Maximum generalized Hasse–Witt invariants and their applications to anabelian geometry

Abstract

Let $X^{\bullet }=(X, D_{X})$ be a pointed stable curve of topological type $(g_{X}, n_{X})$ over an algebraically closed field of characteristic $p>0$. Under certain assumptions, we prove that, if $X^{\bullet }$ is component-generic, then the first generalized Hasse–Witt invariant of every prime-to-p cyclic admissible covering of $X^{\bullet }$ attains maximum. This result generalizes a result of S. Nakajima concerning the ordinariness of prime-to-p cyclic étale coverings of smooth projective generic curves to the case of (possibly ramified) admissible coverings of (possibly singular) pointed stable curves. Moreover, we prove that, if $X^{\bullet }$ is an arbitrary pointed stable curve, then there exists a prime-to-p cyclic admissible covering of $X^{\bullet }$ whose first generalized Hasse–Witt invariant attains maximum. This result generalizes a result of M. Raynaud concerning the new-ordinariness of prime-to-p cyclic étale coverings of smooth projective curves to the case of (possibly ramified) admissible coverings of (possibly singular) pointed stable curves. 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. Those results generalize A. Tamagawa’s results concerning an anabelian formula for topological types and reconstructions of field structures associated to inertia subgroups of marked points of smooth pointed stable curves to the case of arbitrary pointed stable curves.