## Every $BT_1$ group scheme appears in a Jacobian

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- by Rachel Pries and Douglas Ulmer PDF
- Proc. Amer. Math. Soc.
**150**(2022), 525-537 Request permission

## 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 (BarsottiâTate) group. Our main result is that every $BT_1$ group scheme 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.## References

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## Additional Information

**Rachel Pries**- Affiliation: Department of Mathematics, Colorado State University, Fort Collins, Colorado 80523
- MR Author ID: 665775
- Email: pries@math.colostate.edu
**Douglas Ulmer**- Affiliation: Department of Mathematics, University of Arizona, Tucson, Arizona 85721
- MR Author ID: 175900
- ORCID: 0000-0003-1529-4390
- Email: ulmer@math.arizona.edu
- Received by editor(s): January 21, 2021
- Received by editor(s) in revised form: May 9, 2021
- Published electronically: November 4, 2021
- Additional Notes: The first author was partially supported by NSF grant DMS-1901819.

The second author was partially supported by Simons Foundation grants 359573 and 713699. - Communicated by: Matt A. Papanikolas
- © Copyright 2021 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**150**(2022), 525-537 - MSC (2020): Primary 11D41, 11G20, 14F40, 14H40, 14L15; Secondary 11G10, 14G17, 14K15, 14H10
- DOI: https://doi.org/10.1090/proc/15681
- MathSciNet review: 4356165