Abstract
We prove $R = {\mathbf T}$ theorems for certain reducible residual Galois representations. We answer in the positive a question of Gross and Lubin on whether certain Hecke algebras ${\mathbf T}$ are discrete valuation rings. In order to prove these results we determine (using the theory of Breuil modules) when two finite flat group schemes ${\mathscr G}$ and ${\mathscr H}$ of order p over an arbitrarily tamely ramified discrete valuation ring admit an extension not killed by p .
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