On the averages of generalized Hasse–Witt invariants of pointed stable curves in positive characteristic

Abstract

In the present paper, we study fundamental groups of curves in positive characteristic. Let $X^{\bullet }$ be a pointed stable curve of type $(g_{X}, n_{X})$ over an algebraically closed field of characteristic $p>0$, $\Gamma _{X^{\bullet }}$ the dual semi-graph of $X^{\bullet }$, and $\Pi _{X^{\bullet }}$ the admissible fundamental group of $X^{\bullet }$. In the present paper, we study a kind of group-theoretical invariant $\text {Avr}_{p}(\Pi _{X^{\bullet }})$ associated to the isomorphism class of $\Pi _{X^{\bullet }}$ called the limit of p-averages of $\Pi _{X^{\bullet }}$, which plays a central role in the theory of anabelian geometry of curves over algebraically closed fields of positive characteristic. Without any assumptions concerning $\Gamma _{X^{\bullet }}$, we give a lower bound and a upper bound of $\text {Avr}_{p}(\Pi _{X^{\bullet }})$. In particular, we prove an explicit formula for $\text {Avr}_{p}(\Pi _{X^{\bullet }})$ under a certain assumption concerning $\Gamma _{X^{\bullet }}$ which generalizes a formula for $\text {Avr}_{p}(\Pi _{X^{\bullet }})$ obtained by Tamagawa. Moreover, if $X^{\bullet }$ is a component-generic pointed stable curve, we prove an explicit formula for $\text {Avr}_{p}(\Pi _{X^{\bullet }})$ without any assumptions concerning $\Gamma _{X^{\bullet }}$, which can be regarded as an averaged analogue of the results of Nakajima, Zhang, and Ozman–Pries concerning p-rank of abelian étale coverings of projective generic curves for admissible coverings of component-generic pointed stable curves.