Problems from the workshop on Automorphisms of Curves (Leiden, August, 2004)
Rendiconti del Seminario Matematico della Università di Padova, Volume 113 (2005), pp. 129-177.

In the week of August, 16th-20th of 2004, we organized a workshop about “Automorphisms of Curves” at the Lorentz Center in Leiden. The programme included two “problem sessions”. Some of the problems presented at the workshop were written down; this is our edition of these refereed and revised papers. Edited by Gunther Cornelissen and Frans Oort with contributions of I. Bouw; T. Chinburg; G. Cornelissen; C. Gasbarri; D. Glass; C. Lehr; M. Matignon; F. Oort; R. Pries; S. Wewers.

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Cornelissen, Gunther; Oort, Frans. Problems from the workshop on Automorphisms of Curves (Leiden, August, 2004). Rendiconti del Seminario Matematico della Università di Padova, Volume 113 (2005), pp. 129-177. http://archive.numdam.org/item/RSMUP_2005__113__129_0/

[1] E. Arbarello - M. Cornalba - P. Griffith - J. Harris, Geometry of Algebraic Curves, Number 267 in Grundlehren. Springer, 1985. | MR | Zbl

[2] I. I. Bouw - S. Wewers, Alternating groups as monodromy groups in positive characteristic, Math. AG/0402436, to appear in Pacific J. Math. | MR | Zbl

[3] M. D. Fried - E. Klassen - Y. Kopeliovich, Realizing alternating groups as monodromy groups of genus one covers, Proc. Amer. Math. Soc., 129 (2000), pp. 111-119. | MR | Zbl

[4] M. D. Fried - H. Völklein, The inverse Galois problem and rational points on moduli spaces, Math. Ann., 290 (1991), pp. 771-800. | EuDML | MR | Zbl

[5] W. Fulton, Hurwitz schemes and the irreducibility of the moduli of algebraic curves, Ann. of Math. 90 (1969), pp. 542-575. | MR | Zbl

[6] J. Harris - I. Morrison, Moduli of curves, Number 187 in GTM. Springer, 1998. | MR | Zbl

[7] K. Magaard - H. Völklein, The monodromy group of a function on a general curve, Math. AG/0304130, to appear in Israel J. Math, 2003. | MR | Zbl

[8] A. Tamagawa, Finiteness of isomorphism classes of curves in positive characteristic with prescribed fundamental group, Preprint. | MR | Zbl

[9] G. Böckle, DemusÏkin groups with group actions and applications to deformations of Galois representations, Compositio Math., 121 (2000), pp. 109-154. | MR | Zbl

[10] B. De Smit - H. W. Lenstra, Jr., Explicit construction of universal deformation rings, in Modular forms and Fermat's Last Theorem, G. Cornell, J. H. Silverman - G. Stevens (eds.), Springer, New York, Berlin, Heidelberg (1977), pp. 313- 326. | MR | Zbl

[11] J. Bertin, Obstructions locales au relèvement de revêtements galoisiens de courbes lisses, C. R. Acad. Sci. Paris Sér. I Math. 326, no. 1 (1998), pp. 55-58. | MR | Zbl

[12] J. Bertin - A. Mézard, Déformations formelles des revêtements sauvagement ramifiés de courbes algébriques, Invent. Math., 141 (2000), pp. 195-238. | MR | Zbl

[13] I. Bouw - S. Wewers, The local lifting problem for dihedral groups, Math. AG/0409395.

[14] G. Cornelissen - F. Kato, Equivariant deformation of Mumford curves and of ordinary curves in positive characteristic, Duke Math. J., 116 (2003), pp. 431-470. | MR | Zbl

[15] G. Cornelissen - A. Mézard, Relèvements des revêtements de courbes faiblement ramifiés, Math. AG/0412189. | Zbl

[16] M. Matignon, p-groupes abéliens de type (p; Á Á Á ; p) et disques ouverts p-adiques, Manuscripta Math., 99, no. 1 (1999), pp. 93-109. | MR | Zbl

[17] S. Nakajima, p-ranks and automorphism groups of algebraic curves, Trans. Amer. Math. Soc., 303 (1987), pp. 595-607. | MR | Zbl

[18] F. Oort - T. Sekiguchi - N. Suwa, On the deformation of Artin-Schreier to Kummer, Ann. Sci. École Norm. Sup. (4) 22, no. 3 (1989), pp. 345-375. | Numdam | MR | Zbl

[19] G. Pagot, Relèvement des actions de (Z=2)2 , preprint (2004).

[20] D. Gieseker, Flat vector bundles and the fundamental group in non-zero characteristics. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 2 (1975). | Numdam | MR | Zbl

[21] H. Lange - U. Stuhler, Vektorbndel auf Kurven und Darstellungen der algebraischen Fundamentalgruppe, Math. Z., 156, no. 1 (1977), pp. 73-83. | MR | Zbl

[22] M. V. Nori, On the representations of the fundamental group, Compositio Math., 33, no. 1 (1976), pp. 29-41. | Numdam | MR | Zbl

[23] F. Oort, Subvarieties of Moduli Space. Inventiones Mathematicae, 29 (1974), pp. 95-119. | MR | Zbl

[24] C. Faber - G. Van Der Geer, Complete subvarieties of moduli spaces and the Prym map, to appear in J. Reine Angew. Math. | MR | Zbl

[25] D. Glass - R. Pries, Hyperelliptic curves with prescribed p-torsion, accepted by Manuscripta. | Zbl

[26] E. Goren, Lectures on Hilbert modular varieties and modular forms, volume 14 of CRM Monograph Series, American Mathematical Society, Providence, RI, 2002. With the assistance of Marc-Hubert Nicole. | MR | Zbl

[27] J.-I. Igusa, Class number of a definite quaternion with prime discriminant, Proc. Nat. Acad. Sci. USA, 44 (1958), pp. 312-314. | MR | Zbl

[28] E. Kani - M. Rosen, Idempotent relations and factors of Jacobians, Math. Ann., 284 (2) (1989) pp. 307-327. | MR | Zbl

[29] F. Oort, Hyperelliptic supersingular curves. In Arithmetic algebraic geometry (Texel, 1989), volume 89 of Progr. Math., pages 247-284. Birkhäuser Boston, Boston, MA, 1991. | MR | Zbl

[30] N. Yui, On the Jacobian varieties of hyperelliptic curves over fields of characteristic p > 2, J. Algebra, 52 (2) (1978), pp. 378-410. | MR | Zbl

[31] B. Green - M. Matignon, Order p automorphisms of the open disc of a p-adic field, J. Amer. Math. Soc., 12 (1999), pp. 269-303. | MR | Zbl

[32] Y. Henrio, Arbres de Hurwitz et automorphismes d'ordre p des disques et des couronnes p-adiques formels, Thèse Université Bordeaux I, 1999.

[33] B. Huppert, Endliche Gruppen I, Die Grundlehren der Mathematischen Wissenschaften, Band 134 Springer-Verlag, Berlin-New York 1967. | MR | Zbl

[34] C. Lehr, Reduction of p-cyclic Covers of the Projective Line, Manuscripta Math., 106 2 (2001) 151-175. | MR | Zbl

[35] C. Lehr - M. Matignon, Automorphism groups for p-cyclic covers of the affine line, to appear in Compositio Math. http://www.math.u-bordeaux1.fr/ ~matignon/preprints.html | MR | Zbl

[36] C. Lehr - M. Matignon, Curves with a big p-group action, in preparation.

[37] C. Lehr, M. Matignon, Wild monodromy and automorphisms of curves, conference proceedings, Tokyo, (2002) (T. Sekiguchi, N. Suwa, editors), to appear. Available on http://www.math.u-bordeaux1.fr/~matignon/ | MR

[38] C. Lehr - M. Matignon, Wild monodromy and automorphisms of curves, in preparation. | Zbl

[39] M. Matignon, Vers un algorithme pour la réduction stable des revêtements p-cycliques de la droite projective sur un corps p-adique, Mathematische Annalen, 325, (2003), pp. 323-354. | MR | Zbl

[40] S. Nakajima, p-ranks and automorphism groups of algebraic curves, Trans. Amer. Math. Soc., 303 (1987), pp. 595-607. | MR | Zbl

[41] M. Raynaud, p-groupes et réduction semi-stable des courbes, The Grothendieck Festschrift, Vol III, Progress in Mathematics, 88 (1990), Birkhäuser, pp. 179-197. | MR | Zbl

[42] M. Raynaud, Spécialisation des revêtements en caractéristique p > 0, Ann. Scient. Ec. Norm. Sup., 32 (4) (1999), pp. 87-126. | Numdam | MR | Zbl

[43] M. Suzuki, Group theory II, Grundlehren der Mathematischen Wissenschaften 248, Springer-Verlag, New York, 1986. | MR | Zbl

[44] J. Bertin, Obstructions locales au relêvement de revêtements galoisiens de courbes lisses, C. R. Acad. Sci. Paris Sér. I Math., 326, no. 1 (1998), pp. 55-58. | MR | Zbl

[45] I. Bouw, - S. Wewers, The local lifting problem for dihedral groups, preprint Math. AG/0409395. | MR | Zbl

[46] B. Green - M. Matignon, Liftings of Galois covers of smooth curves, Compositio Math., 113 (1998), pp. 239-274. | MR | Zbl

[47] B. Green - M. Matignon, Order p automorphisms of the open disc of a p-adic field, J. Amer. Math. Soc., 12 (1999), pp. 269-303. | MR | Zbl

[48] Y. Henrio, Arbres de Hurwitz et automorphismes d'ordre p des disques et des couronnes p-adiques formels, Thèse Université Bordeaux I, 1999.

[49] Q. Liu, Reduction and lifting of finite covers of curves, Proceedings of the 2003 Workshop on Cryptography and Related Mathematics, Chuo University (2003), pp. 161-180, available at http://www.math.u-bordeaux1.fr/~liu/preprints.html

[50] M. Matignon, p-groupes abeliens de type (p; . . . ; p) et disques ouverts p-adiques, Manuscripta Math., 99 (1999), pp. 93-109. | MR | Zbl

[51] M. Matignon, Autour d'une conjecture de F. Oort sur le relèvement de revêtements galoisiens de courbes, French version for the Limoges seminar (2003) of my MSRI lecture in the workshop on Constructive Galois Theory (1999), available at http://www.math.u-bordeaux1.fr/~matignon/preprints.html

[52] M. Matignon, Lifting (Z=2Z)2 -actions, (2003) available at http://www.math.u-bordeaux1.fr/~matignon/preprints.html

[53] G. Pagot, Fp-espaces vectoriels de formes différentielles logarithmiques en caractéristique > 0 et automorphismes du disque ouvert p-adique, J. Number Theory, 97 no. 1 (2002), pp. 58-94. | MR | Zbl

[54] G. Pagot, Relèvement en caractéristique zéro d'actions de groupes abéliens de type (p; :::; p), Thèse Université Bordeaux I, 2002.

[55] G. Pagot, Relèvement des actions de (Z=2Z)2 , Projet de note aux CRAS (2004).

[56] T. Sekiguchi - N. Suwa, A note on extensions of algebraic and formal groups. IV. Kummer-Artin-Schreier-Witt theory of degree p2 , Tohoku Math. J. (2) 53, no. 2 (2001), pp. 203-240. | MR | Zbl

[57] T. Sekiguchi - N. Suwa, A note on extensions of algebraic and formal groups. V, Japan. J. Math. (N.S.) 29 no. 2 (2003), pp. 221-284. | MR | Zbl

[58] T. Sekiguchi - N. Suwa, On the unified Kummer-Artin-Schreier-Witt theory, Prépublication Université de Bordeaux (1999).

[59] K.-Z. Li - F. Oort, Moduli of supersingular abelian varieties, Lecture Notes Math. 1680, Springer - Verlag, 116 (1998), pp. ??? | MR | Zbl

[60] C. Faber - G. Van Der Geer, Complete subvarieties of moduli spaces and the Prym map, In arXiv: math.AG/0305334 [to appear in J. Reine Angew. Math.] | MR | Zbl

[61] G. Van Der Geer - M. Van Der Vlugt, On the existence of supersingular curves of given genus, J. Reine Angew. Math., 458 (1995), pp. 53-61. | MR | Zbl

[62] D. Glass - R. Pries, Hyperelliptic curves with prescribed p-torsion, [To appear in Masnuscr. Math.] | MR | Zbl

[63] D. Glass - R. Pries, Questions on p-torsion of hyperelliptic curves, [This volume]

[64] N. Katz, Slope filtrations of F-crystals, Journ. Géom. Alg. Rennes 1978. Vol. I, Astérisque 63, Soc. Math. France 1979; pp. 113-163. | MR | Zbl

[65] Yu. I. Manin, The theory of commutative formal groups over fields of finite characteristic, Usp. Math., 18 (1963), pp. 3-90; Russ. Math. Surveys, 18 (1963), pp. 1-80. | MR | Zbl

[66] J. S. Milne, Jacobian varieties, Chapter 7 in: Arithmetic geometry (Ed. G. Cornell - J. H. Silverman. Springer - Verlag, 1986; pp. 167-212. | MR | Zbl

[67] P. Norman - F. Oort, Moduli of abelian varieties, Ann. Math., 112 (1980), pp. 413-439. | MR | Zbl

[68] F. Oort, Hyperelliptic supersingular curves, In: Arithmetic Algebraic Geometry (Texel 1989) (Ed. G. van der Geer, F. Oort, J. Steenbrink). Progress Math. 89, Birkhäuser 1991; pp. 247-284. | MR | Zbl

[69] F. Oort, Newton polygons and formal groups: conjectures by Manin and Grothendieck, Ann. Math., 152 (2000), pp. 183-206. | MR | Zbl

[70] F. Oort, A stratification of a moduli space of polarized abelian varieties, In: Moduli of abelian varieties (Ed. C. Faber, G. van der Geer, F. Oort). Progress Math., 195, Birkhäuser Verlag 2001; pp. 345-416. | MR | Zbl

[71] F. Oort, Newton polygon strata in the moduli space of abelian varieties, In: Moduli of abelian varieties (Ed. C. Faber, G. van der Geer, F. Oort). Progress Math., 195, Birkhäuser Verlag 2001; pp. 417-440. | MR | Zbl

[72] F. Oort - K. Ueno, Principally polarized abelian varieties of dimension two or three are Jacobian varieties, Journ. Fac. Sc. Univ. Tokyo, Sec. IA, 20 (1973), pp. 377-381. | MR | Zbl

[73] R. S. G. Re, Invariants of curves and Jacobians in positive characteristic, PhD-thesis Amsterdam 2004.

[74] J. Scholten - H. J. Zhu, Hyperelliptic curves in characteristic 2, Int. Math. Res. Not. 17 (2002), pp. 905-917. | MR | Zbl

[75] J. Scholten - H. J. Zhu, Families of supersingular curves in characteristic 2, Math. Res. Lett., 9 (2002), pp. 639-650. | MR | Zbl

[76] A. Weil, Zum Beweis des Torellischen Satzes, Nachr. Akad. Göttingen, Math.-Phys. Kl. (1957), pp. 33-53. See êII [1957a]. | MR | Zbl

[77] R. D. M. Accola, On the number of automorphisms of a closed Riemann surface, Transact. AMS 131 (1968), pp. 398-408. | MR | Zbl

[78] R. D. M. Accola, Topics in the theory of Riemann surfaces, Lect. Notes Math. 1595, Springer - Verlag, 1994. | MR | Zbl

[79] A. M. Macbeath, On a theorem of Hurwitz, Proceed. Glasgow Math. Assoc., 5 (1961), pp. 90-96. | MR | Zbl

[80] C. Maclachlan, A bound for the number of automorphisms of a compact Riemann surface, Journ. London Math. Soc. 44 (1969), pp. 265-272. | MR | Zbl