Every $BT_1$ group scheme appears in a Jacobian

TL;DR

This paper proves that every $BT_1$ group scheme over an algebraically closed field of characteristic $p$ can be realized as a direct factor of the $p$-torsion of a Jacobian of an explicit curve over $F_p$, extending the understanding of the structure of Jacobians.

Contribution

The paper demonstrates that all $BT_1$ group schemes appear in Jacobians, providing explicit constructions and using classification, cohomology, and combinatorial methods.

Findings

Every $BT_1$ group scheme occurs as a direct factor of a Jacobian's $p$-torsion.

Explicit curves over $F_p$ realize all $BT_1$ schemes.

The approach combines Kraft classification, Oda's theorem, and Fermat curve cohomology.

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. 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 $\mathbb{F}_p$. We also treat a variant with polarizations. Our main tools are the Kraft classification of $BT_1$ group schemes, a theorem of Oda, and a combinatorial description of the de Rham cohomology of Fermat curves.