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. 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 Tate (unpublished) defined it globally by observing that the logarithmic height $$h_L$$ associated to a symmetric invertible sheaf $$L$$ on an abelian variety $$A$$ is “almost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


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

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


 * $$\hat h_L(P) = h_L(P) + O(1),$$

where the implied $$O(1)$$ constant is independent of $$P$$. If $$L$$ is anti-symmetric, that is $$[-1]^*L=L^{-1}$$, then the analogous limit


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

converges and satisfies $$\hat h_L(P) = h_L(P) + O(1)$$, but in this case $$\hat h_L$$ is a linear function on the Mordell-Weil group. For general invertible sheaves, one writes $$L^{\otimes2} = (L\otimes[-1]^*L)\otimes(L\otimes[-1]^*L^{-1})$$ as a product of a symmetric sheaf and an anti-symmetric sheaf, and then


 * $$\hat h_L(P) = \frac12 \hat h_{L\otimes[-1]^*L}(P) + \frac12 \hat h_{L\otimes[-1]^*L^{-1}}(P)$$

is the unique quadratic function satisfying


 * $$\hat h_L(P) = h_L(P) + O(1) \quad\mbox{and}\quad \hat h_L(0)=0.$$

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, $$\hat h_L$$ 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 and 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 $$A\times\hat A$$, the product of $$A$$ with its dual.

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,...,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 &times; 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


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

(The definitions of elliptic and abelian regulator are not entirely consistent, since if A is an elliptic curve, then the latter is 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)    $$ \hat h(P) \ge c(K) \log\max\bigl\{\operatorname{Norm}_{K/\mathbb{Q}}\operatorname{Disc}(E/K),h(j(E))\bigr\}\quad$$ for all $$E/K$$ and all nontorsion $$P\in E(K).$$
 * (Lehmer)    $$\hat h(P) \ge \frac{c(E/K)}{[K(P):K]}$$ for all nontorsion $$P\in E(\bar K).$$

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 $$[K:\mathbb Q]$$.) 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 $$\hat h(P)\ge c(E/K)/[K(P):K]^{3+\varepsilon}$$ due to Masser. When the elliptic curve has complex multiplication, this has been improved to $$\hat h(P)\ge c(E/K)/[K(P):K]^{1+\varepsilon}$$ 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 $$\hat h(P)\ge c(K)h(A/K)$$, where $$h(A/K)$$ is the Faltings height of $$A/K$$.

Generalizations
A polarized algebraic dynamical system is a triple $$(V,\varphi, L)$$ consisting of a (smooth projective) algebraic variety $$V$$, an endomorphism $$\varphi:V \to V$$, and a line bundle $$L \to V$$ with the property that $$\varphi^*L = L^{\otimes d}$$ for some integer $$d > 1$$. The associated canonical height is given by the Tate limit


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

where $$\varphi^{(n)} = \varphi\circ \cdots \circ \varphi$$ is the n-fold iteration of $$\varphi$$. For example, any morphism $$\varphi: \mathbb{P}^n \to \mathbb{P}^n$$ of degree $$d > 1$$ yields a canonical height associated to the line bundle relation $$\varphi^*\mathcal{O}(1) = \mathcal{O}(n)$$. If $$V$$ is defined over a number field and $$L$$ is ample, then the canonical height is non-negative, and


 * $$ \hat h_{V,\varphi,L}(P) = 0 \Longleftrightarrow  P \text{ is preperiodic for } \varphi.$$

($$P$$ is preperiodic if its forward orbit $$P, \varphi(P), \varphi^2(P), \varphi^3(P),\ldots$$ contains only finitely many distinct points.)