On the ring of cooperations for 2-primary connective topological modular forms

We analyze the ring (Formula presented.) of cooperations for the connective spectrum of topological modular forms (at the prime 2) through a variety of perspectives: (1) the (Formula presented.) -term of the Adams spectral sequence for (Formula presented.) admits a decomposition in terms of (Formula presented.) groups for (Formula presented.) -Brown–Gitler modules, (2) the image of (Formula presented.) in (Formula presented.) admits a description in terms of 2-variable modular forms, and (3) modulo (Formula presented.) -torsion, (Formula presented.) injects into a certain product of copies of (Formula presented.), for various values of (Formula presented.). We explain how these different perspectives are related, and leverage these relationships to give complete information on (Formula presented.) in low degrees. We reprove a result of Davis–Mahowald–Rezk, that a piece of (Formula presented.) gives a connective cover of (Formula presented.), and show that another piece gives a connective cover of (Formula presented.). To help motivate our methods, we also review the existing work on (Formula presented.), the ring of cooperations for (2-primary) connective (Formula presented.) -theory, and in the process give some new perspectives on this classical subject matter.