Néron–Tate height

The elliptic and abelian regulators

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

${\displaystyle ⟨P,Q⟩=\frac{1}{2}(\hat{h}(P+Q)-\hat{h}(P)-\hat{h}(Q)).}$

The elliptic regulator of E/K is

${\displaystyle \mathrm{Reg}(E/K)=det(⟨P_i,P_j⟩{)}_{1\le i,j\le r},}$

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

${\displaystyle \mathrm{Reg}(A/K)=det(⟨Q_i,{\eta }_j{⟩}_P{)}_{1\le i,j\le r}.}$

(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) ${\displaystyle \hat{h}(P)\ge c(K)\log max\{{\mathrm{Norm}}_{K/\mathbb{Q}}\mathrm{Disc}(E/K),h(j(E))\}}$ for all ${\displaystyle E/K}$ and all nontorsion ${\displaystyle P\in E(K).}$
• (Lehmer) ${\displaystyle \hat{h}(P)\ge \frac{c(E/K)}{[K(P):K]}}$ for all nontorsion ${\displaystyle P\in E(\overline{K}).}$

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. The best general result on Lehmer's conjecture is the weaker estimate ${\displaystyle \hat{h}(P)\ge c(E/K)/[K(P):K{]}^{3+\epsilon }}$ due to Masser. When the elliptic curve has complex multiplication, this has been improved to ${\displaystyle \hat{h}(P)\ge c(E/K)/[K(P):K{]}^{1+\epsilon }}$ by Laurent. 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 ${\displaystyle \hat{h}(P)\ge c(K)h(A/K)}$, where ${\displaystyle h(A/K)}$ is the Faltings height of ${\displaystyle A/K}$.

Generalizations

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

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

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

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

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