Hodge theory for elliptic curves and the Hopf element $\nu$

Abstract

We show that the vector bundle on the moduli stack $M_{\mathrm{ell}}$ of elliptic curves associated to the $2$-cell complex $C\nu$ is isomorphic to the de Rham cohomology sheaf ${\mathrm{H}}_{\mathrm{dR}}^1(\mathcal{E}/M_{\mathrm{ell}})$ of the universal elliptic curve $\mathcal{E}\to M_{\mathrm{ell}}$. We use this to calculate the homotopy groups of the ${\mathbf{E}}_1$-quotient $\mathrm{tmf}//\nu$ of $\mathrm{tmf}$ by $\nu$, called the spectrum of "topological quasimodular forms", by relating its Adams-Novikov spectral sequence to the cohomology of the moduli stack of cubic curves with a chosen splitting of the Hodge-de Rham filtration.

10 pages; comments very welcome