Differential equations satisfied by modular forms and K3 surfaces

Abstract

We study differential equations satisfied by modular forms associated to ${\Gamma }_1\times {\Gamma }_2$, where ${\Gamma }_i(i=1,2)$ are genus zero subgroups of $S{L}_2(\mathbf{R})$ commensurable with $S{L}_2(\mathbf{Z})$, e.g., ${\Gamma }_0(N)$ or ${\Gamma }_0(N{)}^{\ast }$. 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''

References in corpus (2)

• Mirror Symmetry and Integral Variations of Hodge Structure Underlying One Parameter Families of Calabi-Yau Threefolds
• Monodromy calculatons of fourth order equations of Calabi-Yau type