The $\mathbb{C}$-motivic Adams-Novikov spectral sequence for topological modular forms

We analyze the $\mathbb{C}$-motivic (and classical) Adams-Novikov spectral sequence for the $\mathbb{C}$-motivic modular forms spectrum $mmf$ (and for the classical topological modular forms spectrum $tmf$). We primarily use purely algebraic techniques, with a few exceptions. Along the way, we settle a previously unresolved detail about the multiplicative structure of the homotopy groups of $tmf$.