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 Tate (unpublished) defined it globally by observing that the logarithmic height $h_{L}$ associated to a symmetric invertible sheaf $L$ on an abelian variety $A$ is “almost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 $L^{\otimes 2}=(L\otimes [-1]^{*}L)\otimes (L\otimes [-1]^{*}L^{-1})$ as a product of a symmetric sheaf and an anti-symmetric sheaf, and then

${\hat {h}}_{L}(P)={\frac {1}{2}}{\hat {h}}_{L\otimes [-1]^{*}L}(P)+{\frac {1}{2}}{\hat {h}}_{L\otimes [-1]^{*}L^{-1}}(P)$

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

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

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

$\operatorname {Reg} (A/K)=\det {\bigl (}\langle Q_{i},\eta _{j}\rangle _{P}{\bigr )}_{1\leq i,j\leq r}.$

(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)     ${\hat {h}}(P)\geq c(K)\log \max {\bigl \{}\operatorname {Norm} _{K/\mathbb {Q} }\operatorname {Disc} (E/K),h(j(E)){\bigr \}}\quad$ for all $E/K$ and all nontorsion $P\in E(K).$
• (Lehmer)     ${\hat {h}}(P)\geq {\frac {c(E/K)}{[K(P):K]}}$ 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 ${\hat {h}}(P)\geq c(E/K)/[K(P):K]^{3+\varepsilon }$ due to Masser. When the elliptic curve has complex multiplication, this has been improved to ${\hat {h}}(P)\geq c(E/K)/[K(P):K]^{1+\varepsilon }$ 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 ${\hat {h}}(P)\geq c(K)h(A/K)$, 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.)

References

General references for the theory of canonical heights

• Bombieri, Enrico; Gubler, Walter (2006). Heights in Diophantine Geometry. New Mathematical Monographs. Vol. 4. Cambridge University Press. ISBN 978-0-521-71229-3. Zbl 1130.11034.
• Hindry, Marc; Silverman, Joseph H. (2000). Diophantine Geometry: An Introduction. Graduate Texts in Mathematics. Vol. 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.