CMI-IMSc Number Theory Seminar
Date: Tuesday, 22 April 2025
Time: 3:30 to 4:30 PM
Venue: Seminar Hall, CMI
Cohomology of families of $(\varphi, \tau)$-modules
Aditya Karnataki
Chennai Mathematical Institute.
22-04-25

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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 a family of $(\varphi, \tau)$-modules to a family of $p$-adic Galois representations over rigid analytic spaces. In this talk, we describe a complex of $(\varphi, \tau)$-modules that explicitly computes the Galois cohomology of these families. This is joint work with Anand Chitrao and Jishnu Ray.