Deformation Conditions for Pseudorepresentations

Abstract

Given a property of representations satisfying a basic stability condition, Ramakrishna developed a variant of Mazur’s Galois deformation theory for representations with that property. We introduce an axiomatic definition of pseudorepresentations with such a property. Among other things, we show that pseudorepresentations with a property enjoy a good deformation theory, generalizing Ramakrishna’s theory to pseudorepresentations.

THE EISENSTEIN IDEAL OF WEIGHT k AND RANKS OF HECKE ALGEBRAS

“…The results from [26] that we use are about the reducibility properties of the T m -valued pseudo-representation ([26, Theorem 5.1.1]) and the index of Eisenstein ideal in T 0 m ([26, Theorem 5.1.2]). Note that, in [28], Wake and Wang-Erickson work with pseudo-representations which are finite flat at p (a notion that they define and study in [27]). But since this condition is not present in weight k > 2, we work with pseudo-representations that are ordinary at p and recover the results of Wake and Wang-Erickson mentioned above using them.…”

Adjoint Selmer groups of automorphic Galois representations of unitary type

“…In fact, these are equivalent, but both definitions are useful. We will prove a new result about the deformation theory of pseudocharacters (Proposition 2.9) using Lafforgue's point of view, while we follow [WWE19] in using Chenevier's definition to impose deformation conditions on pseudocharacters.…”

“…The second innovation is the formulation, by Wake and Wang-Erickson [WWE19], of functors of pseudodeformations satisfying deformation conditions (e.g. conditions arising from p-adic Hodge theory).…”

Potential automorphy over CM fields

“…Remark 4.4.7. Ideas similar to, and more general than, those used in the proof above were developed by Wake-Wang-Erickson [WW17].…”