On 2-adic deformations

TL;DR

This paper computes the versal deformation ring for certain 2-dimensional Galois representations and demonstrates how the Breuil--Mézard conjecture for extensions implies it for split sums, with new results for p=2.

Contribution

It provides the first computation of the versal deformation ring for split generic 2-dimensional Galois representations at p=2 and links the conjecture's validity for extensions to the split case.

Findings

Computed the versal deformation ring for split generic 2-dimensional Galois representations.

Showed that the Breuil--Mézard conjecture for extensions implies it for split sums.

Established results for all primes, including p=2.

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.