We compute the mod $2$ homology of the spectrum $\mathrm{ tmf}$ of topological modular forms by proving a 2-local equivalence $\mathrm{ tmf} \wedge DA(1) \simeq \mathrm{ tmf}_1(3) \simeq BP\left \langle 2\right\rangle$, where $DA(1)$ is an eight cell complex whose cohomology "doubles" the subalgebra $\mathcal{ A}(1)$ of the Steenrod algebra generated by $\mathrm{ Sq}^1$ and $\mathrm{ Sq}^2$. To do so, we give, using the language of stacks, a modular description of the elliptic homology of $DA(1)$ via level three structures. We briefly discuss analogs at odd primes and recover the stack-theoretic description of the Adams-Novikov spectral sequence for $\mathrm{ tmf}$.
Categories: math.AT math.AG