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

TL;DR

This paper extends Sen's theory to semi-stable p-adic Galois representations by constructing a new subspace with a derivation that relates to monodromy and captures geometric variation properties.

Contribution

It introduces a subspace D_{π-Sen}(V) with a derivation for semi-stable representations, linking monodromy and Hodge-theoretic variations.

Findings

Constructed D_{π-Sen}(V) with a derivation abla^{(π)}.

Linked the derivation's action to monodromy operator N.

Described infinitesimal variations of Hodge structures in the geometric case.

Abstract

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.