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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,quad zinmathbb{H}={zinmathbb{C} ,|,operatorname{Im}z>0 },] where $Q_2(z)$ and $Q_3(z)-frac12 Q_2(z)$ are meromorphic modular forms on $mathrm{SL}(2,mathbb{Z})$ of weight $4$ and $6$, respectively. We show that any quasimodular form of depth $2$ on $mathrm{SL}(2,mathbb{Z})$ leads to such a MODE. Conversely, we introduce the so-called Bol representation $hat{rho}: mathrm{SL}(2,mathbb{Z})tomathrm{SL}(3,mathbb{C})$ 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 $mathrm{SU}(3)$ Toda systems. Note that the $mathrm{SU}(N+1)$ Toda systems are the classical Plucker infinitesimal formulas for holomorphic maps from a Riemann surface to $mathbb{CP}^N$.
We describe torsion classes in the first cohomology group of $text{SL}_2(mathbb{Z})$. In particular, we obtain generalized Dicksons invariants for p-power polynomial rings. Secondly, we describe torsion classes in the zero-th homology group of $text{
We prove automorphy lifting results for certain essentially conjugate self-dual $p$-adic Galois representations $rho$ over CM imaginary fields $F$, which satisfy in particular that $p$ splits in $F$, and that the restriction of $rho$ on any decomposi
We provide an explicit set of algebraically independent generators for the algebra of invariant differential operators on the Riemannian symmetric space associated with $SL_n(R)$.
By work of Belyi, the absolute Galois group $G_{mathbb{Q}}=mathrm{Gal}(overline{mathbb{Q}}/mathbb{Q})$ of the field $mathbb{Q}$ of rational numbers can be embedded into $A=mathrm{Aut}(widehat{F_2})$, the automorphism group of the free profinite group
Let $Gamma_n(p)$ be the level-$p$ principal congruence subgroup of $text{SL}_n(mathbb{Z})$. Borel-Serre proved that the cohomology of $Gamma_n(p)$ vanishes above degree $binom{n}{2}$. We study the cohomology in this top degree $binom{n}{2}$. Let $mat