Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

Request Permissions   Purchase Content 
 
 

 

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [BDFL10a] Alina Bucur, Chantal David, Brooke Feigon, and Matilde Lalín, Fluctuations in the number of points on smooth plane curves over finite fields, J. Number Theory 130 (2010), no. 11, 2528-2541. MR 2678860 (2011f:11076), https://doi.org/10.1016/j.jnt.2010.05.009
  • [BDFL10b] Alina Bucur, Chantal David, Brooke Feigon, and Matilde Lalín, Statistics for traces of cyclic trigonal curves over finite fields, Int. Math. Res. Not. IMRN 5 (2010), 932-967. MR 2595014 (2011c:11100), https://doi.org/10.1093/imrn/rnp162
  • [BDFL11] Alina Bucur, Chantal David, Brooke Feigon, and Matilde Lalín, Biased statistics for traces of cyclic $ p$-fold covers over finite fields, WIN--women in numbers, Fields Inst. Commun., vol. 60, Amer. Math. Soc., Providence, RI, 2011, pp. 121-143. MR 2777802 (2012j:11130)
  • [BDFLS12] Alina Bucur, Chantal David, Brooke Feigon, Matilde Lalín, and Kaneenika Sinha, Distribution of zeta zeroes of Artin-Schreier covers, Math. Res. Lett. 19 (2012), no. 6, 1329-1356. MR 3091611, https://doi.org/10.4310/MRL.2012.v19.n6.a12
  • [BK12] Alina Bucur and Kiran S. Kedlaya, The probability that a complete intersection is smooth, J. Théor. Nombres Bordeaux 24 (2012), no. 3, 541-556 (English, with English and French summaries). MR 3010628
  • [CWZ15] GilYoung Cheong, Melanie Matchett Wood, and Azeem Zaman, The distribution of points on superelliptic curves over finite fields, Proc. Amer. Math. Soc. 143 (2015), no. 4, 1365-1375. MR 3314052, https://doi.org/10.1090/S0002-9939-2014-12218-0
  • [Del74] Pierre Deligne, La conjecture de Weil. I, Inst. Hautes Études Sci. Publ. Math. 43 (1974), 273-307 (French). MR 0340258 (49 #5013)
  • [DS94] Persi Diaconis and Mehrdad Shahshahani, On the eigenvalues of random matrices, J. Appl. Probab. 31A (1994), 49-62. Studies in applied probability. MR 1274717 (95m:60011)
  • [Ent12] Alexei Entin, On the distribution of zeroes of Artin-Schreier L-functions, Geom. Funct. Anal. 22 (2012), no. 5, 1322-1360. MR 2989435, https://doi.org/10.1007/s00039-012-0192-5
  • [EW15] Daniel Erman and Melanie Matchett Wood, Semiample Bertini theorems over finite fields, Duke Math. J. 164 (2015), no. 1, 1-38. MR 3299101, https://doi.org/10.1215/00127094-2838327
  • [FR10] Dmitry Faifman and Zeév Rudnick, Statistics of the zeros of zeta functions in families of hyperelliptic curves over a finite field, Compos. Math. 146 (2010), no. 1, 81-101. MR 2581242 (2011b:11124), https://doi.org/10.1112/S0010437X09004308
  • [Gar05] Arnaldo Garcia, On curves over finite fields, Arithmetic, geometry and coding theory (AGCT 2003), Sémin. Congr., vol. 11, Soc. Math. France, Paris, 2005, pp. 75-110 (English, with English and French summaries). MR 2182838 (2006g:11123)
  • [Kat87] Nicholas M. Katz, On the monodromy groups attached to certain families of exponential sums, Duke Math. J. 54 (1987), no. 1, 41-56. MR 885774 (88i:11053), https://doi.org/10.1215/S0012-7094-87-05404-4
  • [KS99] Nicholas M. Katz and Peter Sarnak, Random matrices, Frobenius eigenvalues, and monodromy, American Mathematical Society Colloquium Publications, vol. 45, American Mathematical Society, Providence, RI, 1999. MR 1659828 (2000b:11070)
  • [KR09] Pär Kurlberg and Zeév Rudnick, The fluctuations in the number of points on a hyperelliptic curve over a finite field, J. Number Theory 129 (2009), no. 3, 580-587. MR 2488590 (2009m:14029), https://doi.org/10.1016/j.jnt.2008.09.004
  • [Mon94] Hugh L. Montgomery, Ten lectures on the interface between analytic number theory and harmonic analysis, CBMS Regional Conference Series in Mathematics, vol. 84, published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1994. MR 1297543 (96i:11002)
  • [PZ12] Rachel Pries and Hui June Zhu, The $ p$-rank stratification of Artin-Schreier curves, Ann. Inst. Fourier (Grenoble) 62 (2012), no. 2, 707-726 (English, with English and French summaries). MR 2985514, https://doi.org/10.5802/aif.2692
  • [Ros02] Michael Rosen, Number theory in function fields, Graduate Texts in Mathematics, vol. 210, Springer-Verlag, New York, 2002. MR 1876657 (2003d:11171)
  • [GV92] Gerard van der Geer and Marcel van der Vlugt, Reed-Muller codes and supersingular curves. I, Compositio Math. 84 (1992), no. 3, 333-367. MR 1189892 (93k:14038)
  • [Woo12] Melanie Matchett Wood, The distribution of the number of points on trigonal curves over $ \mathbb{F}_q$, Int. Math. Res. Not. IMRN 23 (2012), 5444-5456. MR 2999148
  • [Xio10a] Maosheng Xiong, The fluctuations in the number of points on a family of curves over a finite field, J. Théor. Nombres Bordeaux 22 (2010), no. 3, 755-769 (English, with English and French summaries). MR 2769344 (2012a:11085)
  • [Xio10b] Maosheng Xiong, Statistics of the zeros of zeta functions in a family of curves over a finite field, Int. Math. Res. Not. IMRN 18 (2010), 3489-3518. MR 2725502 (2011g:11177), https://doi.org/10.1093/imrn/rnq015
  • [Xio15] Maosheng Xiong, Distribution of zeta zeroes for abelian covers of algebraic curves over a finite field, J. Number Theory 147 (2015), 789-823. MR 3276354, https://doi.org/10.1016/j.jnt.2014.08.008
  • [Zhu] Hui June Zhu,
    Some families of supersingular Artin-Schreier curves in characteristic $ >2.$
    Preprint, arXiv:0809.0104.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 11G20, 11M50, 14G15

Retrieve articles in all journals with MSC (2010): 11G20, 11M50, 14G15


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

American Mathematical Society