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The $ a$-numbers of Jacobians of Suzuki curves


Authors: Holley Friedlander, Derek Garton, Beth Malmskog, Rachel Pries and Colin Weir
Journal: Proc. Amer. Math. Soc. 141 (2013), 3019-3028
MSC (2010): Primary 11G20, 14H40; Secondary 14G50
DOI: https://doi.org/10.1090/S0002-9939-2013-11581-9
Published electronically: May 31, 2013
MathSciNet review: 3068955
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Abstract: For $ m \in {\mathbb{N}}$, let $ S_m$ be the Suzuki curve defined over $ \mathbb{F}_{2^{2m+1}}$. It is well-known that $ S_m$ is supersingular, but the $ p$-torsion group scheme of its Jacobian is not known. The $ a$-number is an invariant of the isomorphism class of the $ p$-torsion group scheme. In this paper, we compute a closed formula for the $ a$-number of $ S_m$ using the action of the Cartier operator on $ H^0(S_m,\Omega ^1)$.


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Additional Information

Holley Friedlander
Affiliation: Department of Mathematics, University of Massachusetts–Amherst, Amherst, Massachusetts 01003
Email: holleyf@math.umass.edu

Derek Garton
Affiliation: Department of Mathematics, University of Wisconsin–Madison, Madison, Wisconsin 53706
Address at time of publication: Department of Mathematics, Northwestern University, Evanston, Illinois 60208
Email: garton@math.wisc.edu

Beth Malmskog
Affiliation: Department of Mathematics, Wesleyan University, Middletown, Connecticut 06457
Address at time of publication: Department of Mathematics and Statistics, Colorado College, Colorado Springs, Colorado 80946
Email: emalmskog@wesleyan.edu

Rachel Pries
Affiliation: Department of Mathematics, Colorado State University, Fort Collins, Colorado 80521
Email: pries@math.colostate.edu

Colin Weir
Affiliation: Department of Mathematics and Statistics, University of Calgary, Calgary, AB, Canada T2N 1N4
Email: cjweir@ucalgary.ca

DOI: https://doi.org/10.1090/S0002-9939-2013-11581-9
Keywords: Suzuki curve, maximal curve, Jacobian, p-torsion, $a$-number
Received by editor(s): November 2, 2011
Received by editor(s) in revised form: November 29, 2011
Published electronically: May 31, 2013
Additional Notes: The third author was partially supported by NSF grant DMS-11-01712
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2013 American Mathematical Society

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