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We develop a local model theory for moduli stacks of $2$-dimensionalnon-scalar tame potentially Barsotti--Tate Galois representations of the Galoisgroup of an unramified extension of $\mathbb{Q}_p$. We derive from thisexplicit presentations of potentially Barsotti--Tate deformation rings,allowing us to prove structural results about them, and prove variousconjectures formulated by Caruso--David--M\'ezard.

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