On 2-adic deformations

We compute the versal deformation ring of a split generic 2-dimensional representation χ 1 ⊕ χ 2 of the absolute Galois group of 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...

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Bibliographic Details
Published in:Mathematische Zeitschrift 2017-08, Vol.286 (3-4), p.801-819
Main Author: Paškūnas, Vytautas
Format: Article
Language:English
Subjects:
Deformation
Mathematics
Mathematics and Statistics
Proving
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Summary:We compute the versal deformation ring of a split generic 2-dimensional representation χ 1 ⊕ χ 2 of the absolute Galois group of 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 ⊕ χ 2 . The result is new for p = 2 , the proof works for all primes.
ISSN:0025-5874
1432-1823
DOI:10.1007/s00209-016-1785-8