#### Vol. 298, No. 2, 2019

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Semistable deformation rings in even Hodge–Tate weights

### Lucio Guerberoff and Chol Park

Vol. 298 (2019), No. 2, 299–374
##### 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 ${G}_{{ℚ}_{p}}$ 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 ${G}_{{ℚ}_{p}}$.

##### Keywords
semistable representations, strongly divisible modules, Breuil modules, semistable deformation rings
Primary: 11F80