On $B{T}_1$ group schemes and Fermat Jacobians

Abstract

Let $p$ be a prime number and let $k$ be an algebraically closed field of characteristic $p$. A $B{T}_1$ 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 $B{T}_1$ 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.

v1: 38 pages. v2: Universality result split off as arXiv:2101.07946 and remainder streamlined. 26 pages