View source for 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 [[#References|[3]]], [[#References|[4]]], [[#References|[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 [[#References|[6]]]. The local components of a height constructed there play the role of intersection indices in arithmetic.

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top"> A. Weil, "Number theory and algebraic geometry" , ''Proc. Internat. Congress Mathematicians (Cambridge, 1950)'' , '''2''' , Amer. Math. Soc. (1952) pp. 90–100 {{MR|0045416}} {{ZBL|0049.02802}} </TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top"> S. Lang, "Diophantine geometry" , Interscience (1962) {{MR|0142550}} {{ZBL|0115.38701}} </TD></TR>
<TR><TD valign="top">[3]</TD> <TD valign="top"> D. Mumford, "Abelian varieties" , Oxford Univ. Press (1974) (Appendix in Russian translation: Yu.I. Manin; The Mordell–Weil theorem (in Russian)) {{MR|2514037}} {{MR|1083353}} {{MR|0352106}} {{MR|0441983}} {{MR|0282985}} {{MR|0248146}} {{MR|0219542}} {{MR|0219541}} {{MR|0206003}} {{MR|0204427}} {{ZBL|0326.14012}} </TD></TR>
<TR><TD valign="top">[4]</TD> <TD valign="top"> 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)</TD></TR>
<TR><TD valign="top">[5]</TD> <TD valign="top"> D. Mumford, "A remark on Mordell's conjecture" ''Amer. J. Math.'' , '''87''' (1965) pp. 1007–1016 {{MR|186624}} {{ZBL|}} </TD></TR>
<TR><TD valign="top">[6]</TD> <TD valign="top"> A. Néron, "Quasi-fonctions et hauteurs sur les variétés abéliennes" ''Ann. of Math. (2)'' , '''82''' (1965) pp. 249–331 {{MR|0179173}} {{ZBL|0163.15205}} </TD></TR>
</table>



====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 variety|Abelian varieties]] over number fields, the [[Shafarevich conjecture]] that there are only finitely many isomorphism classes of Abelian varieties over a number field over $K$ of given dimension $g\ge 1$ with good reduction outside a finite set of places $S$ of $K$, and the [[Mordell conjecture]] on the finiteness of the set of rational points $X(K)$ of a smooth curve of genus $g \ge 2$ over a number field $K$. 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====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top"> G. Faltings (ed.) G. Wüstholtz (ed.) , ''Rational points'' , Vieweg (1986) {{MR|0863887}} {{ZBL|0636.14019}} </TD></TR>
</table>

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