In
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777 ...
, the Néron–Tate height (or canonical height) is a
quadratic form
In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example,
:4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong to a ...
on the
Mordell–Weil group In arithmetic geometry, the Mordell–Weil group is an abelian group associated to any abelian variety A defined over a number field K, it is an arithmetic invariant of the Abelian variety. It is simply the group of K-points of A, so A(K) is the Mo ...
of
rational points
In number theory and algebraic geometry, a rational point of an algebraic variety is a point whose coordinates belong to a given field. If the field is not mentioned, the field of rational numbers is generally understood. If the field is the fiel ...
of an
abelian variety defined over a
global field In mathematics, a global field is one of two type of fields (the other one is local field) which are characterized using valuations. There are two kinds of global fields:
* Algebraic number field: A finite extension of \mathbb
*Global function fi ...
. It is named after
André Néron
André Néron (November 30, 1922, La Clayette, France – April 6, 1985, Paris, France) was a French mathematician at the Université de Poitiers who worked on elliptic curves and abelian varieties. He discovered the Néron minimal model of an ...
and
John Tate John Tate may refer to:
* John Tate (mathematician) (1925–2019), American mathematician
* John Torrence Tate Sr. (1889–1950), American physicist
* John Tate (Australian politician) (1895–1977)
* John Tate (actor) (1915–1979), Australian act ...
.
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 Tate (unpublished) defined it globally by observing that the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 associated to a symmetric
invertible sheaf
In mathematics, an invertible sheaf is a coherent sheaf ''S'' on a ringed space ''X'', for which there is an inverse ''T'' with respect to tensor product of ''O'X''-modules. It is the equivalent in algebraic geometry of the topological notion of ...
on an
abelian variety is “almost quadratic,” and used this to show that the limit
:
exists, defines a quadratic form on the Mordell–Weil group of rational points, and satisfies
:
where the implied
constant is independent of
.
[Lang (1997) p.72] If
is anti-symmetric, that is
, then the analogous limit
:
converges and satisfies
, but in this case
is a linear function on the Mordell-Weil group. For general invertible sheaves, one writes
as a product of a symmetric sheaf and an anti-symmetric sheaf, and then
:
is the unique quadratic function satisfying
:
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
in
the
Néron–Severi group In algebraic geometry, the Néron–Severi group of a variety is
the group of divisors modulo algebraic equivalence; in other words it is the group of components of the Picard scheme of a variety. Its rank is called the Picard number. It is nam ...
of
. If the abelian variety
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
. More generally,
induces a positive definite quadratic form on the real vector space
.
On an
elliptic curve
In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If ...
, 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
without reference to a particular line bundle. (However, the height that naturally appears in the statement of the
Birch and Swinnerton-Dyer conjecture
In mathematics, the Birch and Swinnerton-Dyer conjecture (often called the Birch–Swinnerton-Dyer conjecture) describes the set of rational solutions to equations defining an elliptic curve. It is an open problem in the field of number theory an ...
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 In mathematics, a dual abelian variety can be defined from an abelian variety ''A'', defined over a field ''K''.
Definition
To an abelian variety ''A'' over a field ''k'', one associates a dual abelian variety ''A''v (over the same field), which ...
on
, the product of
with its
dual.
The elliptic and abelian regulators
The bilinear form associated to the canonical height
on an elliptic curve ''E'' is
:
The elliptic regulator of ''E''/''K'' is
:
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'' ≅ Pic
0(''A'') be the dual abelian variety to ''A'', and let ''P'' be the
Poincaré line bundle In mathematics, a dual abelian variety can be defined from an abelian variety ''A'', defined over a field ''K''.
Definition
To an abelian variety ''A'' over a field ''k'', one associates a dual abelian variety ''A''v (over the same field), which ...
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
:
(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
In mathematics, the Birch and Swinnerton-Dyer conjecture (often called the Birch–Swinnerton-Dyer conjecture) describes the set of rational solutions to equations defining an elliptic curve. It is an open problem in the field of number theory a ...
.
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)
[Lang (1997) pp.73–74] for all
and all nontorsion
* (Lehmer)
[Lang (1997) pp.243] for all nontorsion
In both conjectures, the constants are positive and depend only on the indicated quantities. (A stronger form of Lang's conjecture asserts that
depends only on the degree