On 2-adic deformations

We compute the versal deformation ring of a split generic $2$-dimensional representation $χ_1\oplus χ_2$ of the absolute Galois group of $\mathbb{Q}_p$. As an application, we show that the Breuil--Mézard conjecture for both non-split extensions of $χ_1$ by $χ_2$ and $χ_2$ by $χ_1$ implies the Breuil--Mézard conjecture for $χ_1\oplus χ_2$. The result is new for $p=2$, the proof works for all primes.

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