{FAMILIES OF GALOIS REPRESENTATIONS AND ($\phi$, $\tau$ )-MODULES}

Let p be a prime, and let K be a finite extension of Qp, with absolute Galois group GK. Let π be a uniformizer of K and let K∞ be the Kummer extension obtained by adjoining to K a system of compatible p n-th roots of π, for all n, and let L be the Galois closure of K∞. Using these extensions, Caruso has constructed étale (φ, τ)-modules, which classify p-adic Galois representations of K. In this paper, we use locally analytic vectors and theories of families of φmodules over Robba rings to prove the overconvergence of (φ, τ)-modules in families. As examples, we also compute some explicit families of (φ, τ)-modules in some simple cases.