On density of modular points in pseudo-deformation rings

TL;DR

This paper investigates the structure of pseudo-deformation rings associated with reducible Galois representations and establishes their relation to Hecke algebras, extending prior results and applying to modular forms and Galois cohomology.

Contribution

It proves an isomorphism between the reduced pseudo-deformation ring and the local Hecke algebra component under certain conditions, extending B"ockle's results to residually reducible cases.

Findings

Maximal reduced quotient of pseudo-deformation ring is isomorphic to local Hecke algebra component.

Application to the structure of Hecke algebras modulo p.

Results on non-optimal levels of newforms lifting residual representations.

Abstract

Given a continuous, odd, reducible and semi-simple $2$-dimensional representation $\bar\rho_0$ of $G_{\mathbb{Q},Np}$ over a finite field of odd characteristic $p$, we study the relation between the universal deformation ring of the pseudo-representation corresponding to $\bar\rho_0$ (pseudo-deformation ring) and the big $p$-adic Hecke algebra to prove that the maximal reduced quotient of the pseudo-deformation ring is isomorphic to the local component of the big $p$-adic Hecke algebra corresponding to $\bar\rho_0$ if a certain global Galois cohomology group has dimension $1$. This partially extends the results of B\"{o}ckle to the case of residually reducible representations. We give an application of our main theorem to the structure of Hecke algebras modulo $p$. As another application of our methods and results, we prove a result about non-optimal levels of newforms lifting $\bar\rho_0$ in the spirit of Diamond-Taylor. This also gives a partial answer to a conjecture of Billerey-Menares.