Local model theory for non-generic tame potentially Barsotti--Tate deformation rings

We develop a local model theory for moduli stacks of $2$-dimensional non-scalar tame potentially Barsotti--Tate Galois representations of the Galois group of an unramified extension of $\mathbb{Q}_p$. We derive from this explicit presentations of potentially Barsotti--Tate deformation rings, allowing us to prove structural results about them, and prove various conjectures formulated by Caruso--David--M\'ezard.
Comment: Various typos fixed