The Kodaira dimension of some moduli spaces of elliptic K3 surfaces

We study the moduli spaces of elliptic K3 surfaces of Picard number at least 3, i.e. $U\oplus ⟨-2k⟩$-polarized K3 surfaces. Such moduli spaces are proved to be of general type for $k\ge 220$. The proof relies on the low-weight cusp form trick developed by Gritsenko, Hulek and Sankaran. Furthermore, explicit geometric constructions of some elliptic K3 surfaces lead to the unirationality of these moduli spaces for $k<11$ and for 19 other isolated values up to $k=64$.