Differential equations satisfied by modular forms and $K3$ surfaces

Abstract

We study differential equations satisfied by modular forms of two variables associated to $\Gamma_1\times \Gamma_2$, where $\Gamma_i$ ($i=1,2$) are genus zero subgroups of $SL_2(\R)$ commensurable with $SL_2(\Z)$, e.g., $\Gamma_0(N)$ or $\Gamma_0(N)^*$ for some $N$. In some examples, these differential equations are realized as the Picard-Fuchs 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 \cite{LY1}, \cite{LY2}.