Néron–Tate height explained

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

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 → infty} (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} ^{-1} , then the analogous limit\hat h _{L(P)} = \lim _{N → infty} (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 ^{⊗ 2} = (L ⊗ [ -1 ] ^{*L) ⊗ (L ⊗ [-1]} ^{*L} ^{-1} ) as a product of a symmetric sheaf and an anti-symmetric sheaf, and then\hat h _{L(P)} = (12)/(\hat) h _{L ⊗ [ -1 ] ^{*L}} (P) + (12)/(\hat) h _{L ⊗ [ -1 ] ^{*L} ^{-1}} (P)

is the unique quadratic function satisfying

\hat h _{L(P)} = h _{L(P)} + O( 1 ) and \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 inthe 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) ⊗ 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 x \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 = (1)/(2) l ( \hat h(P + Q) - \hat h(P) - \hat h(Q) r ) .

The elliptic regulator of E/K is

\operatorname{Reg}(E/K) = \det l ( \langle P _{i,P} _{j\rangle} r ) _{1\le} ,

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 l ( \langle Q _{i,η} _{j\rangle} _{P} r ) _{1\le} .

(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) logmax l \{\operatorname{Norm} _{K/Q} \operatorname{Disc}(E/K),h(j(E)) r \} for all E/K and all nontorsion P\in E(K).

• (Lehmer)^[4]     

\hat h(P) \ge (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: 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+\varepsilon} 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+\varepsilon} 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,\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 ^{⊗} for some integer d > 1 . The associated canonical height is given by the Tate limit^[8] \hat h _{V,\varphi,L} (P) = \lim _{n\toinfty} (h _{V,L} (\varphi ^{(n)} (P)))/(d ^{n}) ,

where

\varphi ^{(n)} = \varphi\circ … \circ \varphi is the n-fold iteration of \varphi . For example, any morphism \varphi: P ^{n} \to P ^{n} of degree d > 1 yields a canonical height associated to the line bundle relation \varphi ^{*l{O}(1)=} l{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 ispreperiodicfor \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

• Book: Enrico . Bombieri . Enrico Bombieri . Walter . Gubler . Heights in Diophantine Geometry . New Mathematical Monographs . 4 . . 2006 . 978-0-521-71229-3 . 1130.11034 .
• Book: Marc . Hindry . Joseph H. . Silverman . Joseph H. Silverman . Diophantine Geometry: An Introduction . . 201 . 2000 . 0-387-98981-1 . 0948.11023 .
• Book: Lang, Serge . Serge Lang

. Serge Lang . Survey of Diophantine Geometry . . 1997 . 3-540-61223-8 . 0869.11051 .

• J. H. Silverman, The Arithmetic of Elliptic Curves,

Notes and References

1. André . Néron . Quasi-fonctions et hauteurs sur les variétés abéliennes . . 82 . 1965 . 2 . 249–331 . 10.2307/1970644 . 1970644 . 0179173 . fr.
2. Lang (1997) p.72
3. Lang (1997) pp.73–74
4. Lang (1997) pp.243
5. Marc . Hindry . Joseph H. . Silverman . Joseph H. Silverman. The canonical height and integral points on elliptic curves . . 93 . 1988 . 419–450 . 0657.14018 . 10.1007/bf01394340 . 2 . 0948108. 121520625 .
6. David W. . Masser . Counting points of small height on elliptic curves . Bull. Soc. Math. France . 117 . 1989 . 247–265 . 2 . 10.24033/bsmf.2120 . 1015810 . free .
7. Michel . Laurent . Minoration de la hauteur de Néron–Tate . Lower bounds of the Nerón-Tate height . fr . Séminaire de théorie des nombres, Paris 1981–82 . Seminar on number theory, Paris 1981–82 . Progress in Mathematics . Birkhäuser . 1983 . 137–151 . 0729165 . 0-8176-3155-0 . Bertin . Marie-José.
8. Gregory S. . Call . Joseph H. . Silverman . Canonical heights on varieties with morphisms . . 89 . 1993 . 163–205 . 2 . 1255693 .