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–Mézard.