## Illinois Journal of Mathematics

- Illinois J. Math.
- Volume 51, Number 2 (2007), 667-696.

### Differential equations satisfied by modular forms and $K3$ surfaces

Yifan Yang and Noriko Yui

#### 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}.

#### Article information

**Source**

Illinois J. Math., Volume 51, Number 2 (2007), 667-696.

**Dates**

First available in Project Euclid: 13 November 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.ijm/1258138437

**Digital Object Identifier**

doi:10.1215/ijm/1258138437

**Mathematical Reviews number (MathSciNet)**

MR2342682

**Zentralblatt MATH identifier**

1211.11053

**Subjects**

Primary: 11F23: Relations with algebraic geometry and topology

Secondary: 11F11: Holomorphic modular forms of integral weight 14D05: Structure of families (Picard-Lefschetz, monodromy, etc.) 14J28: $K3$ surfaces and Enriques surfaces 33C70: Other hypergeometric functions and integrals in several variables

#### Citation

Yang, Yifan; Yui, Noriko. Differential equations satisfied by modular forms and $K3$ surfaces. Illinois J. Math. 51 (2007), no. 2, 667--696. doi:10.1215/ijm/1258138437. https://projecteuclid.org/euclid.ijm/1258138437