Non-abelian cohomology of universal curves in positive characteristic

Abstract

In this paper, we will compute the non-abelian cohomology of the universal complete curve in positive characteristic. This extends Hain’s result on the non-abelian cohomology of generic curves in characteristic zero to positive characteristics. Furthermore, we will prove that the exact sequence of etale fundamental groups of the universal n n -punctured curve in positive characteristic does not split.