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If one has a Galois representation $\overline{\rho}: G_{\mathbb{Q}} \rightarrow GL_2(\mathbb{Z}/p \mathbb{Z})$ where $\overline{\rho} = \left( \begin{array}{cc} \phi & * \\ 0 & 1 \\ \end{array} \right)$ where $\phi^2 \ne 1$, then one cannot have $SL_2$ inside the image. To construct a deformation to $GL_2(\mathbb{Z}_p)$ or a similar big ring of characteristic zero, what condition would one need to impose on the image of $\overline{\rho}$?

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  • $\begingroup$ If $*$ were zero then you could use a Teichmuller lift. In general you could ask that $*$ mapped to zero in the $p$-torsion of $H^2$ of the Teichmuller lift of $\psi$ I guess; then the element of $H^1$ it represented would lift. $\endgroup$ – znt Mar 12 '16 at 21:20
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The conditions are detailed in the main theorem of Hamblen and Ramakrishna " Deformations of Certain Reducible Galois Representations 2" HR. Roughly, $\bar{\rho}$ simply needs to be indecomposable, $p$ must be atleast $3$, $\phi^2\neq 1$, $\phi$ is not the mod $p$ cyclotomic character and its inverse, some conditions on $\bar{\rho}_{\restriction G_{\mathbb{Q}_p}}$ in both the even and odd cases. Then there exists a finitely ramified and irreducible lift to $GL_2(\mathbb{Z}_p)$ (assuming that the residual repn is to $GL_2(\mathbb{F}_p)$). If additionally $\bar{\rho}$ is odd, the lift will be ordinary at $p$ and thus also come from an eigenform as a consequence of Skinner-Wiles (Fontaine-Mazur in the reducible case). You will however have a lift to an eigenform of perhaps larger weight than what you may want (and never get weight $2$ lifts). The level of the lift is also not optimal. Since reducible representations of the kind we are talking about naturally arise from class group data, these lifting theorems are of potential interest to questions about class groups of number fields.

If you're asking for lifts to $\mathbb{Z}_p[|U|]$-algebras, you may wish to regard this modular lift as lying on a Hida-Family. There is some interest in describing deformations to Hida families in the reducible case which have some nice properties R.

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