Non-$p$-ordinary smooth abelian covers of ${\mathbb{P}}^1$

Abstract

Given a family of odd abelian covers of ${\mathbb{P}}^1$ and a prime $p$ of good reduction, in [1], under some assumptions, we computed the generic Newton polygon (resp. Ekedahl-Oort type) in the family (called $p$-ordinary). In this paper, we investigate the existence of non-$p$-ordinary smooth curves in the family. In particular, under some restrictions, we show that when $p$ is sufficiently large, the complement of the $p$-ordinary locus is always nonempty. For $1$-dimensional families satisfying additional auxiliary conditions, we obtain a lower bound for the number of non-$p$-ordinary smooth curves. In specific instances, the above general statement can be improved; as an example, for families of covers of degree at most 7, we establish the non-emptiness of certain non-$p$-ordinary Newton/Ekedahl-Oort strata (called almost $p$-ordinary). Our method relies on further study of the extended Hasse-Witt matrix introduced by Moonen in [2], and initiated in [1], and on known results about the geometry of the mod-$p$ reduction of Shimura varieties of PEL type.