Irregular Loci in the Emerton-Gee Stack for GL2

Let K be a finite extension of Qp. The Emerton-Gee stack for GL2 is a stack of etale (phi, Gamma)-modules of rank two. Its reduced part, X, is an algebraic stack of finite type over a finite field, and can be viewed as a moduli stack of two dimensional mod p representations of the absolute Galois group of K. By the work of Caraiani, Emerton, Gee and Savitt, it is known that in most cases, the locus of mod p representations admitting crystalline lifts with specified regular Hodge-Tate weights is an irreducible component of X. Their work relied on a detailed study of a closely related stack of etale phi-modules which admits a map from a stack of Breuil-Kisin modules with descent data. In our work, we assume K is unramfied and further study this map with a view to studying the loci of mod p representations admitting crystalline lifts with small, irregular Hodge-Tate weights. We identify these loci as images of certain irreducible components of the stack of Breuil-Kisin modules and obtain several inclusions of the non-regular loci into the irreducible components of X. This is joint work with Rebecca Bellovin, Neelima Borade, Anton Hilado, Heejong Lee, Brandon Levin, David Savitt and Hanneke Wiersema.



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