Locally analytic vectors and rings of periods

Abstract

In this paper, we try to extend Berger’s and Colmez’s point of view, using locally analytic vectors in order to generalize classical cyclotomic theory, in higher rings of periods. We also explain how the formalism of locally analytic vectors recovers the ring ${\mathbf{B}}_{\mathrm{Sen}}$ of Colmez, and extends to Sen theory in the de Rham case, and to classical $(\phi ,\Gamma )$-modules theory. We explain what happens when we try to generalize constructions of $(\phi ,\Gamma )$-modules to arbitrary infinitely ramified $p$-adic Lie extensions, and provide a conjecture on the structure of the locally analytic vectors in the corresponding rings. We also highlight the fact that the situation should be very different, depending on wether the $p$-adic Lie extension “contains a cyclotomic extension” or not. Finally, we explain how some of these constructions may be related to the construction of a ring of trianguline periods.