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] Although the global Néron–Tate height is quadratic, the local heights that it is the sum of 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 Tate (unpublished) defined it globally by observing that the logarithmic height $h_L$ associated to a symmetric invertible sheaf $L$ on an abelian variety $A$ is “almost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 $P$ .^[2] If $L$ is anti-symmetric, that is $[-1]^*L=L$ , 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) \qquad\mbox{is the unique quadratic function satisfying}\qquad \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–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,…,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 elliptic Lehmer conjecture, the curve E/K is fixed while the field of definition of the point P varies.

(Lang)^[3] $\hat h(P) \ge c(K) \log(\operatorname{Norm}_{K/\mathbb{Q}}\operatorname{Disc}(E/K))\quad$ for all $E/K$ and all nontorsion $P\in E(K).$
(Lehmer)^[4] $\hat h(P) \ge \frac{c(E/K)}{[K(P):K]}$ 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.^[3]^[5] 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.^[6] 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.^[7]

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^[8]

\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

↑ A. Néron, Quasi-fonctions et hauteurs sur les variétés abéliennes, Ann. of Math. 82 (1965), 249–331
↑ Lang (1997) p.72
↑ 3.0 3.1 Lang (1997) pp.73–74
↑ Lang (1997) pp.243
↑ Hindry, M.; Silverman, J.H. (1988). "The canonical height and integral points on elliptic curves". Invent. Math. 93: 419–450. doi:10.1007/bf01394340. Zbl 0657.14018.
↑ D. Masser, Counting points of small height on elliptic curves, Bull. Soc. Math. France 117 (1989), 247-265
↑ M. Laurent, Minoration de la hauteur de Néron-Tate, Séminaire de Théorie des Nombres (Paris 1981-1982), Progress in Mathematics, Birkhäuser 1983, 137-151
↑ G. Call and J.H. Silverman, Canonical heights on varieties with morphisms, Compositio Mathematica 89 (1993), 163-205

General references for the theory of canonical heights

Bombieri, Enrico; Gubler, Walter (2006). Heights in Diophantine Geometry. New Mathematical Monographs 4. Cambridge University Press. doi:10.2277/0521846153. ISBN 978-0-521-71229-3. Zbl 1130.11034.
Hindry, Marc; Silverman, Joseph H. (2000). Diophantine Geometry: An Introduction. Graduate Texts in Mathematics 201. ISBN 0-387-98981-1. Zbl 0948.11023.
Lang, Serge (1997). Survey of Diophantine Geometry. Springer-Verlag. ISBN 3-540-61223-8. Zbl 0869.11051.
J.H. Silverman, The Arithmetic of Elliptic Curves, ISBN 0-387-96203-4

External links

Canonical height on an elliptic curve at PlanetMath.org.