Height, in Diophantine geometry
A certain numerical function on the set of solutions of a Diophantine equation (cf. Diophantine equations). In the simplest case of a solution in integers of a Diophantine equation, the height is a function of the solution, and equals
. It is encountered in this form in Fermat's method of descent. Let
be a projective algebraic variety defined over a global field
. The height is a class of real-valued functions
defined on the set
of rational points
and depending on a morphism
of the variety
into the projective space
. Each function in this class is also called a height. From the point of view of estimating the number of rational points there are no essential differences between the functions in this class: for any two functions
and
there exist constants
and
, such that
. Such functions are called equivalent, and this equivalence is denoted (here) as
.
Fundamental properties of the height. The function is functorial with respect to
, i.e. for any morphism
and morphism
,
![]() |
If the morphisms ,
and
are defined by invertible sheaves
,
and
, and if
, then
. The set of points
of bounded height is finite in the following sense: If the basic field
is an algebraic number field, the set is finite; if it is an algebraic function field with field of constants
, the elements of
depend on a finite number of parameters from the field
; in particular,
is finite if the field
is finite. Let
run through the set of all norms of
. One may then define the height of a point
of the projective space
with coordinates from
as
![]() | (*) |
This is well defined because of the product formula ,
. Let
be an arbitrary projective variety over
and let
be a closed imbedding of
into the projective space; the height
may then be obtained by transferring the function (*), using the imbedding, to the set
. Various projective imbeddings, corresponding to the same sheaf
, define equivalent functions on
. A linear extension yields the desired function
. The function
is occasionally replaced by its logarithm — the so-called logarithmic height.
The above estimates may sometimes follow from exact equations [3], [4], [5]. There is a variant of the height function — the Néron–Tate height — which is defined on Abelian varieties and behaves as a functor with respect to the morphisms of Abelian varieties preserving the zero point. For the local aspect see [6]. The local components of a height constructed there play the role of intersection indices in arithmetic.
References
[1] | A. Weil, "Number theory and algebraic geometry" , Proc. Internat. Congress Mathematicians (Cambridge, 1950) , 2 , Amer. Math. Soc. (1952) pp. 90–100 MR0045416 Zbl 0049.02802 |
[2] | S. Lang, "Diophantine geometry" , Interscience (1962) MR0142550 Zbl 0115.38701 |
[3] | D. Mumford, "Abelian varieties" , Oxford Univ. Press (1974) (Appendix in Russian translation: Yu.I. Manin; The Mordell–Weil theorem (in Russian)) MR2514037 MR1083353 MR0352106 MR0441983 MR0282985 MR0248146 MR0219542 MR0219541 MR0206003 MR0204427 Zbl 0326.14012 |
[4] | Yu.I. Manin, "Height of theta points on an Abelian manifold, their variants and applications" Izv. Akad. Nauk SSSR Ser. Mat. , 28 (1964) pp. 1363–1390 (In Russian) |
[5] | D. Mumford, "A remark on Mordell's conjecture" Amer. J. Math. , 87 (1965) pp. 1007–1016 MR186624 |
[6] | A. Néron, "Quasi-fonctions et hauteurs sur les variétés abéliennes" Ann. of Math. (2) , 82 (1965) pp. 249–331 MR0179173 Zbl 0163.15205 |
Comments
The notion of height is a major tool in arithmetic algebraic geometry. It plays an important role in Faltings' proof of the Tate conjecture on endomorphisms of Abelian varieties over number fields, the Shafarevich conjecture that there are only finitely many isomorphism classes of Abelian varieties over a number field over of given dimension
with good reduction outside a finite set of places
of
, and the Mordell conjecture on the finiteness of the set of rational points
of a smooth curve of genus
over a number field
. Heights also play an important role in Arakelov intersection theory, which via moduli spaces of algebraic curves has also become important in string theory in mathematical physics.
References
[a1] | G. Faltings (ed.) G. Wüstholtz (ed.) , Rational points , Vieweg (1986) MR0863887 Zbl 0636.14019 |
Height, in Diophantine geometry. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Height,_in_Diophantine_geometry&oldid=18249