On $BT_1$ group schemes and Fermat Jacobians

TL;DR

This paper compares different classifications of $BT_1$ group schemes over algebraically closed fields of characteristic $p$, applies these to Fermat quotient curves, and computes their $p$-torsion invariants.

Contribution

It provides a detailed comparison of three classification methods for $BT_1$ group schemes and applies this to determine invariants of Fermat quotient curves.

Findings

Determined the Ekedahl--Oort types of Fermat quotient curves.

Computed four invariants of the $p$-torsion group schemes of these curves.

Established connections between classifications and geometric properties.

Abstract

Let $p$ be a prime number and let $k$ be an algebraically closed field of characteristic $p$. A $BT_1$ group scheme over $k$ is a finite commutative group scheme which arises as the kernel of $p$ on a $p$-divisible (Barsotti--Tate) group. We compare three classifications of $BT_1$ group schemes, due in large part to Kraft, Ekedahl, and Oort, and defined using words, canonical filtrations, and permutations. Using this comparison, we determine the Ekedahl--Oort types of Fermat quotient curves and we compute four invariants of the $p$-torsion group schemes of these curves.