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New maximal curves as ray class fields over Deligne-Lusztig curves

Author: Dane C. Skabelund
Journal: Proc. Amer. Math. Soc. 146 (2018), 525-540
MSC (2010): Primary 11G20; Secondary 14H25
Published electronically: August 30, 2017
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Abstract: We construct new covers of the Suzuki and Ree curves which are maximal with respect to the Hasse-Weil bound over suitable finite fields. These covers are analogues of the Giulietti-Korchmáros curve, which covers the Hermitian curve and is maximal over a base field extension. We show that the maximality of these curves implies that of certain ray class field extensions of each of the Deligne-Lusztig curves. Moreover, we show that the Giulietti-Korchmáros curve is equal to the above-mentioned ray class field extension of the Hermitian curve.

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  • [1] Wieb Bosma, John Cannon, and Catherine Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3-4, 235-265, Computational algebra and number theory (London, 1993). MR 1484478
  • [2] P. Deligne and G. Lusztig, Representations of reductive groups over finite fields, Ann. of Math. (2) 103 (1976), no. 1, 103-161. MR 0393266
  • [3] Abdulla Eid and Iwan Duursma, Smooth embeddings for the Suzuki and Ree curves, Algorithmic arithmetic, geometry, and coding theory, Contemp. Math., vol. 637, Amer. Math. Soc., Providence, RI, 2015, pp. 251-291. MR 3364452
  • [4] Arnaldo Garcia, Henning Stichtenoth, and Chao-Ping Xing, On subfields of the Hermitian function field, Compositio Math. 120 (2000), no. 2, 137-170. MR 1739176
  • [5] Arnaldo Garcia and Saeed Tafazolian, On additive polynomials and certain maximal curves, J. Pure Appl. Algebra 212 (2008), no. 11, 2513-2521. MR 2440263
  • [6] Massimo Giulietti and Gábor Korchmáros, A new family of maximal curves over a finite field, Math. Ann. 343 (2009), no. 1, 229-245. MR 2448446
  • [7] Johan P. Hansen, Deligne-Lusztig varieties and group codes, Coding theory and algebraic geometry (Luminy, 1991), Lecture Notes in Math., vol. 1518, Springer, Berlin, 1992, pp. 63-81. MR 1186416
  • [8] Johan P. Hansen and Jens Peter Pedersen, Automorphism groups of Ree type, Deligne-Lusztig curves and function fields, J. Reine Angew. Math. 440 (1993), 99-109. MR 1225959 (94h:14024)
  • [9] Johan P. Hansen and Henning Stichtenoth, Group codes on certain algebraic curves with many rational points, Appl. Algebra Engrg. Comm. Comput. 1 (1990), no. 1, 67-77. MR 1325513
  • [10] Hans-Wolfgang Henn, Funktionenkörper mit grosser Automorphismengruppe, J. Reine Angew. Math. 302 (1978), 96-115. MR 511696
  • [11] J. W. P. Hirschfeld, G. Korchmáros, and F. Torres, Algebraic curves over a finite field, Princeton Series in Applied Mathematics, Princeton University Press, Princeton, NJ, 2008. MR 2386879 (2008m:14040)
  • [12] Bertram Huppert and Norman Blackburn, Finite groups. III, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 243, Springer-Verlag, Berlin-New York, 1982. MR 662826
  • [13] Yasutaka Ihara, Some remarks on the number of rational points of algebraic curves over finite fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), no. 3, 721-724 (1982). MR 656048
  • [14] Gábor Korchmáros and Fernando Torres, Embedding of a maximal curve in a Hermitian variety, Compositio Math. 128 (2001), no. 1, 95-113. MR 1847666
  • [15] Gilles Lachaud, Sommes d'Eisenstein et nombre de points de certaines courbes algébriques sur les corps finis, C. R. Acad. Sci. Paris Sér. I Math. 305 (1987), no. 16, 729-732. MR 920053
  • [16] Kristin Lauter, Deligne-Lusztig curves as ray class fields, Manuscripta Math. 98 (1999), no. 1, 87-96. MR 1669591
  • [17] M. Montanucci and G. Zini, Some Ree and Suzuki curves are not Galois covered by the Hermitian curve, (2016), Pre-print, arXiv:1603.06706.
  • [18] Jens Peter Pedersen, A function field related to the Ree group, Coding theory and algebraic geometry (Luminy, 1991), Lecture Notes in Math., vol. 1518, Springer, Berlin, 1992, pp. 122-131. MR 1186420 (93k:14036)
  • [19] Henning Stichtenoth, Algebraic function fields and codes, 2nd ed., Springer Publishing Company, Incorporated, 2008.

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Additional Information

Dane C. Skabelund
Affiliation: Department of Mathematics, University of Illinois, Urbana, Illinois 61801

Received by editor(s): June 29, 2016
Received by editor(s) in revised form: March 30, 2017
Published electronically: August 30, 2017
Communicated by: Romyar T. Sharifi
Article copyright: © Copyright 2017 American Mathematical Society

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