Semi-stable deformation rings in even Hodge–Tate weights: The residually reducible case

Abstract

Let [Formula: see text] be a prime number and [Formula: see text] a positive even integer less than [Formula: see text]. In this paper, we find the strongly divisible modules corresponding to the Galois stable lattices in each 2-dimensional semi-stable non-crystalline representation of [Formula: see text] with Hodge–Tate weights [Formula: see text] whose mod-[Formula: see text] reductions are corresponding to nontrivial extensions of two distinct characters. We use these results to construct the irreducible components of the semi-stable deformation rings in Hodge–Tate weights [Formula: see text] of the non-split reducible residual representations of [Formula: see text].