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

We show that the vector bundle on the moduli stack $M_{\mathrm{e}\mathrm{l}\mathrm{l}}$ of elliptic curves associated to the $2$-cell complex $C\nu$ is isomorphic to the de Rham cohomology sheaf ${\mathrm{H}}_{\mathrm{d}\mathrm{R}}^1(\mathcal{E}\mathrm{/}{M}_{\mathrm{e}\mathrm{l}\mathrm{l}})$ of the universal elliptic curve $\mathcal{E}\to M_{\mathrm{e}\mathrm{l}\mathrm{l}}$. We use this to calculate the homotopy groups of the ${\mathbf{E}}_1$-quotient $$\mathrm{t}\mathrm{m}\mathrm{f}$ /!!/ \nu$of$$\mathrm{t}\mathrm{m}\mathrm{f}$$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.