For p=2, the support of patched modules meets every irreducible component of the potentially semi-stable deformation ring, yielding the Breuil-Mezard conjecture in the case where the residual representation is a twist of an extension of 1 by 1.
On the density of supercuspidal points of fixed regular weight in local deformation rings and global Hecke algebras
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abstract
We study the Zariski closure of points in local deformation rings corresponding to potential semi-stable representations with certain prescribed $p$-adic Hodge theoretic properties. We show in favourable cases that the closure is equal to a union of irreducible components of the deformation space. We also study an analogous question for global Hecke algebras.
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On the modularity of 2-adic potentially semi-stable deformation rings
For p=2, the support of patched modules meets every irreducible component of the potentially semi-stable deformation ring, yielding the Breuil-Mezard conjecture in the case where the residual representation is a twist of an extension of 1 by 1.