Rice University

## When

## Where

**Title:** Components of the Moduli Stack of Galois Representations

**Abstract:** The Emerton-Gee stack for GL_2 serves as a moduli space for 2-dimensional representations of the absolute Galois group of K, where K is a finite, unramified extension of Q_p. This stack is of significant interest because it is expected to play the role of the stack of L-parameters in the conjectural categorical p-adic Langlands correspondence for GL_2(K). In this talk, I will present recent joint work with Kalyani Kansal, where we determine which of the irreducible components of the Emerton-Gee stack are smooth. Among the non-smooth components, we also identify those which are normal or Cohen-Macaulay. This allows us to show that the normalization of every component has fairly mild (resolution-rational) singularities. The talk will begin with a review of Galois representations and modular forms, followed by a discussion of key ideas in the construction of the Emerton-Gee stack. Finally, I will describe how our results update expectations about the categorical p-adic Langlands conjecture.