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

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 $(\varphi ,N,L/K,E)$-modules.

Final version. To appear in Trans. A.M.S