Mathematics Seminar
Date: Wednesday, 13 November 2024
Time: 3:30 PM
Venue: Seminar Hall
Families of $(\varphi, \tau)$-modules
Aditya Karnataki
Chennai Mathematical Institute.
13-11-24

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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 thus desirable to establish properties of $(\varphi, \tau)$-modules parallel to the cyclotomic case. In joint work with L\'{e}o Poyeton, we constructed 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. In this talk, we will explain this construction and indicate some future directions that are the content of ongoing work with Anand Chitrao and Jishnu Ray.