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We study the Zariski closure of points in local deformation ringscorresponding to potential semi-stable representations with certain prescribed$p$-adic Hodge theoretic properties. We show in favourable cases that theclosure is equal to a union of irreducible components of the deformation space.We also study an analogous question for global Hecke algebras.

On the density of supercuspidal points of fixed regular weight in local deformation rings and global Hecke algebras