On the Vinberg Family of K3 Surfaces

We study orthogonal modular forms associated with moduli spaces of lattice-polarized K3 surfaces whose generic transcendental lattices are of the form $T = H \oplus H \oplus L(-1)$ where $L$ is a root lattice of type $A_n$ or $D_n$. In Picard numbers $10$ through $17$, we use explicit Jacobian elliptic fibrations to construct modular forms on type IV domains associated with orthogonal groups $\mathrm{O}^+(T)$. We show that the coefficients of suitable Weierstrass models naturally realize generators for the corresponding graded rings of orthogonal modular forms.