On the Existence, Geometry and $p$-Ranks of Vertical Fibers of Coverings of Curves (Algebraic Number Theory and Related Topics 2015)

"Algebraic Number Theory and Related Topics 2015". November 30 - December 4, 2015. edited by Hiroki Takahashi, Yasuo Ohno and Takahiro Tsushima. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.Let R be a complete DVR with algebraically closed residue field of characteristic p> 0 and X, Y stable curves over R with smooth generic fibers. Let f : Y arrow X be a morphism over R such that the morphism of generic fibers induced by f is a Galois étale covering. A closed point x of X is called a vertical point if dim f^{-1}(x) = 1 . In this case, f^{-1}(x) is called the vertical fiber associated to x . We study the existence, the geometry, and the p-ranks of vertical fibers under certain assumptions