Let \(X\) be a polarized \(K3\) surface defined over a number field \(k\). Suppose \(X\) has good reduction at a place \(\mathfrak{p}\). Denote by \(X_{\mathfrak{p}}\) the special fibre of a smooth model of \(X\) over the ring of integers of \(k_{\mathfrak{p}}\), by \(\overline{X}:=X\times_{\text{Spec}\, k} \text{Spec}\, \bar{k}\) and by \(\overline{X_{\mathfrak{p}}}\) the base change of \(X_{\mathfrak{p}}\) to the residue field of \(\mathfrak{p}\). Let \(sp: \mathrm{NS}(\overline{X}) \to \mathrm{NS}(\overline{X_{\mathfrak{p}}})\) be the induced specialization of the Néron-Severi groups.
The paper under review concerns two questions: when the map \(sp\) is surjective; and how to compute the geometric Picard number \(\rho(X)\) of \(X\).
Due to a theorem of Tankeev on Mumford-Tate conjecture, let \(T:=\mathrm{NS}(X)^\perp_{H^2(X,\mathbb{Q})}\) be the transcendental part (in a Hodge decomposition) of \(X\), and \(E\) the field of endomorphims of \(T\). The field \(E\) is either CM or totally real due to Zarhin.
The main results are Theorems 1 and 5, where obstruction conditions for the map \(sp\) being avoided to be surjective, lower bound of the geometric Picard number of \(X_{\mathfrak{p}}\), and finally, an algorithm for computation of the geometric Picard number of \(X\) are respectively given in cases on the part \(T\) and the field \(E\).