Generic ordinarity for abelian coverings of the projective line

We show that abelian coverings of the projective line of order prime to $p$ are generically $\mu$-ordinary in characteristic $p$.

The images of the irreducible components of Hurwitz spaces of abelian coverings of the projective line by the Torelli morphism lie in some Shimura varieties.

The stratification by Newton polygons of these varieties is known, and we show that the generic Newton polygon for the Hurwitz space coincides with the generic (or $\mu$-ordinary) Newton polygon of the smallest Shimura variety that contains its image.

In order to do this, we compute the generic Newton polygons for $L$-functions associated to multiplicative character sums over the projective line.