p-groups, p-rank, and semistable reduction of coverings of curves

We prove various explicit formulas concerning $p$-rank of $p$-coverings of pointed semistable curves over discrete valuation rings. In particular, we obtain a full generalization of Raynaud’s formula for $p$-rank of fibers over nonmarked smooth closed points in the case of arbitrary closed points. As an application, for abelian $p$-coverings, we give an affirmative answer to an open problem concerning boundedness of $p$-rank asked by Saïdi more than twenty years ago.