# Height, in Diophantine geometry

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

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 [2] S. Lang, "Diophantine geometry" , Interscience (1962) [3] D. Mumford, "Abelian varieties" , Oxford Univ. Press (1974) (Appendix in Russian translation: Yu.I. Manin; The Mordell–Weil theorem (in Russian)) [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 [6] A. Néron, "Quasi-fonctions et hauteurs sur les variétés abéliennes" Ann. of Math. (2) , 82 (1965) pp. 249–331