Jinyue Luo : Criteria for pseudorepresentations to arise from genuine representations

Posted by Frederick Saia , part of the Number Theory Seminar.

At

Feb. 21, 2025, 1 p.m.

In

636 SEO

Abstract

In the study of connections between Galois representations and modular forms, one often seeks an R=T theorem, which asserts that the deformation ring R is isomorphic to a (localized) Hecke algebra T. However, sometimes only the framed deformation ring exists. With the framing variables, it is obviously larger than the Hecke algebra. Pseudorepresentations, which is a generalization of the notion of traces of representations, was invented to get around this issue. We will introduce the notion of pseudorepresentations and discuss the criteria for pseudorepresentations to arise from genuine representations. Next, we will introduce the algorithm used to explicitly compute usual deformation rings and pseudodeformation rings for finitely presented groups, which leads to the discovery of a counterexample.