Families of two dimensional modular (, ) -modules

Let F/ Qₚ be a finite unramified extension, let k be a finite extension of the residue field of F. We provide explicit constructions of integral structures for all rank two \'etale Lubin-Tate (, OF^) -modules over k. We construct algebraic families of such integral structures and show that these comprehensively reflect the degeneration behaviour of (, OF^) -modules. These results reveal new combinatorial structures of the moduli stack of (, OF^) -modules, and allow us, in particular, to rederive the fact that the Serre weights assigned to a two dimensional Gal (F/F) -representation over k can be read off from the geometry of the stack.