Delta characters in positive characteristic and Galois representations

In this article we develop the theory of ?-characters of Anderson modules. Given any Anderson module E (satisfying certain conditions), using the theory of ?-geometry, we construct a canonical z-isocrystal H?(E) with a Hodge-Pink structure. As an application, we show that when E is a Drinfeld module, our constructed z-isocrystal H?(E) is weakly admissible given that a ?-parameter is non-zero. Therefore the equal characteristic analogue of the Fontaine functor associates a local shtuka and hence a crystalline z-adic Galois representation to the ?-geometric object H?(E). It is also well known that there is a natural local shtuka attached to E. In the case of Carlitz modules, we show that the Galois representation associated to H?(E) is indeed the usual one coming from the Tate module. Hence this article further raises the question of how the above two apparently different Galois representations compare with each other for a Drinfeld module of arbitrary rank.