Modular Ordinary Differential Equations on SL(2, ℤ) of Third Order and Applications

Abstract

In this paper, we study third-order modular ordinary differential equations (MODE for short) of the following form 𝑦′′′ + 𝑄₂(𝓏)𝑦′ + 𝑄₃(𝓏)𝑦 = 0, 𝓏 ∈ ℍ = {𝓏 ∈ ℂ | Im𝓏 > 0}, where 𝑄₂(𝓏) and 𝑄₃(𝓏) − 1/2𝑄′₂(𝓏) are meromorphic modular forms on SL(2, ℤ) of weight 4 and 6, respectively. We show that any quasimodular form of depth 2 on SL(2, ℤ) leads to such a MODE. Conversely, we introduce the so-called Bol representation 𝜌^: SL(2, ℤ) → SL(3, 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 SU(3) Toda systems. Note that the SU(𝑁 + 1) Toda systems are the classical Plücker infinitesimal formulas for holomorphic maps from a Riemann surface to ℂℙᴺ.