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) x Gamma(2), where Gamma(i) (i = 1, 2) are genus zero subgroups of SL2(R) commensurable with SL2(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 [14], [15].