Let L be an even hyperbolic lattice and denote by $\mathcal{F}_L$ the moduli space of L-polarized K3 surfaces. This parametrizes K3 surfaces $X$ together with a primitive embedding of lattices $L \hookrightarrow \mathrm{NS}(X)$ and, when $L = \langle 2d \rangle $, one recovers the classical moduli spaces of 2d-polarized K3 surfaces. In this talk, I will introduce a simple criterion to decide whether a given $\overline{ \mathbb{Q}}$-point of  $\mathcal{F}_L$ has generic Néron-Severi lattice (that is, $\mathrm{NS}(X) \cong L$). The criterion is of arithmetic nature and only uses properties of covering maps between Shimura varieties.