Representability of formal group laws by oriented ring spectra

Determine which formal group laws are representable by classical complex-oriented ring spectra and, in the motivic setting, by GL-oriented ring spectra; i.e., classify the formal group laws that arise as the associated formal group laws of such oriented ring spectra.

Background

In discussing representability of formal ternary laws (FTL) via symplectically oriented motivic ring spectra, the authors highlight that, beyond grading constraints, the problem of determining representable laws is largely open.

They point out that even for the classical case of formal group laws (FGL) in topology and their GL-oriented motivic analogues, it remains unclassified which FGLs occur as those of oriented ring spectra. Resolving this would clarify the landscape of orientations and associated characteristic classes across homotopy-theoretic contexts.