## Rational points on K3 surfaces: A new canonical height.(English)Zbl 0754.14023

A marked $$K3$$ surface $$S$$ is a complex projective $$K3$$ surface with automorphism group containing a subgroup $${\mathcal A}$$ isomorphic to the free product $$\mathbb{Z}_ 2*\mathbb{Z}_ 2$$ of two cyclic groups of order 2 together with a fixed isomorphism from $${\mathcal A}$$ onto $$\mathbb{Z}_ 2*\mathbb{Z}_ 2$$. The surface $$S$$ and the action of the automorphism subgroup $${\mathcal A}$$ are assumed to be defined over a number field $$K$$. The orbit of a point $$P\in S(K)$$ $${\mathcal C}={\mathcal C}(P)=\{\varphi P;\varphi\in{\mathcal A}\}$$ is called a chain (of $$K$$-rational points on $$S)$$. The aim of the paper is to describe the points in a chain and the collection of chains in $$S(K)$$. For this purpose a Weil height function $$\hat h$$ and a canonical height $$\hat H$$ are introduced. The latter does not depend on the special choice of a point in a given chain $${\mathcal C}$$. So $$\hat H({\mathcal C})$$ is well-defined. The following basic result is proved:
Theorem 1.2. The chain $${\mathcal C}$$ is finite iff $$\hat H({\mathcal C})=0$$ iff $$\hat h(P)=0$$ for all $$P\in{\mathcal C}$$. For any constant $$B$$, the set {chains $${\mathcal C}$$ in $$S(K)$$; $$\hat H({\mathcal C})<B\}$$ is finite. There are only finitely many chains $${\mathcal C}$$ in $$S(K)$$ with finite members.
The next theorems give useful estimations for infinite chains $${\mathcal C}$$ in $$S(K)$$: $2\sqrt{\hat H({\mathcal C})}\leq\min_{P\in C}\hat h(P)\leq 2\alpha\sqrt{\hat H({\mathcal C})},$ where $$\alpha=2+\sqrt 3$$. With $$\mu({\mathcal C})=\#\{\sigma\in{\mathcal A};\sigma Q=Q\}$$ for $$Q\in{\mathcal C}$$ one gets $\left|\#\{P\in{\mathcal C};\hat h(P)\leq B\}-{1\over\mu({\mathcal C})}\log_ \alpha{B^ 2\over 4\hat H({\mathcal C})}\right|\leq 4$ if $$B^ 2\geq 2\sqrt{\hat H({\mathcal C})}$$. For any ample divisor $$D$$ of $$\text{Pic}(S)$$ there is an approximation $\#\{P\in{\mathcal C};\;h_ D(P)\leq B\}={1\over\mu({\mathcal C})}\log_ \alpha{B^ 2\over 4\hat H({\mathcal C})}+O(1),\;B\to\infty.$
$\#\{P\in S(K);\;\hat h(P)\leq B\}=S(K)_{\text{fin}}+\sum\left\{{1\over\mu({\mathcal C})}\log_ \alpha{B^ 2\over 4\hat H({\mathcal C})}+\delta({\mathcal C})\right\},$ where $$S(K)_{\text{fin}}=\{P\in S(K);\;\hat h(P)=0\}=\{P\in S(K)$$; $${\mathcal C}(P)$$ is finite}, $$\#S(K)_{\text{fin}}$$ is finite $$|\delta({\mathcal C})|\leq 4$$, and the sum runs over all chains $${\mathcal C}$$ of $$S(K)$$ with $$0<4\hat H({\mathcal C})\leq B^ 2$$. If the chain $${\mathcal C}(P)$$ of a point $$P\in S(\overline K)$$ is $$\text{Gal}(\overline K/K)$$-stable then it holds that $$P\in S(K)$$, $${\mathcal C}(P)$$ is $$K$$-rational, $${\mathcal C}(P)$$ is finite and $$[K(P):K]=2$$. After the proofs the author presents in section 5 explicit calculations of heights of points in a chain on a randomly chosen $$K3$$ surface defined over $$\mathbb{Q}$$. In the last section the calculations are connected with a $$K3$$ version of a recent conjecture of Vojta generalizing a famous theorem of Siegel on the growth of logarithmic heights of integral points on elliptic curves.

### MSC:

 14J28 $$K3$$ surfaces and Enriques surfaces 14G05 Rational points 14G40 Arithmetic varieties and schemes; Arakelov theory; heights 14G27 Other nonalgebraically closed ground fields in algebraic geometry 14C22 Picard groups
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