Modular Ordinary Differential Equations on ${\rm SL}(2,\mathbb{Z})$ of Third Order and Applications
Journal
SIGMA 18 (2022), 013, 50 pages
Journal Volume
18
Date Issued
2022
Author(s)
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 Pl\"ucker infinitesimal formulas for holomorphic maps from a
Riemann surface to $\mathbb{CP}^N$.
Subjects
modular differential equations; quasimodular forms; Toda system; GENERALIZED LAME EQUATION; VERTEX OPERATOR-ALGEBRAS; FORMS; CLASSIFICATION; GEOMETRY; Mathematics - Number Theory; Mathematics - Number Theory
Publisher
NATL ACAD SCI UKRAINE, INST MATH
Type
journal article