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arXiv · 2101.07946

Every $BT_1$ group scheme appears in a Jacobian

Rachel Pries·Douglas Ulmer
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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.

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Rachel Pries, Douglas Ulmer. 2021-01-20. Every $BT_1$ group scheme appears in a Jacobian. https://arxiv.org/abs/2101.07946

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