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Register a new userTorsions in Cohomology of $text{SL}_2(mathbb{Z})$ and Congruence of Modular Forms
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Taiwang Deng
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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 $text{SL}_2(mathbb{Z})$ as a module over the torsion invariants. As application, we obtain various congruences between cuspidal forms of level one and Eisenstein series.
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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 $p geq 5$.
Generalizing a result of cite{Z1991, CPZ} about elliptic modular forms, we give a closed formula for the sum of all Hilbert Hecke eigenforms over a totally real number field with strict class number $1$, multiplied by their period polynomials, as a single product of the Kronecker series.
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 $mathbb{CP}^N$.
Let $F$ be a totally real field in which $p$ is unramified. We prove that, if a cuspidal overconvergent Hilbert cuspidal form has small slopes under $U_p$-operators, then it is classical. Our method follows the original cohomological approach of Coleman. The key ingredient of the proof is giving an explicit description of the Goren-Oort stratification of the special fiber of the Hilbert modular variety. A byproduct of the proof is to show that, at least when $p$ is inert, of the rigid cohomology of the ordinary locus has the same image as the classical forms in the Grothendieck group of Hecke modules.
We establish an isomorphism between certain complex-valued and vector-valued modular form spaces of half-integral weight, generalizing the well-known isomorphism between modular forms for $Gamma_0(4)$ with Kohnens plus condition and modular forms for the Weil representation associated to the discriminant form for the lattice with Gram matrix $(2)$. With such an isomorphism, we prove the Zagier duality and write down the Borcherds lifts explicitly.
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