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 Tate (unpublished) defined it globally by observing that the logarithmic height [math]\displaystyle{ h_L }[/math] associated to a symmetric invertible sheaf [math]\displaystyle{ L }[/math] on an abelian variety [math]\displaystyle{ A }[/math] is “almost quadratic,” and used this to show that the limit

[math]\displaystyle{ \hat h_L(P) = \lim_{N\rightarrow\infty}\frac{h_L(NP)}{N^2} }[/math]

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

[math]\displaystyle{ \hat h_L(P) = h_L(P) + O(1), }[/math]

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

[math]\displaystyle{ \hat h_L(P) = \lim_{N\rightarrow\infty}\frac{h_L(NP)}{N} }[/math]

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

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

is the unique quadratic function satisfying

[math]\displaystyle{ \hat h_L(P) = h_L(P) + O(1) \quad\mbox{and}\quad \hat h_L(0)=0. }[/math]

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 [math]\displaystyle{ L }[/math] in the Néron–Severi group of [math]\displaystyle{ A }[/math]. If the abelian variety [math]\displaystyle{ A }[/math] 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 [math]\displaystyle{ A(K) }[/math]. More generally, [math]\displaystyle{ \hat h_L }[/math] induces a positive definite quadratic form on the real vector space [math]\displaystyle{ A(K)\otimes\mathbb{R} }[/math].

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 [math]\displaystyle{ \hat h }[/math] 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 [math]\displaystyle{ A\times\hat A }[/math], the product of [math]\displaystyle{ A }[/math] with its dual.

The elliptic and abelian regulators

The bilinear form associated to the canonical height [math]\displaystyle{ \hat h }[/math] on an elliptic curve E is

[math]\displaystyle{ \langle P,Q\rangle = \frac{1}{2} \bigl( \hat h(P+Q) - \hat h(P) - \hat h(Q) \bigr) . }[/math]

The elliptic regulator of E/K is

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

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

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

(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 elliptic Lehmer conjecture, the curve E/K is fixed while the field of definition of the point P varies.

• (Lang)     [math]\displaystyle{ \hat h(P) \ge c(K) \log\max\bigl\{\operatorname{Norm}_{K/\mathbb{Q}}\operatorname{Disc}(E/K),h(j(E))\bigr\}\quad }[/math] for all [math]\displaystyle{ E/K }[/math] and all nontorsion [math]\displaystyle{ P\in E(K). }[/math]
• (Lehmer)     [math]\displaystyle{ \hat h(P) \ge \frac{c(E/K)}{[K(P):K]} }[/math] for all nontorsion [math]\displaystyle{ P\in E(\bar K). }[/math]

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

Generalizations

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

[math]\displaystyle{ \hat h_{V,\varphi,L}(P) = \lim_{n\to\infty} \frac{h_{V,L}(\varphi^{(n)}(P))}{d^n}, }[/math]

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

[math]\displaystyle{ \hat h_{V,\varphi,L}(P) = 0 ~~ \Longleftrightarrow ~~ P \text{ is preperiodic for } \varphi. }[/math]

([math]\displaystyle{ P }[/math] is preperiodic if its forward orbit [math]\displaystyle{ P, \varphi(P), \varphi^2(P), \varphi^3(P),\ldots }[/math] contains only finitely many distinct points.)

References

General references for the theory of canonical heights

• Bombieri, Enrico; Gubler, Walter (2006). Heights in Diophantine Geometry. New Mathematical Monographs. 4. Cambridge University Press. ISBN 978-0-521-71229-3. 
• Hindry, Marc; Silverman, Joseph H. (2000). Diophantine Geometry: An Introduction. Graduate Texts in Mathematics. 201. ISBN 0-387-98981-1. 
• Lang, Serge (1997). Survey of Diophantine Geometry. Springer-Verlag. ISBN 3-540-61223-8. 
• J. H. Silverman, The Arithmetic of Elliptic Curves, ISBN 0-387-96203-4

External links

• "Canonical height on an elliptic curve". http://planetmath.org/?op=getobj&from=objects&id={{{id}}}.