New York Journal of Mathematics
Volume 27 (2021), 705-739


Rachel Pries and Douglas Ulmer

On BT1 group schemes and Fermat curves

view    print

Published: April 25, 2021.
Keywords: Curve, finite field, Jacobian, abelian variety, Fermat curve, Frobenius, Verschiebung, group scheme, Ekedahl--Oort type, de Rham cohomology, Dieudonné module.
Subject: Primary 11D41, 11G20, 14F40, 14H40, 14L15; Secondary 11G10, 14G17, 14K15, 14H10.

Let p be a prime number and let k be an algebraically closed field of characteristic p. A BT1 group scheme over k is a finite commutative group scheme which arises as the kernel of p on a p-divisible (Barsotti--Tate) group. We compare three classifications of BT1 group schemes, due in large part to Kraft, Ekedahl, and Oort, and defined using words, canonical filtrations, and permutations. Using this comparison, we determine the Ekedahl--Oort types of Fermat quotient curves and we compute four invariants of the p-torsion group schemes of these curves.


Author RP was partially supported by NSF grant DMS-1901819, and author DU was partially supported by Simons Foundation grants 359573 and 713699. We would like to thank the referee for a thoughtful and very fast report.

Author information

Rachel Pries:
Department of Mathematics
Colorado State University
Fort Collins, CO 80523, USA


Douglas Ulmer:
Department of Mathematics
University of Arizona
Tucson, AZ 85721, USA