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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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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 (Barsotti–Tate) 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.
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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