Time: 14h30, Saturday, 1/10/2022

Abstract: There are many interesting phenomena in the case of positive or mixed characteristics. For instance, unlike characteristic zero, there exist covers of curves whose number of branch points are different but still lie in the same flat family. In this talk, we briefly discuss the process of showing that a flat equal-characteristic p deformation of a cyclic sub-covering extends to that of the whole tower. The result indicates that the p-fibers of the canonical maps between the moduli space of cyclic coverings and one of the sub-coverings are surjective at any closure. The crucial technique is a study of local covering’s degeneration using Kato-Saito-Abbes’ refined Swan conductor, which generalizes the classical perfect residue case.

If time permits, we will introduce the Hasse-Weil zeta functions for Z/p^n-coverings (of curves) and some results of Wan and his colleagues on their associated Newton polygons.