Prolongations of $t$-Motives and Algebraic Independence of Periods

Abstract

In this article we show that the coordinates of a period lattice generator of the n -th tensor power of the Carlitz module are algebraically independent, if n is prime to the characteristic. The main part of the paper, however, is devoted to a general construction for t -motives which we call prolongation, and which gives the necessary background for our proof of the algebraic independence. Another incredient is a theorem which shows hypertranscendence for the Anderson-Thakur function \omega(t) , i.e. that \omega(t) and all its hyperderivatives with respect to t are algebraically independent.