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Transactions of the American Mathematical Society

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Algebraic curves with a large non-tame automorphism group fixing no point


Authors: M. Giulietti and G. Korchmáros
Journal: Trans. Amer. Math. Soc. 362 (2010), 5983-6001
MSC (2010): Primary 14H37
DOI: https://doi.org/10.1090/S0002-9947-2010-05025-1
Published electronically: June 10, 2010
MathSciNet review: 2661505
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $ \mathbb{K}$ be an algebraically closed field of characteristic $ p>0$, and let $ \mathcal{X}$ be a curve over $ \mathbb{K}$ of genus $ g\ge 2$. Assume that the automorphism group $ \mathrm{Aut}(\mathcal{X})$ of $ \mathcal{X}$ over $ \mathbb{K}$ fixes no point of $ \mathcal{X}$. The following result is proven. If there is a point $ P$ on $ \mathcal{X}$ whose stabilizer in $ \mathrm{Aut}(\mathcal{X})$ contains a $ p$-subgroup of order greater than $ gp/(p-1)$, then $ \mathcal{X}$ is birationally equivalent over $ \mathbb{K}$ to one of the irreducible plane curves (II), (III), (IV), (V) listed in the Introduction.


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  • 1. M. Aschbacher, Finite group theory, Corrected reprint of the 1986 original. Cambridge Studies in Advanced Mathematics, 10. Cambridge University Press, Cambridge, 1993. x+274 pp. MR 1264416 (95b:20002)
  • 2. E. Çakçak and F. Özbudak, Subfields of the function field of the Deligne-Lusztig curve of Ree type, Acta Arith. 115 (2004), 133-180. MR 2099835 (2005i:11168)
  • 3. P. Dembowski, Finite geometries. Reprint of the 1968 original. Classics in Mathematics. Springer-Verlag, Berlin, 1997. xii+375 pp. MR 1434062 (97i:51005)
  • 4. M. Giulietti and G. Korchmáros, A new family of maximal curves over a finite field, arXiv: 0711.0445(math.AG), 2007.
  • 5. -, A new family of maximal curves over a finite field, Math. Ann., 343 (2009), 229-245. MR 2448446
  • 6. -, A note on cyclic semiregular subgroups of some $ 2$-transitive permutation groups, to appear in Discrete Math, see also arXiv: 0808.4109(math.GR), 2008.
  • 7. -, On Nakajima's remark on Henn's proof, arXiv: 0808.4108(math.AG), 2008.
  • 8. M. Giulietti, G. Korchmáros and F. Torres, Quotient curves of the Suzuki curve, Acta Arith., 122 (2006), 245-274. MR 2239917 (2007g:11069)
  • 9. R. Griess, Schur multipliers of the known finite simple groups. Bull. Amer. Math. Soc. 78 (1972), 68-71. MR 0289635 (44:6823)
  • 10. -, Schur multipliers of the known finite simple groups. II. The Santa Cruz Conference on Finite Groups (Univ. California, Santa Cruz, Calif., 1979), pp. 279-282, Proc. Sympos. Pure Math., 37, Amer. Math. Soc., Providence, R.I., 1980. MR 604594 (82g:20025)
  • 11. J.P. Hansen and J.P. Pedersen, Automorphism group of Ree type, Deligne-Lusztig curves and function fields, J. Reine Angew. Math. 440 (1993), 99-109. MR 1225959 (94h:14024)
  • 12. H.W. Henn, Funktionenkörper mit gro$ \beta$er Automorphismengruppe, J. Reine Angew. Math. 302 (1978), 96-115. MR 511696 (80a:14012)
  • 13. J.W.P. Hirschfeld, G. Korchmáros and F. Torres, Algebraic Curves Over a Finite Field, Princeton Univ. Press, Princeton and Oxford, 2008, xx+696 pp. MR 2386879 (2008m:14040)
  • 14. A.R. Hoffer, On unitary collineation groups, J. Algebra 22 (1972), 211-218. MR 0301624 (46:780)
  • 15. D.R. Hughes and F.C. Piper, Projective Planes, Graduate Texts in Mathematics 6, Springer, New York, 1973, x+291 pp. MR 0333959 (48:12278)
  • 16. B. Huppert, Endliche Gruppen. I, Grundlehren der Mathematischen Wissenschaften 134, Springer, Berlin, 1967, xii+793 pp. MR 0224703 (37:302)
  • 17. B. Huppert and B.N. Blackburn, Finite groups. III, Grundlehren der Mathematischen Wissenschaften 243, Springer, Berlin, 1982, ix+454 pp. MR 662826 (84i:20001b)
  • 18. W.M. Kantor, M. O'Nan and G.M. Seitz, $ 2$-transitive groups in which the stabilizer of two points is cyclic, J. Algebra 21 (1972), 17-50. MR 0357561 (50:10029)
  • 19. C. Lehr and M. Matignon, Automorphism groups for $ p$-cyclic covers of the affine line, Compos. Math. 141 (2005), 1213-1237. MR 2157136 (2006f:14029)
  • 20. M. Matignon and M. Rocher, Smooth curves having a large automorphism $ p$-group in characteristic $ p>0$, Algebra Number Theory 2 (2008), 887-926. MR 2457356
  • 21. P. Mihauilescu, Primary cyclotomic units and a proof of Catalan's conjecture, J. Reine Angew. Math. 572 (2004), 167-195. MR 2076124 (2005f:11051)
  • 22. S. Nakajima, $ p$-ranks and automorphism groups of algebraic curves, Trans. Amer. Math. Soc. 303 (1987), 595-607. MR 902787 (88h:14037)
  • 23. M. Rocher, Large $ p$-group actions with a $ p$-elementary abelian derived group, J. Algebra 321 (2009), 704-740. MR 2483289 (2010a:14059)
  • 24. -, Large $ p$-groups actions with $ \vert G\vert /g^2 > 4/ (p^2-1)^2$, arXiv:0801.3494v1[math.A.G.], 2008.
  • 25. P. Roquette, Abschätzung der Automorphismenanzahl von Funktionenkörpern bei Primzahlcharakteristik, Math. Z. 117 (1970), 157-163. MR 0279100 (43:4826)
  • 26. J.-P. Serre, Local Fields, Graduate Texts in Mathematics 67, Springer, New York, 1979. viii+241 pp. MR 554237 (82e:12016)
  • 27. H. Stichtenoth, Über die Automorphismengruppe eines algebraischen Funktionenkörpers von Primzahlcharakteristik. I. Eine Abschätzung der Ordnung der Automorphismengruppe, Arch. Math. 24 (1973), 527-544. MR 0337980 (49:2749)
  • 28. -, Über die Automorphismengruppe eines algebraischen Funktionenkörpers von Primzahlcharakteristik. II. Ein spezieller Typ von Funktionenkörpern, Arch. Math. 24 (1973), 615-631. MR 0404265 (53:8068)
  • 29. K.O. Stöhr and J.F. Voloch, Weierstrass points and curves over finite fields, Proc. London Math. Soc. 52 (1986), 1-19. MR 812443 (87b:14010)

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Additional Information

M. Giulietti
Affiliation: Dipartimento di Matematica e Informatica, Università degli Studi di Perugia, Via Vanvitelli, 1, 06123 Perugia, Italy
Email: giuliet@dipmat.unipg.it

G. Korchmáros
Affiliation: Dipartimento di Matematica, Università della Basilicata, Contrada Macchia Romana, 85100 Potenza, Italy
Email: gabor.korchmaros@unibas.it

DOI: https://doi.org/10.1090/S0002-9947-2010-05025-1
Keywords: Algebraic curves, positive characteristic, automorphism groups
Received by editor(s): August 29, 2008
Received by editor(s) in revised form: February 19, 2009
Published electronically: June 10, 2010
Additional Notes: This research was supported by the Italian Ministry MURST, Strutture geometriche, combinatoria e loro applicazioni
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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