Automorphisms of curves and Weierstrass semigroups for Harbater–Katz–Gabber covers
HTML articles powered by AMS MathViewer
- by Sotiris Karanikolopoulos and Aristides Kontogeorgis PDF
- Trans. Amer. Math. Soc. 371 (2019), 6377-6402 Request permission
Abstract:
We study $p$-group Galois covers $X \rightarrow \mathbb {P}^1$ with only one fully ramified point in characteristic $p>0$. These covers are important because of the Harbater–Katz–Gabber compactification theorem of Galois actions on complete local rings. The sequence of ramification jumps is related to the Weierstrass semigroup of the global cover at the stabilized point. We determine explicitly the jumps of the ramification filtrations in terms of pole numbers. We give applications for curves with zero $p$-rank: we focus on curves that admit a big action. Moreover, we initiate the study of the Galois module structure of polydifferentials.References
- Nurdagül Anbar, Henning Stichtenoth, and Seher Tutdere, On ramification in the compositum of function fields, Bull. Braz. Math. Soc. (N.S.) 40 (2009), no. 4, 539–552. MR 2563130, DOI 10.1007/s00574-009-0026-8
- José Bertin and Ariane Mézard, Déformations formelles des revêtements sauvagement ramifiés de courbes algébriques, Invent. Math. 141 (2000), no. 1, 195–238 (French, with English summary). MR 1767273, DOI 10.1007/s002220000071
- Frauke M. Bleher, Ted Chinburg, Bjorn Poonen, and Peter Symonds, Automorphisms of Harbater-Katz-Gabber curves, Math. Ann. 368 (2017), no. 1-2, 811–836. MR 3651589, DOI 10.1007/s00208-016-1490-2
- Niels Borne, Cohomology of $G$-sheaves in positive characteristic, Adv. Math. 201 (2006), no. 2, 454–515. MR 2211535, DOI 10.1016/j.aim.2005.03.002
- Helmut Boseck, Zur Theorie der Weierstrasspunkte, Math. Nachr. 19 (1958), 29–63 (German). MR 106221, DOI 10.1002/mana.19580190103
- Nigel P. Byott and G. Griffith Elder, A valuation criterion for normal bases in elementary abelian extensions, Bull. Lond. Math. Soc. 39 (2007), no. 5, 705–708. MR 2365217, DOI 10.1112/blms/bdm036
- T. Chinburg, R. Guralnick, and D. Harbater, Oort groups and lifting problems, Compos. Math. 144 (2008), no. 4, 849–866. MR 2441248, DOI 10.1112/S0010437X08003515
- Ted Chinburg, Robert Guralnick, and David Harbater, The local lifting problem for actions of finite groups on curves, Ann. Sci. Éc. Norm. Supér. (4) 44 (2011), no. 4, 537–605 (English, with English and French summaries). MR 2919977, DOI 10.24033/asens.2150
- Gunther Cornelissen, Fumiharu Kato, and Aristides Kontogeorgis, Discontinuous groups in positive characteristic and automorphisms of Mumford curves, Math. Ann. 320 (2001), no. 1, 55–85. MR 1835062, DOI 10.1007/PL00004470
- Stefania Fanali and Massimo Giulietti, Quotient curves of the GK curve, Adv. Geom. 12 (2012), no. 2, 239–268. MR 2911149, DOI 10.1515/advgeom.2011.046
- R. Fröberg, C. Gottlieb, and R. Häggkvist, On numerical semigroups, Semigroup Forum 35 (1987), no. 1, 63–83. MR 880351, DOI 10.1007/BF02573091
- Arnaldo García, On Weierstrass points on Artin-Schreier extensions of $k(x)$, Math. Nachr. 144 (1989), 233–239. MR 1037171, DOI 10.1002/mana.19891440116
- Arnaldo García, On Weierstrass points on certain elementary abelian extensions of $k(x)$, Comm. Algebra 17 (1989), no. 12, 3025–3032. MR 1030607, DOI 10.1080/00927878908823891
- Arnaldo Garcia, Cem Güneri, and Henning Stichtenoth, A generalization of the Giulietti-Korchmáros maximal curve, Adv. Geom. 10 (2010), no. 3, 427–434. MR 2660419, DOI 10.1515/ADVGEOM.2010.020
- 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, DOI 10.1007/s00208-008-0270-z
- Massimo Giulietti and Gábor Korchmáros, Algebraic curves with a large non-tame automorphism group fixing no point, Trans. Amer. Math. Soc. 362 (2010), no. 11, 5983–5983.
- David M. Goldschmidt, Algebraic functions and projective curves, Graduate Texts in Mathematics, vol. 215, Springer-Verlag, New York, 2003. MR 1934359, DOI 10.1007/b97844
- David Harbater, Moduli of $p$-covers of curves, Comm. Algebra 8 (1980), no. 12, 1095–1122. MR 579791, DOI 10.1080/00927878008822511
- 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, DOI 10.1515/9781400847419
- A. Hurwitz, Ueber algebraische Gebilde mit eindeutigen Transformationen in sich, Math. Ann. 41 (1892), no. 3, 403–442 (German). MR 1510753, DOI 10.1007/BF01443420
- Nicholas M. Katz, Local-to-global extensions of representations of fundamental groups, Ann. Inst. Fourier (Grenoble) 36 (1986), no. 4, 69–106 (English, with French summary). MR 867916, DOI 10.5802/aif.1069
- A. Kontogeorgis, Polydifferentials and the deformation functor of curves with automorphisms, J. Pure Appl. Algebra 210 (2007), no. 2, 551–558. MR 2320018, DOI 10.1016/j.jpaa.2006.10.015
- A. Kontogeorgis, The ramification sequence for a fixed point of an automorphism of a curve and the Weierstrass gap sequence, Math. Z. 259 (2008), no. 3, 471–479. MR 2395122, DOI 10.1007/s00209-007-0231-3
- Manish Kumar, On the compositum of wildly ramified extensions, J. Pure Appl. Algebra 218 (2014), no. 8, 1528–1536. MR 3175037, DOI 10.1016/j.jpaa.2013.12.004
- Claus Lehr and Michel Matignon, Automorphism groups for $p$-cyclic covers of the affine line, Compos. Math. 141 (2005), no. 5, 1213–1237. MR 2157136, DOI 10.1112/S0010437X05001296
- Pedro Ricardo López-Bautista and Gabriel Daniel Villa-Salvador, On the Galois module structure of semisimple holomorphic differentials, Israel J. Math. 116 (2000), 345–365. MR 1759412, DOI 10.1007/BF02773225
- Daniel J. Madden, Arithmetic in generalized Artin-Schreier extensions of $k(x)$, J. Number Theory 10 (1978), no. 3, 303–323. MR 506641, DOI 10.1016/0022-314X(78)90027-6
- Michel Matignon and Magali Rocher, Smooth curves having a large automorphism $p$-group in characteristic $p>0$, Algebra Number Theory 2 (2008), no. 8, 887–926. MR 2457356, DOI 10.2140/ant.2008.2.887
- Ian Morrison and Henry Pinkham, Galois Weierstrass points and Hurwitz characters, Ann. of Math. (2) 124 (1986), no. 3, 591–625. MR 866710, DOI 10.2307/2007094
- Sh\B{o}ichi Nakajima, Equivariant form of the Deuring-Šafarevič formula for Hasse-Witt invariants, Math. Z. 190 (1985), no. 4, 559–566. MR 808922, DOI 10.1007/BF01214754
- Sh\B{o}ichi Nakajima, $p$-ranks and automorphism groups of algebraic curves, Trans. Amer. Math. Soc. 303 (1987), no. 2, 595–607. MR 902787, DOI 10.1090/S0002-9947-1987-0902787-6
- Andrew Obus, The (local) lifting problem for curves, Galois-Teichmüller theory and arithmetic geometry, Adv. Stud. Pure Math., vol. 63, Math. Soc. Japan, Tokyo, 2012, pp. 359–412. MR 3051249, DOI 10.2969/aspm/06310359
- Andrew Obus and Stefan Wewers, Cyclic extensions and the local lifting problem, Ann. of Math. (2) 180 (2014), no. 1, 233–284. MR 3194815, DOI 10.4007/annals.2014.180.1.5
- Florian Pop, The Oort conjecture on lifting covers of curves, Ann. of Math. (2) 180 (2014), no. 1, 285–322. MR 3194816, DOI 10.4007/annals.2014.180.1.6
- J. L. Ramírez Alfonsín, The Diophantine Frobenius problem, Oxford Lecture Series in Mathematics and its Applications, vol. 30, Oxford University Press, Oxford, 2005. MR 2260521, DOI 10.1093/acprof:oso/9780198568209.001.0001
- Magali Rocher, Large $p$-group actions with a $p$-elementary abelian derived group, J. Algebra 321 (2009), no. 2, 704–740. MR 2483289, DOI 10.1016/j.jalgebra.2008.09.030
- Peter Roquette, On the history of Artin’s $L$-functions and conductors, Mitt. Math. Ges. Hamburg 19 (2000), 5–50.
- J. C. Rosales and P. A. García-Sánchez, Numerical semigroups, Developments in Mathematics, vol. 20, Springer, New York, 2009. MR 2549780, DOI 10.1007/978-1-4419-0160-6
- Martha Rzedowski-Calderón and Gabriel Villa-Salvador, Function field extensions with null Hasse-Witt map, Int. Math. J. 2 (2002), no. 4, 361–371. MR 1891121
- Martha Rzedowski-Calderón, Gabriel Villa-Salvador, and Manohar L. Madan, Galois module structure of holomorphic differentials in characteristic $p$, Arch. Math. (Basel) 66 (1996), no. 2, 150–156. MR 1367157, DOI 10.1007/BF01273346
- Jean-Pierre Serre, Local fields, Graduate Texts in Mathematics, vol. 67, Springer-Verlag, New York-Berlin, 1979. Translated from the French by Marvin Jay Greenberg. MR 554237, DOI 10.1007/978-1-4757-5673-9
- H. J. Smith, Fractions of numerical semigroups, 2010. Thesis (Ph.D.)–University of Tennessee, https://trace.tennessee.edu/utk_graddiss/750.
- Henning Stichtenoth, Über die Automorphismengruppe eines algebraischen Funktionenkörpers von Primzahlcharakteristik. I. Eine Abschätzung der Ordnung der Automorphismengruppe, Arch. Math. (Basel) 24 (1973), 527–544 (German). MR 337980, DOI 10.1007/BF01228251
- Henning Stichtenoth, Über die Automorphismengruppe eines algebraischen Funktionenkörpers von Primzahlcharakteristik. II. Ein spezieller Typ von Funktionenkörpern, Arch. Math. (Basel) 24 (1973), 615–631. MR 404265, DOI 10.1007/BF01228261
- Hidegorô Nakano, Über Abelsche Ringe von Projektionsoperatoren, Proc. Phys.-Math. Soc. Japan (3) 21 (1939), 357–375 (German). MR 94
- Karl-Otto Stöhr, On the moduli spaces of Gorenstein curves with symmetric Weierstrass semigroups, J. Reine Angew. Math. 441 (1993), 189–213. MR 1228616
- Karl-Otto Stöhr and José Felipe Voloch, Weierstrass points and curves over finite fields, Proc. London Math. Soc. (3) 52 (1986), no. 1, 1–19. MR 812443, DOI 10.1112/plms/s3-52.1.1
- Tuneo Tamagawa, On unramified extensions of algebraic function fields, Proc. Japan Acad. 27 (1951), 548–551. MR 47705
- Lara Thomas, A valuation criterion for normal basis generators in equal positive characteristic, J. Algebra 320 (2008), no. 10, 3811–3820. MR 2457723, DOI 10.1016/j.jalgebra.2008.05.024
- Robert C. Valentini, Representations of automorphisms on differentials of function fields of characteristic $p$, J. Reine Angew. Math. 335 (1982), 164–179. MR 667465, DOI 10.1515/crll.1982.335.164
- Robert C. Valentini and Manohar L. Madan, A hauptsatz of L. E. Dickson and Artin-Schreier extensions, J. Reine Angew. Math. 318 (1980), 156–177. MR 579390
- Robert C. Valentini and Manohar L. Madan, Automorphism groups of algebraic function fields, Math. Z. 176 (1981), no. 1, 39–52. MR 606170, DOI 10.1007/BF01258903
- Robert C. Valentini and Manohar L. Madan, Automorphisms and holomorphic differentials in characteristic $p$, J. Number Theory 13 (1981), no. 1, 106–115. MR 602451, DOI 10.1016/0022-314X(81)90032-9
- Gabriel Daniel Villa Salvador, Topics in the theory of algebraic function fields, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 2006. MR 2241963
- Qingquan Wu and Renate Scheidler, The ramification groups and different of a compositum of Artin-Schreier extensions, Int. J. Number Theory 6 (2010), no. 7, 1541–1564. MR 2740721, DOI 10.1142/S1793042110003617
Additional Information
- Sotiris Karanikolopoulos
- Affiliation: Institut für Mathematik, Freie Universität Berlin, Arnimallee 3, 14195 Berlin, Germany
- MR Author ID: 981743
- Email: skaran@zedat.fu-berlin.de
- Aristides Kontogeorgis
- Affiliation: Department of Mathematics, National and Kapodistrian University of Athens, Panepistimioupolis, 15784 Athens, Greece
- MR Author ID: 638098
- ORCID: 0000-0002-6869-8367
- Email: kontogar@math.uoa.gr
- Received by editor(s): September 24, 2015
- Received by editor(s) in revised form: December 22, 2016, July 27, 2017, and November 27, 2017
- Published electronically: February 1, 2019
- Additional Notes: The first author was supported by a Dahlem Research School and Marie Curie Cofund fellowship; he is also a member of the SFB 647 project Space–Time–Matter, Analytic and Geometric Structures.
The second author was supported by the European Union (European Social Fund—ESF) and Greek national funds through the Operational Program “Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF)—Research Funding Program: THALIS - © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 371 (2019), 6377-6402
- MSC (2010): Primary 14H37, 14H55, 11G20; Secondary 20M14, 14H10
- DOI: https://doi.org/10.1090/tran/7562
- MathSciNet review: 3937329