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

TL;DR

This paper proves the overconvergence of families of $(, au)$-modules associated with $p$-adic Galois representations over a finite extension of $Q_p$, using locally analytic vectors and Robba ring theories, with explicit examples.

Contribution

It establishes the overconvergence property of $(, au)$-modules in families and provides explicit computations in simple cases.

Findings

Overconvergence of $(, au)$-modules in families is proven.

Explicit examples of $(, au)$-modules are computed.

The use of locally analytic vectors and Robba rings is key to the results.

Abstract

Let $p$ be a prime, and let $K$ be a finite extension of $\mathbf{Q}_p$, with absolute Galois group $\cal{G}_K$. 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 \'etale $(\varphi,\tau)$-modules, which classify $p$-adic Galois representations of $K$. In this paper, we use locally analytic vectors and theories of families of $\varphi$-modules over Robba rings to prove the overconvergence of $(\varphi,\tau)$-modules in families. As examples, we also compute some explicit families of $(\varphi,\tau)$-modules in some simple cases.