On Density of Modular Points in Pseudo-Deformation Rings

Abstract

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 Böckle 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.