hat h(P) \ge \frac{c(E/K)}{[K(P):K]} \quad{\rm for~all}~P\in E(\bar K).$

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.^[2] The best general result on Lehmer's conjecture is the weaker estimate $\hat h(P)\ge c(E/K)/[K(P):K]^{3+\epsilon}$ due to Masser.^[3] When the elliptic curve has complex multiplication, this has been improved to $\hat h(P)\ge c(E/K)/[K(P):K]^{1+\epsilon}$ by Laurent.^[4]

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 $\phi^*L = L^{\otimes d}$ for some integer d > 1. The associated canonical height is given by the Tate limit^[5]

$\hat h_{V,\phi,L}(P) = \lim_{n\to\infty} \frac{h_{V,L}(\phi^{(n)}(P))}{d^n},$

where φ⁽ⁿ⁾ = φ o φ o … o φ is the n-fold iteration of φ. For example, any morphism φ : P^N → P^N 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

$\hat h_{V,\phi,L}(P) = 0 ~~ \Longleftrightarrow ~~ P~{\rm is~preperiodic~for~}\phi.$

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

References

1. ^ A. Néron, Quasi-fonctions et hauteurs sur les variétés abéliennes, Ann. of Math. 82 (1965), 249–331
2. ^ M. Hindry and J.H. Silverman, The canonical height and integral points on elliptic curves, Invent. Math. 93 (1988), 419-450
3. ^ D. Masser, Counting points of small height on elliptic curves, Bull. Soc. Math. France 117 (1989), 247-265
4. ^ M. Laurent, Minoration de la hauteur de Néron-Tate, Séminaire de Théorie des Nombres (Paris 1981-1982), Progress in Mathematics, Birkhäuser 1983, 137-151
5. ^ G. Call and J.H. Silverman, Canonical heights on varieties with morphisms, Compositio Math. 89 (1993), 163-205

General references for the theory of canonical heights

• E. Bombieri and W. Gubler Heights in Diophantine Geometry, ISBN 0521846153, DOI 10.2277/0521846153
• M. Hindry and J.H. Silverman, Diophantine Geometry, ISBN 0-387-98981-1
• S. Lang, Fundamentals of Diophantine Geometry, ISBN 0-387-90837-4
• J.H. Silverman, The Arithmetic of Elliptic Curves, ISBN 0-387-96203-4

External links

• Canonical height on an elliptic curve on PlanetMath