### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 013, 50 pages      arXiv:2106.12438      https://doi.org/10.3842/SIGMA.2022.013

### Modular Ordinary Differential Equations on ${\rm SL}(2,\mathbb{Z})$ of Third Order and Applications

Zhijie Chen a, Chang-Shou Lin b and Yifan Yang c
a) Department of Mathematical Sciences, Yau Mathematical Sciences Center, Tsinghua University, Beijing, 100084, China
b) Center for Advanced Study in Theoretical Sciences, National Taiwan University, Taipei 10617, Taiwan
c) Department of Mathematics, National Taiwan University and National Center for Theoretical Sciences, Taipei 10617, Taiwan

Received June 24, 2021, in final form February 13, 2022; Published online February 22, 2022

Abstract
In this paper, we study third-order modular ordinary differential equations (MODE for short) of the following form $y'''+Q_2(z)y'+Q_3(z)y=0$, $z\in\mathbb{H}=\{z\in\mathbb{C} \,|\,\operatorname{Im}z>0 \}$, where $Q_2(z)$ and $Q_3(z)-\frac12 Q_2'(z)$ are meromorphic modular forms on ${\rm SL}(2,\mathbb{Z})$ of weight $4$ and $6$, respectively. We show that any quasimodular form of depth $2$ on ${\rm SL}(2,\mathbb{Z})$ leads to such a MODE. Conversely, we introduce the so-called Bol representation $\hat{\rho}\colon {\rm SL}(2,\mathbb{Z})\to{\rm SL}(3,\mathbb{C})$ for this MODE and give the necessary and sufficient condition for the irreducibility (resp. reducibility) of the representation. We show that the irreducibility yields the quasimodularity of some solution of this MODE, while the reducibility yields the modularity of all solutions and leads to solutions of certain ${\rm SU}(3)$ Toda systems. Note that the ${\rm SU}(N+1)$ Toda systems are the classical Plücker infinitesimal formulas for holomorphic maps from a Riemann surface to $\mathbb{CP}^N$.

Key words: modular differential equations; quasimodular forms; Toda system.

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