Differential equations satisfied by modular forms and K3 surfaces

Details Differential equations satisfied by modular forms and K3 surfaces Differential equations satisfied by modular forms and K3 surfaces

2005-06-28

arxiv.org/abs/math/0506576v2

Mathematics

Number Theory

Some revisions are incorporated, in particular, replaced the terminology ''bi-modular'' by ''modular''

Scientific paper

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 $SL_2(\mathbf R)$ commensurable with $SL_2(\mathbf Z)$, e.g., $\Gamma_0(N)$ or $\Gamma_0(N)^*$. 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].

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