Density of certain classes of potentially crystalline representations in local and global Galois deformation rings

Density of certain classes of potentially crystalline representations in local and global Galois deformation rings

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Matthew Emerton, University of Chicago
Fine Hall 214

In this talk I will explain some results (joint with Vytas Paskunas) showing that certain classes of potentially crystalline representations (e.g. in the case of two-dimensional representations: crystabelline potentially Barsotti--Tate representations, or potentially Barsotti--Tate representations of supercuspidal type) are Zariski dense in local or global Galois deformation space. The arguments combine methods from the p-adic representation theory of p-adic reductive groups with techniques from the p-adic Langlands program, which allow one to relate representation theory of p-adic groups to Galois representations. (Related results have been proved by Hellmann and Schraen.)