2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/102077We study differential equations satisfied by modular forms associated to $Γ_1\timesΓ_2$, where $Γ_i (i=1,2)$ are genus zero subgroups of $SL_2(\mathbf R)$ commensurable with $SL_2(\mathbf Z)$, e.g., $Γ_0(N)$ or $Γ_0(N)^*$. In some examples, these differential equations are realized as the Picard--Fuch differential equations of families of K3 surfaces with large Picard numbers, e.g., $19, 18, 17, 16$. Our method rediscovers some of the Lian--Yau examples of ``modular relations'' involving power series solutions to the second and the third order differential equations of Fuchsian type in [14, 15].Some revisions are incorporated, in particular, replaced the terminology ''bi-modular'' by ''modular''Number TheoryAlgebraic Geometry11F03, 11F11, 14D05, 14J28text