Finiteness of isomorphism classes of curves in positive characteristic with prescribed fundamental groups
Author:
Akio Tamagawa
Journal:
J. Algebraic Geom. 13 (2004), 675-724
DOI:
https://doi.org/10.1090/S1056-3911-04-00376-5
Published electronically:
February 25, 2004
MathSciNet review:
2073193
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Abstract: We prove that there are only finitely many isomorphism classes of smooth, hyperbolic curves over an algebraic closure of the finite prime field $\mathbb {F}_{p}$, whose (tame) fundamental group is isomorphic to a prescribed profinite group. This is a generalization of partial results by Pop, Saïdi and Raynaud. The key ingredient of the proof is Raynaud’s theory of theta divisors. In course of the proof, we also obtain some results concerning gonalities of coverings of curves and concerning the infinitesimal Torelli problem for generalized Prym varieties, which are applicable to arbitrary (not necessarily positive) characteristic and may be of some interest independent of the study of fundamental groups of curves in positive characteristic.
[AI]AI G. W. Anderson and R. Indik, On primes of degree one in function fields, Proc. Amer. Math. Soc. 94 (1985), 31–32.
[ACGH]ACGH E. Arbarello, M. Cornalba, P. A. Griffiths and J. Harris, Geometry of algebraic curves, I, Grundlehren der Mathematischen Wissenschaften, 267, Springer-Verlag, 1985.
[A]Asada M. Asada, The faithfulness of the monodromy representations associated with certain families of algebraic curves, J. Pure Appl. Algebra 159 (2001), 123–147.
[B]Belyi G. V. Belyĭ, Galois extensions of a maximal cyclotomic field, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), 267–276, 479 (Russian), English translation in Math. USSR-Izv. 14 (1980), 247–256.
[BLR]BLR S. Bosch, W. Lütkebohmert and M. Raynaud, Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 21, Springer-Verlag, 1990.
[D1]Deligne1 P. Deligne, Le lemme de Gabber, in Séminaire sur les pinceaux arithmétiques: la conjecture de Mordell, Astérisque, 127 (L. Szpiro, ed.), Société Mathématique de France, 1985, pp. 131–150.
[D2]Deligne2 ---, Le groupe fondamental de la droite projective moins trois points, in Galois groups over $\mathbb {Q}$ (Berkeley, 1987), Math. Sci. Res. Inst. Publ., 16 (Y. Ihara, K. Ribet and J.-P. Serre, eds.), Springer-Verlag, 1989, pp. 79–297.
[E]Eisenbud D. Eisenbud, Commutative algebra, With a view toward algebraic geometry, Graduate Texts in Mathematics, 150, Springer-Verlag, 1995.
[FK]FK H. M. Farkas and I. Kra, Riemann surfaces, Graduate Texts in Mathematics, 71, Springer-Verlag, 1980.
[FJ]FJ M. D. Fried and M. F. Jarden, Field arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 11, Springer-Verlag, 1986.
[Harb1]Harbater1 D. Harbater, Galois groups with prescribed ramification, in Arithmetic geometry (Tempe, 1993), Contemp. Math., 174 (N. Childress and J. W. Jones, eds.), Amer. Math. Soc., 1994, pp. 35–60.
[Harb2]Harbater2 ---, Fundamental groups of curves in characteristic $p$, in Proceedings of the International Congress of Mathematicians, 1, 2 (Zürich, 1994), Birkhäuser, 1995, pp. 656–666.
[Hart]Hartshorne R. Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, 52, Springer-Verlag, 1977.
[Hr]Hrushovski E. Hrushovski, The Mordell-Lang conjecture for function fields, J. Amer. Math. Soc. 9 (1996), 667–690.
[K]Kempf G. Kempf, Schubert methods with an application to algebraic curves, Publ. Math. Centrum, 1972.
[KL1]KL1 S. L. Kleiman and D. Laksov, On the existence of special divisors, Amer. J. Math. 94 (1972), 431–436.
[KL2]KL2 ---, Another proof of the existence of special divisors, Acta Math. 132 (1974), 163–176.
[Mar]Martens H. H. Martens, On the varieties of special divisors on a curve, J. Reine Angew. Math. 227 (1967), 111–120.
[Mat]Matsumoto M. Matsumoto, Galois representations on profinite braid groups on curves, J. Reine Angew. Math. 474 (1996), 169–219.
[MT]MT M. Matsumoto and A. Tamagawa, Mapping-class-group action versus Galois action on profinite fundamental groups, Amer. J. Math. 122 (2000), 1017–1026.
[Me]Merindol J.-Y. Mérindol, Variétés de Prym d’un revêtement galoisien, J. Reine Angew. Math. 461 (1995), 49–61.
[Mi]Milne J. S. Milne, Abelian varieties, in Arithmetic geometry (Storrs, 1984) (G. Cornell and J. H. Silverman, eds.), Springer-Verlag, 1986, pp. 103–150.
[Mu1]Mumford1 D. Mumford, Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, 5, Oxford University Press, 1970.
[Mu2]Mumford2 ---, Prym varieties, I, in Contributions to analysis, A collection of papers dedicated to Lipman Bers (L. V. Ahlfors, I. Kra, B. Maskit and L. Nirenberg, eds.), Academic Press, 1974, pp. 325–350.
[NS]NS K. V. Nguyen and M.-H. Saito, $d$-gonality of modular curves and bounding torsions, preprint (1996), arXiv:alg-geom/9603024.
[OS]OS F. Oort and J. Steenbrink, The local Torelli problem for algebraic curves, in Journées de Géometrie Algébrique d’Angers, Juillet 1979/Algebraic Geometry, Angers, 1979, Variétés de petite dimension (Angers, 1979) (A. Beauville, ed.), Sijthoff & Noordhoff, 1980, pp. 157–204.
[PR]PR R. Pink and D. Roessler, On $\psi$-invariant subvarieties of semiabelian varieties and the Manin-Mumford conjecture, preprint (2002).
[PS]PS F. Pop and M. Saïdi, On the specialization homomorphism of fundamental groups of curves in positive characteristic, in Galois groups and fundamental groups, Math. Sci. Res. Inst. Publ., 41 (L. Schneps, ed.), Cambridge University Press, 2003, pp. 107–118.
[R1]Raynaud1 M. Raynaud, Sections des fibrés vectoriels sur une courbe, Bull. Soc. Math. France 110 (1982), 103–125.
[R2]Raynaud2 ---, Sous-variétés d’une variété abélienne et points de torsion, in Arithmetic and geometry, Papers dedicated to I. R. Shafarevich on the occasion of his sixtieth birthday, I, Arithmetic, Progr. Math., 35 (M. Artin and J. Tate, eds.), Birkhäuser, 1983, pp. 327–352.
[R3]Raynaud3 ---, Revêtements des courbes en caractéristique $p>0$ et ordinarité, Compositio Math. 123 (2000), 73–88.
[R4]Raynaud4 ---, Sur le groupe fondamental d’une courbe complète en caractéristique $p>0$, in Arithmetic fundamental groups and noncommutative algebra (Berkeley, 1999), Proc. Sympos. Pure Math., 70 (M. D. Fried and Y. Ihara, eds.), Amer. Math. Soc., 2002, pp. 335–351.
[S1]Shiho1 A. Shiho, Crystalline fundamental groups, I, Isocrystals on log crystalline site and log convergent site, J. Math. Sci. Univ. Tokyo 7 (2000), 509–656.
[S2]Shiho2 ---, Crystalline fundamental groups, II, Log convergent cohomology and rigid cohomology, J. Math. Sci. Univ. Tokyo 9 (2002), 1–163.
[T1]Tamagawa1 A. Tamagawa, The Grothendieck conjecture for affine curves, Compositio Math. 109 (1997), 135–194.
[T2]Tamagawa2 ---, On the fundamental groups of curves over algebraically closed fields of characteristic $>0$, Internat. Math. Res. Notices (16) (1999), 853–873.
[T3]Tamagawa3 ---, On the tame fundamental groups of curves over algebraically closed fields of characteristic $>0$, in Galois groups and fundamental groups, Math. Sci. Res. Inst. Publ., 41 (L. Schneps, ed.), Cambridge University Press, 2003, pp. 47–105.
[T4]Tamagawa4 ---, Fundamental groups and geometry of curves in positive characteristic, in Arithmetic fundamental groups and noncommutative algebra (Berkeley, 1999), Proc. Sympos. Pure Math., 70 (M. D. Fried and Y. Ihara, eds.), Amer. Math. Soc., 2002, pp. 297–333.
[V]Voevodskii V. A. Voevodskiĭ, Galois representations connected with hyperbolic curves, Izv. Akad. Nauk SSSR Ser. Mat. 55 (1991), 1331–1342 (Russian), English translation in Math. USSR-Izv. 39 (1992), 1281–1291.
[EGA2]EGA2 A. Grothendieck et al, Éléments de géométrie algébrique II, Étude globale élémentaire de quelques classes de morphismes, Publications Mathématiques, 8, IHES, 1961.
[EGA4]EGA4 A. Grothendieck et al, Éléments de géométrie algébrique IV, Étude locale des schémas et des morphismes de schémas, Publications Mathématiques, 20, 24, 28, 32, IHES, 1964–1967.
[SGA1]SGA1 A. Grothendieck et al, Revêtements étales et groupe fondamental, Séminaire de géométrie algébrique du Bois Marie 1960/61 (SGA 1), Lecture Notes in Mathematics, 224, Springer-Verlag, 1971.
[SGA3]SGA3 M. Demazure, A. Grothendieck et al, Schémas en groupes I, II, III, Séminaire de géométrie algébrique du Bois Marie 1962/64 (SGA 3), Lecture Notes in Mathematics, 151, 152, 153, Springer-Verlag, 1970.
[SGA7]SGA7 A. Grothendieck, P. Deligne, N. Katz et al, Groupes de monodromie en géométrie algébrique, Séminaire de géométrie algébrique du Bois Marie 1967–1969 (SGA 7 I, II), Lecture Notes in Mathematics, 288, 340, Springer-Verlag, 1972, 1973.
[AI]AI G. W. Anderson and R. Indik, On primes of degree one in function fields, Proc. Amer. Math. Soc. 94 (1985), 31–32.
[ACGH]ACGH E. Arbarello, M. Cornalba, P. A. Griffiths and J. Harris, Geometry of algebraic curves, I, Grundlehren der Mathematischen Wissenschaften, 267, Springer-Verlag, 1985.
[A]Asada M. Asada, The faithfulness of the monodromy representations associated with certain families of algebraic curves, J. Pure Appl. Algebra 159 (2001), 123–147.
[B]Belyi G. V. Belyĭ, Galois extensions of a maximal cyclotomic field, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), 267–276, 479 (Russian), English translation in Math. USSR-Izv. 14 (1980), 247–256.
[BLR]BLR S. Bosch, W. Lütkebohmert and M. Raynaud, Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 21, Springer-Verlag, 1990.
[D1]Deligne1 P. Deligne, Le lemme de Gabber, in Séminaire sur les pinceaux arithmétiques: la conjecture de Mordell, Astérisque, 127 (L. Szpiro, ed.), Société Mathématique de France, 1985, pp. 131–150.
[D2]Deligne2 ---, Le groupe fondamental de la droite projective moins trois points, in Galois groups over $\mathbb {Q}$ (Berkeley, 1987), Math. Sci. Res. Inst. Publ., 16 (Y. Ihara, K. Ribet and J.-P. Serre, eds.), Springer-Verlag, 1989, pp. 79–297.
[E]Eisenbud D. Eisenbud, Commutative algebra, With a view toward algebraic geometry, Graduate Texts in Mathematics, 150, Springer-Verlag, 1995.
[FK]FK H. M. Farkas and I. Kra, Riemann surfaces, Graduate Texts in Mathematics, 71, Springer-Verlag, 1980.
[FJ]FJ M. D. Fried and M. F. Jarden, Field arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 11, Springer-Verlag, 1986.
[Harb1]Harbater1 D. Harbater, Galois groups with prescribed ramification, in Arithmetic geometry (Tempe, 1993), Contemp. Math., 174 (N. Childress and J. W. Jones, eds.), Amer. Math. Soc., 1994, pp. 35–60.
[Harb2]Harbater2 ---, Fundamental groups of curves in characteristic $p$, in Proceedings of the International Congress of Mathematicians, 1, 2 (Zürich, 1994), Birkhäuser, 1995, pp. 656–666.
[Hart]Hartshorne R. Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, 52, Springer-Verlag, 1977.
[Hr]Hrushovski E. Hrushovski, The Mordell-Lang conjecture for function fields, J. Amer. Math. Soc. 9 (1996), 667–690.
[K]Kempf G. Kempf, Schubert methods with an application to algebraic curves, Publ. Math. Centrum, 1972.
[KL1]KL1 S. L. Kleiman and D. Laksov, On the existence of special divisors, Amer. J. Math. 94 (1972), 431–436.
[KL2]KL2 ---, Another proof of the existence of special divisors, Acta Math. 132 (1974), 163–176.
[Mar]Martens H. H. Martens, On the varieties of special divisors on a curve, J. Reine Angew. Math. 227 (1967), 111–120.
[Mat]Matsumoto M. Matsumoto, Galois representations on profinite braid groups on curves, J. Reine Angew. Math. 474 (1996), 169–219.
[MT]MT M. Matsumoto and A. Tamagawa, Mapping-class-group action versus Galois action on profinite fundamental groups, Amer. J. Math. 122 (2000), 1017–1026.
[Me]Merindol J.-Y. Mérindol, Variétés de Prym d’un revêtement galoisien, J. Reine Angew. Math. 461 (1995), 49–61.
[Mi]Milne J. S. Milne, Abelian varieties, in Arithmetic geometry (Storrs, 1984) (G. Cornell and J. H. Silverman, eds.), Springer-Verlag, 1986, pp. 103–150.
[Mu1]Mumford1 D. Mumford, Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, 5, Oxford University Press, 1970.
[Mu2]Mumford2 ---, Prym varieties, I, in Contributions to analysis, A collection of papers dedicated to Lipman Bers (L. V. Ahlfors, I. Kra, B. Maskit and L. Nirenberg, eds.), Academic Press, 1974, pp. 325–350.
[NS]NS K. V. Nguyen and M.-H. Saito, $d$-gonality of modular curves and bounding torsions, preprint (1996), arXiv:alg-geom/9603024.
[OS]OS F. Oort and J. Steenbrink, The local Torelli problem for algebraic curves, in Journées de Géometrie Algébrique d’Angers, Juillet 1979/Algebraic Geometry, Angers, 1979, Variétés de petite dimension (Angers, 1979) (A. Beauville, ed.), Sijthoff & Noordhoff, 1980, pp. 157–204.
[PR]PR R. Pink and D. Roessler, On $\psi$-invariant subvarieties of semiabelian varieties and the Manin-Mumford conjecture, preprint (2002).
[PS]PS F. Pop and M. Saïdi, On the specialization homomorphism of fundamental groups of curves in positive characteristic, in Galois groups and fundamental groups, Math. Sci. Res. Inst. Publ., 41 (L. Schneps, ed.), Cambridge University Press, 2003, pp. 107–118.
[R1]Raynaud1 M. Raynaud, Sections des fibrés vectoriels sur une courbe, Bull. Soc. Math. France 110 (1982), 103–125.
[R2]Raynaud2 ---, Sous-variétés d’une variété abélienne et points de torsion, in Arithmetic and geometry, Papers dedicated to I. R. Shafarevich on the occasion of his sixtieth birthday, I, Arithmetic, Progr. Math., 35 (M. Artin and J. Tate, eds.), Birkhäuser, 1983, pp. 327–352.
[R3]Raynaud3 ---, Revêtements des courbes en caractéristique $p>0$ et ordinarité, Compositio Math. 123 (2000), 73–88.
[R4]Raynaud4 ---, Sur le groupe fondamental d’une courbe complète en caractéristique $p>0$, in Arithmetic fundamental groups and noncommutative algebra (Berkeley, 1999), Proc. Sympos. Pure Math., 70 (M. D. Fried and Y. Ihara, eds.), Amer. Math. Soc., 2002, pp. 335–351.
[S1]Shiho1 A. Shiho, Crystalline fundamental groups, I, Isocrystals on log crystalline site and log convergent site, J. Math. Sci. Univ. Tokyo 7 (2000), 509–656.
[S2]Shiho2 ---, Crystalline fundamental groups, II, Log convergent cohomology and rigid cohomology, J. Math. Sci. Univ. Tokyo 9 (2002), 1–163.
[T1]Tamagawa1 A. Tamagawa, The Grothendieck conjecture for affine curves, Compositio Math. 109 (1997), 135–194.
[T2]Tamagawa2 ---, On the fundamental groups of curves over algebraically closed fields of characteristic $>0$, Internat. Math. Res. Notices (16) (1999), 853–873.
[T3]Tamagawa3 ---, On the tame fundamental groups of curves over algebraically closed fields of characteristic $>0$, in Galois groups and fundamental groups, Math. Sci. Res. Inst. Publ., 41 (L. Schneps, ed.), Cambridge University Press, 2003, pp. 47–105.
[T4]Tamagawa4 ---, Fundamental groups and geometry of curves in positive characteristic, in Arithmetic fundamental groups and noncommutative algebra (Berkeley, 1999), Proc. Sympos. Pure Math., 70 (M. D. Fried and Y. Ihara, eds.), Amer. Math. Soc., 2002, pp. 297–333.
[V]Voevodskii V. A. Voevodskiĭ, Galois representations connected with hyperbolic curves, Izv. Akad. Nauk SSSR Ser. Mat. 55 (1991), 1331–1342 (Russian), English translation in Math. USSR-Izv. 39 (1992), 1281–1291.
[EGA2]EGA2 A. Grothendieck et al, Éléments de géométrie algébrique II, Étude globale élémentaire de quelques classes de morphismes, Publications Mathématiques, 8, IHES, 1961.
[EGA4]EGA4 A. Grothendieck et al, Éléments de géométrie algébrique IV, Étude locale des schémas et des morphismes de schémas, Publications Mathématiques, 20, 24, 28, 32, IHES, 1964–1967.
[SGA1]SGA1 A. Grothendieck et al, Revêtements étales et groupe fondamental, Séminaire de géométrie algébrique du Bois Marie 1960/61 (SGA 1), Lecture Notes in Mathematics, 224, Springer-Verlag, 1971.
[SGA3]SGA3 M. Demazure, A. Grothendieck et al, Schémas en groupes I, II, III, Séminaire de géométrie algébrique du Bois Marie 1962/64 (SGA 3), Lecture Notes in Mathematics, 151, 152, 153, Springer-Verlag, 1970.
[SGA7]SGA7 A. Grothendieck, P. Deligne, N. Katz et al, Groupes de monodromie en géométrie algébrique, Séminaire de géométrie algébrique du Bois Marie 1967–1969 (SGA 7 I, II), Lecture Notes in Mathematics, 288, 340, Springer-Verlag, 1972, 1973.
Additional Information
Akio Tamagawa
Affiliation:
Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan
MR Author ID:
362316
Email:
tamagawa@kurims.kyoto-u.ac.jp
Received by editor(s):
May 1, 2002
Published electronically:
February 25, 2004
