The algebraic Novikov spectral sequence for topological modular forms

We compute the $\mathbb{C}$-motivic Adams spectral sequence for $mmf\mathrm{/}\tau$. Up to reindexing, this spectral sequence is isomorphic to the algebraic Novikov spectral sequence for topological modular forms. We give a full analysis of the inclusion and projection maps which occur in the long exact sequence induced by multiplication by $\tau$ on Adams $E_2$-pages. Our computation is totally algebraic and can be used to obtain the $E_2$-page of the Adams-Novikov spectral sequence for topological modular forms in a way that is independent from previous computations using the elliptic curves Hopf algebroid.