## 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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