title: q-difference equations and p-adic monodromy (joint work with Y. Andre')

abstract: After some a quick introduction to complex q-difference euqation theory, we are going to discuss p-adic monodromy of q-difference equations. We will explain how it allows for the construction of a canonical deformation functor for differential equation with strong Frobenius structure. We will give a particular attention to the case of q-difference/differential equation of rang 1, in which the functor can be described completely explicitly.