# 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 $(x_0,\ldots,x_n)$ of a Diophantine equation, the height is a function of the solution, and equals $\max\{|x_i|\}$. It is encountered in this form in Fermat's method of descent. Let $X$ be a projective algebraic variety defined over a global field $K$. The height is a class of real-valued functions $h_L(P)$ defined on the set $X(K)$ of rational points $P$ and depending on a morphism $L:X\rightarrow P^n$ of the variety $X$ into the projective space $P^n$. 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 $h'$ and $h''$ there exist constants $c',c'' > 0$, such that $c' h' \le h'' \le c''h'$. Such functions are called equivalent, and this equivalence is denoted (here) as $\cong$.

Fundamental properties of the height. The function $h_L(P)$ is functorial with respect to $P$, i.e. for any morphism $f:X \rightarrow Y$ and morphism $L : Y \rightarrow P^n$, $$h_{f*L}(P) \cong h_L(f(P))\,\ \ P \in X(K) \ .$$

If the morphisms $L$, $L_1$ and $L_2$ are defined by invertible sheaves $\mathcal{L}$, $\mathcal{L}_1$ and $\mathcal{L}_2$, and if $\mathcal{L} = \mathcal{L}_1 \otimes \mathcal{L}_2$, then $h_L \cong h_{L_1} h_{L_2}$. The set of points $P \in X(K)$ of bounded height is finite in the following sense: If the basic field $K$ is an algebraic number field, the set is finite; if it is an algebraic function field with field of constants $k$, the elements of $X(K)$ depend on a finite number of parameters from the field $k$; in particular, $X(K)$ is finite if the field $k$ is finite. Let $|\cdot|_\nu$ run through the set of all norms of $K$. One may then define the height of a point $(x_0:\cdots:x_n)$ of the projective space $P^n$ with coordinates from $K$ as \begin{equation}\label{eq:1} \prod_\nu \max(|x_0|_\nu,\ldots,|x_n|_\nu) \ . \end{equation}

This is well defined because of the product formula $\prod_nu |x|_\nu = 1$ for $x \in K$. Let $X$ be an arbitrary projective variety over $K$ and let $L$ be a closed imbedding of $X$ into the projective space; the height $h_L$ may then be obtained by transferring the function \eqref{eq:1}, using the imbedding $L$, to the set $X(K)$. Various projective imbeddings, corresponding to the same sheaf $\mathcal{L}$, define equivalent functions on $X(K)$. A linear extension yields the desired function $h_L$. The function $h_L$ is occasionally replaced by its logarithm — the so-called logarithmic height.

The above estimates may sometimes follow from exact equations , , . 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 . The local components of a height constructed there play the role of intersection indices in arithmetic.

How to Cite This Entry:
Height, in Diophantine geometry. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Height,_in_Diophantine_geometry&oldid=53892
This article was adapted from an original article by A.N. Parshin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article