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.


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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