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 Tate (unpublished) defined it globally by observing that the logarithmic height formula_1 associated to a symmetric invertible sheaf formula_2 on an abelian variety formula_3 is “almost quadratic,” and used this to show that the limit
formula_4
exists, defines a quadratic form on the Mordell–Weil group of rational points, and satisfies
formula_5
where the implied formula_6 constant is independent of formula_7. If formula_2 is anti-symmetric, that is formula_9, then the analogous limit
formula_10
converges and satisfies formula_11, but in this case formula_12 is a linear function on the Mordell-Weil group. For general invertible sheaves, one writes formula_13 as a product of a symmetric sheaf and an anti-symmetric sheaf, and then
formula_14
is the unique quadratic function satisfying
formula_15
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 formula_2 in
the Néron–Severi group of formula_3. If the abelian variety formula_3 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 formula_19. More generally, formula_12 induces a positive definite quadratic form on the real vector space formula_21.
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 formula_22 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 formula_23, the product of formula_3 with its dual.
The elliptic and abelian regulators.
The bilinear form associated to the canonical height formula_22 on an elliptic curve "E" is
formula_26
The elliptic regulator of "E"/"K" is
formula_27
where "P"1...,"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 "Q"1...,"Q""r" for the Mordell–Weil group "A"("K") modulo torsion and a basis "η"1...,"η""r" for the Mordell–Weil group "B"("K") modulo torsion and setting
formula_28
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.
In both conjectures, the constants are positive and depend only on the indicated quantities. (A stronger form of Lang's conjecture asserts that formula_34 depends only on the degree formula_35.) 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 formula_36 due to Masser. When the elliptic curve has complex multiplication, this has been improved to formula_37 by Laurent. There are analogous conjectures for abelian varieties, with the nontorsion condition replaced by the condition that the multiples of formula_7 form a Zariski dense subset of formula_3, and the lower bound in Lang's conjecture replaced by formula_40, where formula_41 is the Faltings height of formula_42.
Generalizations.
A polarized algebraic dynamical system is a triple formula_43 consisting of a (smooth projective) algebraic variety formula_44, an endomorphism formula_45, and a line bundle formula_46 with the property that formula_47 for some integer formula_48. The associated canonical height is given by the Tate limit
formula_49
where formula_50 is the "n"-fold iteration of formula_51. For example, any morphism formula_52 of degree formula_48 yields a canonical height associated to the line bundle relation formula_54. If formula_44 is defined over a number field and formula_2 is ample, then the canonical height is non-negative, and
formula_57
References.
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General references for the theory of canonical heights