#### Number Theory Seminar

##### Venue: Microsoft Teams (Online)

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.

Contact: +91 (80) 2293 2711, +91 (80) 2293 2265 ;     E-mail: chair.math[at]iisc[dot]ac[dot]in
Last updated: 03 Feb 2023