Algebraic independence of the Carlitz period and its hyperderivatives

Abstract

This paper deals with the fundamental period π˜ of the Carlitz module. The main theorem states that the Carlitz period and all its hyperderivatives are algebraically independent over the base field Fq(θ). Our approach also reveals a connection of these hyperderivatives with the coordinates of a period lattice generator of the tensor powers of the Carlitz module which was already observed by M. Papanikolas in a yet unpublished paper. Namely, these coordinates can be obtained by explicit polynomial expressions in π˜ and its hyperderivatives. Papanikolas also gave various presentations of these expressions which we also prove here.