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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h_L} associated to a symmetric invertible sheaf Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L} on an abelian variety Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A(K)} . More generally, ${\hat {h}}_{L}$ induces a positive definite quadratic form on the real vector space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 $A\times {\hat {A}}$, 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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{Reg}(A/K) = \det\bigl( \langle Q_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)     Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E/K} and all nontorsion $P\in E(K).$
• (Lehmer)     ${\hat {h}}(P)\geq {\frac {c(E/K)}{[K(P):K]}}$ for all nontorsion Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 $c$ depends only on the degree Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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. The best general result on Lehmer's conjecture is the weaker estimate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat h(P)\ge c(E/K)/[K(P):K]^{3+\varepsilon}} due to Masser. When the elliptic curve has complex multiplication, this has been improved to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat h(P)\ge 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} , and the lower bound in Lang's conjecture replaced by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat h(P)\ge c(K)h(A/K)} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h(A/K)} is the Faltings height of $A/K$.

Generalizations

A polarized algebraic dynamical system is a triple Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (V,\varphi, L)} consisting of a (smooth projective) algebraic variety Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} , an endomorphism $\varphi :V\to V$, and a line bundle Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L \to V} with the property that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi^*L = L^{\otimes d}} for some integer Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d > 1} . The associated canonical height is given by the Tate limit

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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.