Carl Erickson (Brandeis University)
Modularity of ordinary Galois representations via coarse moduli

Abstract: Ordinary Galois representations are among those Galois representations expected to arise from modular forms as part of the Langlands philosophy. For Galois representations that are residually irreducible, this follows from various "R=T theorems," where R is a universal ordinary deformation ring for the residual representation and T is an ordinary Hecke algebra. When the residual representation is reducible, the modularity of ordinary Galois representations is still often known, but generally there is no universal ordinary deformation ring R to correspond to T.

This talk will introduce joint work with Preston Wake. We produce a universal deformation ring R for ordinary pseudorepresentations that is a candidate for comparison with T in the residually reducible case. In many cases, we prove that R=T when T is the Eisenstein component of a Hida Hecke algebra, and derive some consequences. We will introduce this notion of ordinary pseudorepresentation, and will explain how it can be thought of as the coarse moduli of ordinary representations.