# Effect of increasing the ramification on pseudo-deformation rings

## TL;DR

This paper explores how increasing ramification affects pseudo-deformation rings of 2-dimensional Galois representations, establishing new relations and properties, including a big R=T theorem, under various conditions.

## Contribution

It extends the understanding of pseudo-deformation rings by analyzing the impact of added ramification and proves new structural results, including non-local complete intersection properties.

## Key findings

- Analogues of Boston and B"ockle's theorems for reduced pseudo-deformation rings
- Universal deformation rings are not local complete intersections when certain conditions hold
- Established a big R=T theorem as an application

## Abstract

Given a continuous, odd, semi-simple $2$-dimensional representation of $G_{\mathbb{Q},Np}$ over a finite field of odd characteristic $p$ and a prime $\ell$ not dividing $Np$, we study the relation between the universal deformation rings of the corresponding pseudo-representation for the groups $G_{\mathbb{Q},N\ell p}$ and $G_{\mathbb{Q},Np}$. As a related problem, we investigate when the universal pseudo-representation arises from an actual representation over the universal deformation ring. Under some hypotheses, we prove analogues of theorems of Boston and B\"{o}ckle for the reduced pseudo-deformation rings. We improve these results when the pseudo-representation is unobstructed and $p$ does not divide $\ell^2-1$. When the pseudo-representation is unobstructed and $p$ divides $\ell+1$, we prove that the universal deformation rings in characteristic $0$ and $p$ of the pseudo-representation for $G_{\mathbb{Q},N\ell p}$ are not local complete intersection rings. As an application of our main results, we prove a big $R=\mathbb{T}$ theorem.