Generalization of the theory of Sen in the semi-stable representation case

For a semi-stable representation V, we will construct a subspace D_{\pi-Sen}(V) of C_p\otimes_{Q_p}V endowed with a linear derivation \nabla^{(\pi)}. The action of \nabla^{(\pi)} on D_{\pi-Sen}(V) is closely related to the action of the monodromy operator N on D_{st}(V). Furthermore, in the geometric case, the action of \nabla^{(\pi)} on D_{\pi-Sen}(V) describes an analogy of the infinitesimal variations of Hodge structures and satisfies formulae similar to the Griffiths transversality and the local monodromy theorem.