Every $B{T}_1$ group scheme appears in a Jacobian

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. Our main result is that every $B{T}_1$ 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 $B{T}_1$ group schemes, a theorem of Oda, and a combinatorial description of the de Rham cohomology of Fermat curves.

13 pages. This paper is derived from arxiv:2010.15160 which has been divided and streamlined