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Register a new userModular ordinary differential equations on $mathrm{SL}(2,mathbb{Z})$ of third order and applications
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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 $mathbb{CP}^N$.
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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 $widehat{F_2}$ on two generators. The image of $G_{mathbb{Q}}$ lies inside $widehat{GT}$, the Grothendieck-Teichmuller group. While it is known that every abelian representation of $G_{mathbb{Q}}$ can be extended to $widehat{GT}$, Lochak and Schneps put forward the challenge of constructing irreducible non-abelian representations of $widehat{GT}$. We do this virtually, namely by showing that a rich class of arithmetically defined representations of $G_{mathbb{Q}}$ can be extended to finite index subgroups of $widehat{GT}$. This is achieved, in fact, by extending these representations all the way to finite index subgroups of $A=mathrm{Aut}(widehat{F_2})$. We do this by developing a profinite version of the work of Grunewald and Lubotzky, which provided a rich collection of representations for the discrete group $mathrm{Aut}(F_d)$.
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 $mathcal{T}_n(mathbb{Q})$ denote the Tits building of $text{SL}_n(mathbb{Q})$. Lee-Szczarba conjectured that $H^{binom{n}{2}}(Gamma_n(p))$ is isomorphic to $widetilde{H}_{n-2}(mathcal{T}_n(mathbb{Q})/Gamma_n(p))$ and proved that this holds for $p=3$. We partially prove and partially disprove this conjecture by showing that a natural map $H^{binom{n}{2}}(Gamma_n(p)) rightarrow widetilde{H}_{n-2}(mathcal{T}_n(mathbb{Q})/Gamma_n(p))$ is always surjective, but is only injective for $p leq 5$. In particular, we completely calculate $H^{binom{n}{2}}(Gamma_n(5))$ and improve known lower bounds for the ranks of $H^{binom{n}{2}}(Gamma_n(p))$ for $p geq 5$.
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