"Scheme-theoretic images" of morphisms of stacks

We give criteria for certain morphisms from an algebraic stack to a (not necessarily algebraic) stack to admit an (appropriately defined) scheme-theoretic image. We apply our criteria to show that certain natural moduli stacks of local Galois representations are algebraic (or Ind-algebraic) stacks.Comment: 132 pages; final version, to appear in Algebraic Geometr

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

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.Comment: v3. Minor changes; final version; the title is modifie