Modular Ordinary Differential Equations on $\mathrm{S}\mathrm{L}(2,\mathbb{Z})$ of Third Order and Applications

In this paper, we study third-order modular ordinary differential equations (MODE for short) of the following form $y^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}+Q_2(z)y^{\mathrm{\prime }}+Q_3(z)y=0$, $z\in \mathbb{H}=\{z\in \mathbb{C}\mathrm{\mid }\mathrm{Im}z>0\}$, where $Q_2(z)$ and $Q_3(z)-\frac{1}{2}{Q}_2^{\mathrm{\prime }}(z)$ are meromorphic modular forms on ${\rm SL}(2,$\mathbb{Z}$)$ofweight$4$and$6$, respectively. We show that any quasimodular form of depth $2$ on $\mathrm{S}\mathrm{L}(2,\mathbb{Z})$ leads to such a MODE. Conversely, we introduce the so-called Bol representation $\widehat{\rho }\text{\hspace{0.17em}⁣}:\mathrm{S}\mathrm{L}(2,\mathbb{Z})\to \mathrm{S}\mathrm{L}(3,\mathbb{C})$ 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 $\mathrm{S}\mathrm{U}(3)$ Toda systems. Note that the $\mathrm{S}\mathrm{U}(N+1)$ Toda systems are the classical Pl"ucker infinitesimal formulas for holomorphic maps from a Riemann surface to ${\mathbb{C}\mathbb{P}}^N$.