On 2-adic deformations

Abstract

We compute the versal deformation ring of a split generic $2$-dimensional representation $\chi_1\oplus \chi_2$ of the absolute Galois group of $\mathbb{Q}_p$. As an application, we show that the Breuil--M\'ezard conjecture for both non-split extensions of $\chi_1$ by $\chi_2$ and $\chi_2$ by $\chi_1$ implies the Breuil--M\'ezard conjecture for $\chi_1\oplus \chi_2$. The result is new for $p=2$, the proof works for all primes.