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

Abstract:

Let $p$ be a prime, and let $K$ be a finite extension of $\mathbf {Q}_p$, with absolute Galois group $\mathcal {G}_K$. Let $\pi$ be a uniformizer of $K$ and let $K_\infty$ be the Kummer extension obtained by adjoining to $K$ a system of compatible $p^n$-th roots of $\pi$, for all $n$, and let $L$ be the Galois closure of $K_\infty$. Using these extensions, Caruso has constructed é tale $(\phi ,\tau )$-modules, which classify $p$-adic Galois representations of $K$. In this paper, we use locally analytic vectors and theories of families of $\phi$-modules over Robba rings to prove the overconvergence of $(\phi ,\tau )$-modules in families. As examples, we also compute some explicit families of $(\phi ,\tau )$-modules in some simple cases.