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

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