Twisted divided powers and applications

Abstract 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 p th 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.