Abstract

Let $p$ be a prime number and $r$ a positive even integer less than $p-1$. In this paper, we find a Galois stable lattice in each two-dimensional semistable noncrystalline representation of $G_{{\mathbb{Q}}_p}$ with Hodge–Tate weights $\left(0,r\right)$ by constructing the corresponding strongly divisible module. We also compute the Breuil modules corresponding to the mod $p$ reductions of these strongly divisible modules, and determine the semisimplification of the mod $p$ reduction of the original representations. We use these results to construct the irreducible components of the semistable deformation rings in Hodge–Tate weights $\left(0,r\right)$ of the absolutely irreducible residual representations of $G_{{\mathbb{Q}}_p}$.

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Mathematical Subject Classification 2010

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