  In order to give a formal treatment of differential equations in positive
characteristic p, it is necessary to use divided powers. One runs into an
analog problem in the theory of q-difference equations when q is a pth root of
unity. We introduce here a notion of twisted divided powers (relative to q) and
show that one can recover the twisted Weyl algebra and obtain a twisted
p-curvature map that describes the center of the twisted Weyl algebra. We also
build a divided p-Frobenius that will give, by duality, a formal Azumaya
splitting of the twisted Weyl algebra as well as a twisted Simpson
correspondence.
