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 encapsulates the important notion of Breuil-Kisin modules among others. It is thus desirable to establish properties of $(\varphi, \tau)$
-modules parallel to the cyclotomic case. In this talk, we explain the construction of a functor that associates to a family of $p$
-adic Galois representations a family of $(\varphi, \tau)$
-modules. The analogous functor in the $(\varphi, \Gamma)$
-modules case was constructed by Berger and Colmez . This is joint work with Leo Poyeton.