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

TL;DR

This paper establishes a geometric link between a specific vector bundle on the moduli stack of elliptic curves and de Rham cohomology, enabling the calculation of homotopy groups of a spectrum related to topological modular forms.

Contribution

It introduces a novel geometric interpretation of the vector bundle associated to the Hopf element and computes the homotopy groups of the spectrum of topological quasimodular forms.

Findings

The vector bundle on $M_{ell}$ is isomorphic to the de Rham cohomology sheaf.

Homotopy groups of $ ext{tmf} / u$ are explicitly calculated.

The Adams-Novikov spectral sequence is related to the cohomology of cubic curves.

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 $\mathrm{H}^1_\mathrm{dR}(\mathcal{E}/M_\mathrm{ell})$ 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.