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 [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.

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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 $K$ 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.

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