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 Tate (unpublished) defined it globally by observing that the logarithmic height ${\displaystyle h_{L}}$ associated to a symmetric invertible sheaf ${\displaystyle L}$ on an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

${\displaystyle {\hat {h}}_{L}(P)=\lim _{N\rightarrow \infty }{\frac {h_{L}(NP)}{N^{2}}}}$

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

${\displaystyle {\hat {h}}_{L}(P)=h_{L}(P)+O(1),}$

where the implied ${\displaystyle O(1)}$ constant is independent of ${\displaystyle P}$.[2] If ${\displaystyle L}$ is anti-symmetric, that is ${\displaystyle [-1]^{*}L=L^{-1}}$, then the analogous limit

${\displaystyle {\hat {h}}_{L}(P)=\lim _{N\rightarrow \infty }{\frac {h_{L}(NP)}{N}}}$

converges and satisfies ${\displaystyle {\hat {h}}_{L}(P)=h_{L}(P)+O(1)}$, but in this case ${\displaystyle {\hat {h}}_{L}}$ is a linear function on the Mordell-Weil group. For general invertible sheaves, one writes ${\displaystyle L^{\otimes 2}=(L\otimes [-1]^{*}L)\otimes (L\otimes [-1]^{*}L^{-1})}$ as a product of a symmetric sheaf and an anti-symmetric sheaf, and then

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

is the unique quadratic function satisfying

${\displaystyle {\hat {h}}_{L}(P)=h_{L}(P)+O(1)\quad {\mbox{and}}\quad {\hat {h}}_{L}(0)=0.}$

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

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 ${\displaystyle {\hat {h}}}$ 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 ${\displaystyle A\times {\hat {A}}}$, the product of ${\displaystyle A}$ with its dual.

The elliptic and abelian regulators

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

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

The elliptic regulator of E/K is

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

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 ≅ Pic^0(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

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

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

In both conjectures, the constants are positive and depend only on the indicated quantities. (A stronger form of Lang's conjecture asserts that ${\displaystyle c}$ depends only on the degree ${\displaystyle [K:\mathbb {Q} ]}$.) 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 ${\displaystyle {\hat {h}}(P)\geq c(E/K)/[K(P):K]^{3+\varepsilon }}$ due to Masser.[6] When the elliptic curve has complex multiplication, this has been improved to ${\displaystyle {\hat {h}}(P)\geq c(E/K)/[K(P):K]^{1+\varepsilon }}$ by Laurent.[7] There are analogous conjectures for abelian varieties, with the nontorsion condition replaced by the condition that the multiples of ${\displaystyle P}$ form a Zariski dense subset of ${\displaystyle A}$, and the lower bound in Lang's conjecture replaced by ${\displaystyle {\hat {h}}(P)\geq c(K)h(A/K)}$, where ${\displaystyle h(A/K)}$ is the Faltings height of ${\displaystyle A/K}$.

Generalizations

A polarized algebraic dynamical system is a triple ${\displaystyle (V,\varphi ,L)}$ consisting of a (smooth projective) algebraic variety ${\displaystyle V}$, an endomorphism ${\displaystyle \varphi :V\to V}$, and a line bundle ${\displaystyle L\to V}$ with the property that ${\displaystyle \varphi ^{*}L=L^{\otimes d}}$ for some integer ${\displaystyle d>1}$. The associated canonical height is given by the Tate limit[8]

${\displaystyle {\hat {h}}_{V,\varphi ,L}(P)=\lim _{n\to \infty }{\frac {h_{V,L}(\varphi ^{(n)}(P))}{d^{n}}},}$

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

${\displaystyle {\hat {h}}_{V,\varphi ,L}(P)=0~~\Longleftrightarrow ~~P{\text{ is preperiodic for }}\varphi .}$

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

References

1. Néron, André (1965). "Quasi-fonctions et hauteurs sur les variétés abéliennes". Ann. of Math. (in French). 82 (2): 249–331. doi:10.2307/1970644. JSTOR 1970644. MR 0179173.
2. Lang (1997) p.72
3. Lang (1997) pp.73–74
4. Lang (1997) pp.243
5. Hindry, Marc; Silverman, Joseph H. (1988). "The canonical height and integral points on elliptic curves". Invent. Math. 93 (2): 419–450. doi:10.1007/bf01394340. MR 0948108. S2CID 121520625. Zbl 0657.14018.
6. Masser, David W. (1989). "Counting points of small height on elliptic curves". Bull. Soc. Math. France. 117 (2): 247–265. doi:10.24033/bsmf.2120. MR 1015810.
7. Laurent, Michel (1983). "Minoration de la hauteur de Néron–Tate" [Lower bounds of the Nerón-Tate height]. In Bertin, Marie-José (ed.). 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.
8. 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. 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.