Potentially diagonalizable modular lifts of arbitrarily large weight

Abstract: I will begin this talk by recalling the notion of "potential diagonalizability", due to Barnet-Lamb, Gee, Geraghty and Taylor, and how (and why) this notion appears in Automorphy Lifting Theorems (in the work of the aforementioned authors). Then I will present the main result of this talk, which is joint work with Iván Blanco: existence of modular lifts of arbitrarily large weight (of a given residual modular Galois representations) which are potentially diagonalizable. In the non-ordinary case, the proof of this result requires a combination of local and global results for Galois deformation rings, a local to global principle due to Böckle and a potential variant of the result we want to prove due to Barnet-Lamb, Gee, Geraghty and Taylor. I will also explain how this result is a useful tool in the proof of some cases of Langlands functoriality.

number theoryrepresentation theory

Audience: researchers in the topic

Number Theory and Representations in Valparaiso