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

Let X-center dot= (X, D-X) be a pointed stable curve of topological type (gX, nX) over an algebraically closed field of characteristic p > 0. Under certain assumptions, we prove that, if X-center dot is component-generic, then the first generalized Hasse-Witt invariant of every prime-to- p cyclic admissible covering of X-center dot attains maximum. This result generalizes a result of S. Nakajima concerning the ordinariness of prime-to- p cyclic etale 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-center dot is an arbitrary pointed stable curve, then there exists a prime-to-p cyclic admissible covering of X-center dot whose first generalized Hasse-Witt invariant attains maximum. This result generalizes a result ofM. Raynaud concerning the new-ordinariness of prime-top cyclic etale 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.