On Density of Modular Points in Pseudo-Deformation Rings

Abstract Given a continuous, odd, reducible, and semi-simple $2$-dimensional representation $\bar \rho _{0}$ of $G_{{\mathbb{Q}},Np}$ over finite field odd characteristic $p$, we study the relation between universal deformation ring pseudo-representation corresponding to (pseudo-deformation ring) big $p$-adic Hecke algebra prove that maximal reduced quotient pseudo-deformation is isomorphic local component if certain global Galois cohomology group has dimension $1$. This partially extends results Böckle case residually reducible representations. We give an application our main theorem structure algebras modulo $p$. As another methods results, result about non-optimal levels newforms lifting in spirit Diamond–Taylor. also gives partial answer conjecture Billerey–Menares.