Families of $(\varphi, \tau)$-modules and Galois representations

Abstract:Let $K$ be a finite extension of $\mathbb{Q}_p$. The theory of
$(\varphi, \Gamma)$-modules constructed by Fontaine provides a good
category to study $p$-adic representations of the absolute Galois group
$Gal(\bar{K}/K)$. This theory arises from a ``devissage'' of the extension
$\bar{K}/K$ through an intermediate extension $K_{\infty}/K$ which is the
cyclotomic extension of $K$. The notion of $(\varphi, \tau)$-modules
generalizes Fontaine's constructions by using Kummer extensions other than
the cyclotomic one. It is desirable to establish properties of $(\varphi,
\tau)$-modules parallel to the cyclotomic case. In this talk, we explain
construction of a functor that associates to a family of $p$-adic Galois
representations a family of $(\varphi, \tau)$-modules, analogous to a
construction of Berger and Colmez in the $(\varphi, \Gamma)$-modules case.
This is joint work with L\'{e}o Poyeton.