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 Tate (unpublished) defined it globally by observing that the logarithmic height $h_L$ associated to a symmetric invertible sheaf $L$ on an abelian variety $A$ is “almost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

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–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] $\hat h(P) \ge c(K) \log\max\bigl\{\operatorname{Norm}_{K/\mathbb{Q}}\operatorname{Disc}(E/K),h(j(E))\bigr\}\quad$ for all $E/K$ and all nontorsion $P\in E(K).$
(Lehmer)^[4] $\hat h(P) \ge \frac{c(E/K)}{[K(P):K]}$ 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.^[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] 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,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

↑ Néron, André (1965). "Quasi-fonctions et hauteurs sur les variétés abéliennes". Ann. of Math. (in French). 82: 249–331. MR 0179173. doi:10.2307/1970644.
↑ Lang (1997) p.72
1 2 Lang (1997) pp.73–74
↑ Lang (1997) pp.243
↑ Hindry, Marc; Silverman, Joseph H. (1988). "The canonical height and integral points on elliptic curves". Invent. Math. 93 (2): 419–450. MR 0948108. Zbl 0657.14018. doi:10.1007/bf01394340.
↑ Masser, David W. (1989). "Counting points of small height on elliptic curves". Bull. Soc. Math. France. 117 (2): 247–265. MR 1015810.
↑ Laurent, Michel (1983). "Minoration de la hauteur de Néron-Tate" [Lower bounds of the Nerón-Tate height]. In Bertin, Marie-José. Séminaire de théorie des nombres, Paris 1981–82 [Seminar on number theory, Paris 1981–82]. Progress in Mathematics (in French). Birkhäuser. pp. 137–151. ISBN 0-8176-3155-0. MR 0729165.
↑ Call, Gregory S.; Silverman, Joseph H. (1993). "Canonical heights on varieties with morphisms". Compositio Mathematica. 89 (2): 163–205. MR 1255693.

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. Zbl 1130.11034. doi:10.2277/0521846153.
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". PlanetMath.

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