We prove R = T theorems for certain reducible residual Galois representations. We answer in the positive a question of Gross and Lubin oil whether certain Hecke algebras 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 G and H of order p over an arbitrarily tamely ramified discrete valuation ring admit all extension not killed by p.
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