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

Abstract

Let p be a prime number and r a positive even integer less than p-1. 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 Gal(Q¯p/Qp) with Hodge-Tate weights (0,r) whose mod-p 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 (0,r) of the non-split reducible residual representations of Gal(Q¯p/Qp). © 2022 World Scientific Publishing Company.