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Variation of Néron-Severi ranks of reductions of \(K3\) surfaces. (English) Zbl 1311.14039

Let \(X\) be a \(K3\) surface defined over a number field \(k\). Suppose that for a finite place \(\mathfrak{p}\) of \(k\), \(X\) has a good reduction \(X_{\mathfrak{p}}\). There is a natural specialization homomorphism that induces an inequality of geometric Picard numbers of \(X\) and \(X_{\mathfrak{p}}\): \(\rho(\overline{X}) \leq \rho(\overline{X}_{\mathfrak{p}})\). In fact, if we define \(\eta(\overline{X})\) by \[ \eta(\bar{X}) := \begin{cases} 0 & \text{if } E_X \text{ is a CM-field or } \dim_{E_X} T_X \text{ is even},\\ [E_X: \mathbb{Q}] & \text{if } E_X \text{ is a totally real field and } \dim_{E_X} T_X \text{ is odd}, \end{cases} \] where \(T_X\) is the transcendental lattice of \(X\) and \(E_X\) is the endomorphism algebra of the Hodge structure underlying \(T_X\), then, we have \(\rho(\overline{X}) + \eta(\overline{X})\leq \rho(\overline{X}_{\mathfrak{p}})\). Let \[ \Pi_{\text{jump}} := \left\{ \mathfrak{p} \, | \, \rho(\overline{X}) +\eta(\overline{X}) < \rho(\overline{X}_{\mathfrak{p}}) \right\} \] and \[ \gamma(X, B) := \frac{\# \{ p\leq B \text{ and } p\in \Pi_{\text{jump}}\}}{\# \{ p\leq B\}}. \] The aim of the paper under review is to study the amount \(\gamma(X, B)\).
The authors calculate \(\rho(\overline{X})\) and \(\rho(\overline{X}_p)\) for \(2<p<2^{16}\) and \(\gamma(X, B)\) for \(B<2^{16}\), and then plot the results \(\gamma(X, B)\) for several quartic \(K3\) surfaces \(X\) using Kedlaya’s algorithm by studying Frobenius action on some cohomology group throughout the reduction process. With this computer-aided computation experiments, in the Main Result, the authors get evidences that \(\displaystyle\lim \sup _{B\to \infty} \gamma(X, B)\) is at least \(1/2\) if \(\rho(\overline{X}) = 2\), and that \(\rho(\overline{X}_p)\) jumps with probability proportional to \(1/\sqrt{p}\) if \(\rho(\overline{X}) = 1\) and \(E_X = \mathbb{Q}\).

MSC:

14J28 \(K3\) surfaces and Enriques surfaces
14C22 Picard groups
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References:

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