There are two explicit constructions of topological modular forms. One is in the 12th section of the book $Topological$ $Modular$ $Forms$, and the other is Lurie's Elliptic Cohomology II in his homepage. More precisely speaking, they both defined the presheaf of $\mathbb{E}_\infty$-ring spectra over the moduli stack of elliptic curves $\mathcal{M}_\mathrm{Ell}$ or $\overline{\mathcal{M}_\mathrm{Ell}}$.
In the first book, the editor Behrens studied the sheaf over the $p$-completion part of the moduli stack first and then tried to glue them together by virtue of some arithmetic data. Then when he wanted to solve the first problem, he also considered ordinary elliptic curves and supersingular ones separately.
Lurie's way is much more direct. He introduced the notion of orientation in the derived world and could thus construct some $\mathbb{E}_\infty$-ring spectra for nice elliptic curves, which means that he gave us the local section. The only thing he needed to do was he should prove that his definition satisfied our basic assumptions.
I believe both ways are right. But I am now curious about the relationship between these two ways. Could I say that the procedure of Lurie's verification is almost the same as Behrens' decomposition? Simultaneously, how did Lurie avoid the difference between ordinary curves and supersingular curves? I didn't study these things for a long time, so maybe I made a mistake or put up a trivial question. I would appreciate it if I could get a meaningful answer.