  Let $Y\to X$ be an unramified Galois cover of curves over a perfect field $k$
of characteristic $p>0$ with $\mathrm{Gal}(Y/X)\cong\mathbb{Z}/p\mathbb{Z}$,
and let $J_X$ and $J_Y$ be the Jacobians of $X$ and $Y$ respectively. We
consider the $p$-torsion subgroup schemes $J_X[p]$ and $J_Y[p]$. The three main
themes are: the Galois-module structure of $J_Y[p]$; restrictions this
structure imposes on $J_Y[p]$ (for example, as manifested in its Ekedahl--Oort
type) taking $J_X[p]$ as given; and methods for explicitly computing the group
schemes $J_X[p]$ and $J_Y[p]$.
