We use a new idea to construct a theory of iterated Coleman functions on overconvergent spaces with good reduction in any
dimension. A Coleman function in this theory consists of a unipotent differential equation, a functional on the underlying
bundle and a solution to the equation on a residue class. The new idea is to use the theory of Tannakian categories and the
action of Frobenius to analytically continue solutions of the differential equation to all residue classes.
Mathematics Subject Classification (2000): 11S80, 11G25, 14F30, 14G22
Received: 22 November 2000 / Revised version: 28 March 2001 / Published online: 24 September 2001