Rank two filtered $(\phi, N)$-modules with Galois descent data and coefficients

TL;DR

This paper classifies two-dimensional F-semistable Galois representations over p-adic fields by explicitly describing associated filtered $(, N, L/K, E)$-modules, incorporating Galois descent data and coefficients.

Contribution

It provides a complete classification of rank two filtered $(, N, L/K, E)$-modules with Galois descent data, extending previous work to include coefficients and descent structures.

Findings

Explicit list of isomorphism classes of rank two weakly admissible modules

Classification includes Galois descent data and coefficients

Framework for understanding two-dimensional Galois representations

Abstract

Let $K$ be any finite extension of $Q_{p}$, $L$ any finite Galois extension of $K$ and $E$ any finite large enough coefficient field containing $L$. We classify two-dimensional, F-semistable $E$-representations of $G_{K}$, by listing the isomorphism classes of rank two weakly admissible filtered $(\phi,N,L/K,E)$-modules.