Maximums of generalized Hasse-Witt invariants and their applications to anabelian geometry

TL;DR

This paper proves that the generalized Hasse-Witt invariants of certain coverings of pointed stable curves reach their maximum, enabling group-theoretic reconstruction of geometric data and advancing the understanding of moduli spaces in anabelian geometry.

Contribution

It establishes the maximum of generalized Hasse-Witt invariants for prime-to-p cyclic coverings and derives an anabelian formula for the topological type, with applications to fundamental group theory.

Findings

Maximum generalized Hasse-Witt invariants attained for coverings.

Group-theoretic reconstruction of field structures from fundamental groups.

Applications to moduli spaces of fundamental groups.

Abstract

Let $(X, D_{X})$ be an arbitrary pointed stable curve of topological type $(g_{X}, n_{X})$ over an algebraically closed field of characteristic $p>0$. We prove that the generalized Hasse-Witt invariants of prime-to-$p$ cyclic admissible coverings of $(X, D_{X})$ attain maximum. As applications, we obtain an anabelian formula for $(g_{X}, n_{X})$, and prove that the field structures associated to inertia subgroups of marked points can be reconstructed group-theoretically from open continuous homomorphisms of admissible fundamental groups. Moreover, the formula for maximum generalized Hasse-Witt invariants and the result concerning reconstructions of field structures play important roles in the theory of moduli spaces of fundamental groups developed by the author of the present paper.