Actions infinit\'esimales dans la correspondance de Langlands locale p-adique

TL;DR

This paper explores the infinitesimal actions in the p-adic Langlands correspondence for GL_2(Q_p), linking differential equations, Sen polynomials, and the structure of associated Galois representations.

Contribution

It establishes a new connection between the infinitesimal action on locally analytic vectors and the differential equations of Fontaine and Berger, providing a new proof of Colmez's theorem.

Findings

Linked infinitesimal actions to Fontaine-Berger differential equations

Connected Sen polynomial properties to locally algebraic vectors

Provided a new proof of Colmez's theorem on potential semi-stability

Abstract

Let V be a two-dimensional absolutely irreducible p-adic Galois representation and let Pi be the p-adic Banach space representation associated to V via Colmez's p-adic Langlands correspondence. We establish a link between the infinitesimal action of GL_2(Q_p) on the locally analytic vectors of Pi, the differential equation associated to V via the theory of Fontaine and Berger, and the Sen polynomial of V. This answers a question of Harris and gives a new proof of a theorem of Colmez: Pi has nonzero locally algebraic vectors if and only if V is potentially semi-stable with distinct Hodge-Tate weights.