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Statistics for ordinary Artin-Schreier covers and other $ p$-rank strata


Authors: Alina Bucur, Chantal David, Brooke Feigon and Matilde Lalín
Journal: Trans. Amer. Math. Soc. 368 (2016), 2371-2413
MSC (2010): Primary 11G20; Secondary 11M50, 14G15
DOI: https://doi.org/10.1090/tran/6410
Published electronically: July 10, 2015
MathSciNet review: 3449243
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Abstract: We study the distribution of the number of points and of the zeroes of the zeta function in different $ p$-rank strata of Artin-Schreier covers over $ \mathbb{F}_q$ when $ q$ is fixed and the genus goes to infinity. The $ p$-rank strata considered include the ordinary family, the whole family, and the family of covers with $ p$-rank equal to $ p-1.$ While the zeta zeroes always approach the standard Gaussian distribution, the number of points over $ \mathbb{F}_q$ has a distribution that varies with the specific family.


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Additional Information

Alina Bucur
Affiliation: Department of Mathematics, University of California at San Diego, 9500 Gilman Drive $#$0112, La Jolla, California 92093
Email: alina@math.ucsd.edu

Chantal David
Affiliation: Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve West, Montreal, QC H3G 1M8, Canada
Email: cdavid@mathstat.concordia.ca

Brooke Feigon
Affiliation: Department of Mathematics, The City College of New York, CUNY, NAC 8/133, New York, New York 10031
Email: bfeigon@ccny.cuny.edu

Matilde Lalín
Affiliation: Département de Mathématiques et de Statistique, Université de Montréal, CP 6128, succ. Centre-ville, Montreal, QC H3C 3J7, Canada
Email: mlalin@dms.umontreal.ca

DOI: https://doi.org/10.1090/tran/6410
Keywords: Artin-Schreier curves, finite fields, distribution of number of points, distribution of zeroes of $L$-functions of curves.
Received by editor(s): May 5, 2013
Received by editor(s) in revised form: January 13, 2014
Published electronically: July 10, 2015
Additional Notes: The first author was supported by the Simons Foundation #244988 and the UCSD Hellman Fellows Program (2012–2013 Hellman Fellowship)
The second and fourth authors were supported by the Natural Sciences and Engineering Research Council of Canada (Discovery Grant 155635-2008 to the second author, 355412-2008 to the fourth author), and by the Fonds de recherche du Québec - Nature et technologies (144987 to the fourth author, 166534 to the second and fourth authors)
The third author was supported by the National Science Foundation (DMS-1201446) and PSC-CUNY
Article copyright: © Copyright 2015 American Mathematical Society