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–Mézard conjecture for both non-split extensions of \(\chi _1\) by \(\chi _2\) and \(\chi _2\) by \(\chi _1\) implies the Breuil–Mézard conjecture for \(\chi _1\oplus \chi _2\). The result is new for \(p=2\), the proof works for all primes.
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Acknowledgments
A large part of the paper was written while visiting Michael Spieß at the University of Bielefeld. I thank SFB 701 for generous support and for providing an excellent working environment during my visit.
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Paškūnas, V. On 2-adic deformations. Math. Z. 286, 801–819 (2017). https://doi.org/10.1007/s00209-016-1785-8
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DOI: https://doi.org/10.1007/s00209-016-1785-8