Non-$\mu$-ordinary smooth cyclic covers of $\mathbb{P}^1$

TL;DR

This paper explores the existence and abundance of non-$f$-ordinary smooth cyclic covers of the projective line over finite fields, extending understanding of their Newton polygons and Ekedahl--Oort types.

Contribution

It demonstrates the non-emptiness of non-$f$-ordinary loci in families of cyclic covers for large primes and provides bounds on their number, advancing the classification of these curves.

Findings

Non-$f$-ordinary loci are non-empty for large primes.

Lower bounds are established for the number of non-$f$-ordinary curves in certain families.

All codimension 1 non-$f$-ordinary strata are non-empty in specific cases.

Abstract

Given a family of cyclic covers of $\mathbb{P}^1$ and a prime $p$ of good reduction, by [12] the generic Newton polygon (resp. Ekedahl--Oort type) in the family ($\mu$-ordinary) is known. In this paper, we investigate the existence of non-$\mu$-ordinary smooth curves in the family. In particular, under some auxiliary conditions, we show that when $p$ is sufficiently large the complement of the $\mu$-ordinary locus is always non empty, and for $1$-dimensional families with condition on signature type, we obtain a lower bound for the number of non-$\mu$-ordinary smooth curves. In specific examples, for small $m$, the above general statement can be improved, and we establish the non emptiness of all codimension 1 non-$\mu$-ordinary Newton/Ekedahl--Oort strata ({\em almost} $\mu$-ordinary). Our method relies on further study of the extended Hasse-Witt matrix initiated in [12].