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Canonical Vector Heights on Algebraic K3 Surfaces with Picard Number Two |
Canad. Math. Bull. 46(2003), 495-508
Published:2003-12-01
Printed: Dec 2003
Printed: Dec 2003
- Arthur Baragar
Abstract
Let $V$ be an algebraic K3 surface defined over a number field $K$.
Suppose $V$ has Picard number two and an infinite group of
automorphisms $\mathcal{A} = \Aut(V/K)$. In this paper, we
introduce the notion of a vector height $\mathbf{h} \colon V \to
\Pic(V) \otimes \mathbb{R}$ and show the existence of a canonical
vector height $\widehat{\mathbf{h}}$ with the following properties:
\begin{gather*}
\widehat{\mathbf{h}} (\sigma P) = \sigma_* \widehat{\mathbf{h}} (P) \\
h_D (P) = \widehat{\mathbf{h}} (P) \cdot D + O(1),
\end{gather*}
where $\sigma \in \mathcal{A}$, $\sigma_*$ is the pushforward of
$\sigma$ (the pullback of $\sigma^{-1}$), and $h_D$ is a Weil
height associated to the divisor $D$. The bounded function implied
by the $O(1)$ does not depend on $P$. This allows us to attack
some arithmetic problems. For example, we show that the number of
rational points with bounded logarithmic height in an
$\mathcal{A}$-orbit satisfies
$$
N_{\mathcal{A}(P)} (t,D) = \# \{Q \in \mathcal{A}(P) : h_D (Q)
| MSC Classifications: |
11G50 - Heights [See also 14G40, 37P30] 14J28 - $K3$ surfaces and Enriques surfaces 14G40 - Arithmetic varieties and schemes; Arakelov theory; heights [See also 11G50, 37P30] 14J50 - Automorphisms of surfaces and higher-dimensional varieties 14G05 - Rational points |