On $BT_1$ group schemes and Fermat Jacobians

Abstract

PRIES AND DOUGLAS ULMER A. 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. Our main result is that every BT 1 scheme group over k occurs as a direct factor of the p-torsion group scheme of the Jacobian of an explicit curve defined over F p . To prove this, we give a careful account of three classifications of BT 1 group schemes, due in large part to Kraft, Ekedahl, and Oort, and we apply these classifications to study the p-torsion group schemes of Jacobians of Fermat curves.

Every $BT_1$ group scheme appears in a Jacobian

“…Remark. In [PU20], we give a more refined analysis and compute the number of a ∈ S with w a = w for any w, p, and ℓ. It turns out that the restriction on p in part (2) is essential.…”

“…Our arguments use parts of three classifications of BT 1 group schemes (in terms of words, the "canonical filtration", and permutations), largely due to Kraft, Ekedahl, and Oort. In a companion paper [PU20], we provide a complete translation between these classifications, and we apply them to give a detailed study of the p-torsion subgroups of Jacobians of Fermat curves, including well-known invariants like the p-rank and a-number, as well as two other invariants related to supersingular elliptic curves.…”

“…Oort proved [Oor01, § §2, 5, 9] (see also [PU20,Cor. 4.2.3]) that any self-dual BT 1 module can be given a unique polarization: 2.5.…”