$p$-torsion for unramified Artin--Schreier covers of curves

TL;DR

This paper investigates the structure of the p-torsion subgroup schemes of Jacobians of unramified Artin--Schreier covers of curves over fields of characteristic p, revealing restrictions imposed by the Galois-module structure.

Contribution

It provides a detailed analysis of the Galois-module structure of J_Y[p] and establishes restrictions based on the known structure of J_X[p], including implications for Ekedahl--Oort types.

Findings

Determined the Galois-module structure of J_Y[p] for unramified Artin--Schreier covers.

Identified restrictions on J_Y[p] imposed by the structure of J_X[p].

Connected the Galois-module structure to Ekedahl--Oort types.

Abstract

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 $J_Y[p]$, analyze the Galois-module structure of $J_Y[p]$, and find restrictions this structure imposes on $J_Y[p]$ (for example, as manifested in its Ekedahl--Oort type) taking $J_X[p]$ as given.