Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)

   
 

 

Rational points of universal curves


Author: Richard Hain
Journal: J. Amer. Math. Soc. 24 (2011), 709-769
MSC (2010): Primary 14G05, 14G27, 14H10, 14H25; Secondary 11G30, 14G32
Published electronically: January 25, 2011
MathSciNet review: 2784328
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Suppose that $ k$ is a field of characteristic zero and that $ g+n>2$. The universal curve $ C$ of type $ (g,n)$ is the restriction of the universal curve to the generic point $ \operatorname{Spec} k(\mathcal{M}_{g,n})$ of the moduli stack $ \mathcal{M}_{g,n}$ of $ n$-pointed smooth projective curves of genus $ g$. In this paper we prove that if $ g \ge 3$, then its set of rational points $ C(k(\mathcal{M}_{g,n}))$ consists only of the $ n$ tautological points. We then prove that if $ g\ge 5$ and $ n=0$, then Grothendieck's Section Conjecture holds for $ C$ when, for example, $ k$ is a number field or a non-archimedean local field. When $ n>0$, we consider a modified version of Grothendieck's conjecture in which the geometric fundamental group of $ C$ is replaced by its $ \ell$-adic unipotent completion. We prove that if $ k$ is a number field or a non-archimedean local field, then this modified version of the Section Conjecture holds for all $ g \ge 5$ and $ n \ge 1$.


References [Enhancements On Off] (What's this?)

  • 1. Michael P. Anderson, Exactness properties of profinite completion functors, Topology 13 (1974), 229–239. MR 0354882
  • 2. Joan S. Birman, Braids, links, and mapping class groups, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1974. Annals of Mathematics Studies, No. 82. MR 0375281
  • 3. Fedor Alekseivich Bogomolov, Sur l’algébricité des représentations 𝑙-adiques, C. R. Acad. Sci. Paris Sér. A-B 290 (1980), no. 15, A701–A703 (French, with English summary). MR 574307
  • 4. P. Deligne, Cohomologie étale, Lecture Notes in Mathematics, Vol. 569, Springer-Verlag, Berlin-New York, 1977. Séminaire de Géométrie Algébrique du Bois-Marie SGA 41\over2; Avec la collaboration de J. F. Boutot, A. Grothendieck, L. Illusie et J. L. Verdier. MR 0463174
  • 5. Pierre Deligne, Phillip Griffiths, John Morgan, and Dennis Sullivan, Real homotopy theory of Kähler manifolds, Invent. Math. 29 (1975), no. 3, 245–274. MR 0382702
  • 6. Clifford J. Earle and Irwin Kra, On sections of some holomorphic families of closed Riemann surfaces, Acta Math. 137 (1976), no. 1-2, 49–79. MR 0425183
  • 7. J. Ellenberg: $ 2$-nilpotent quotients of fundamental groups of curves, unpublished manuscript.
  • 8. William Fulton and Joe Harris, Representation theory, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991. A first course; Readings in Mathematics. MR 1153249
  • 9. Bert van Geemen and Frans Oort, A compactification of a fine moduli space of curves, Resolution of singularities (Obergurgl, 1997) Progr. Math., vol. 181, Birkhäuser, Basel, 2000, pp. 285–298. MR 1748624
  • 10. Mark Goresky and Robert MacPherson, Stratified Morse theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 14, Springer-Verlag, Berlin, 1988. MR 932724
  • 11. A. Grothendieck, Éléments de géométrie algébrique. II. Étude globale élémentaire de quelques classes de morphismes, Inst. Hautes Études Sci. Publ. Math. 8 (1961), 222. MR 0217084
  • 12. A. Grothendieck: Letter to Faltings dated June 27, 1983. Available at http:// people.math.jussieu.fr/$ \sim$leila/grothendieckcircle/letters.php.
  • 13. Richard M. Hain, Completions of mapping class groups and the cycle 𝐶-𝐶⁻, Mapping class groups and moduli spaces of Riemann surfaces (Göttingen, 1991/Seattle, WA, 1991) Contemp. Math., vol. 150, Amer. Math. Soc., Providence, RI, 1993, pp. 75–105. MR 1234261, 10.1090/conm/150/01287
  • 14. Richard M. Hain, Torelli groups and geometry of moduli spaces of curves, Current topics in complex algebraic geometry (Berkeley, CA, 1992/93), Math. Sci. Res. Inst. Publ., vol. 28, Cambridge Univ. Press, Cambridge, 1995, pp. 97–143. MR 1397061
  • 15. Richard Hain, Infinitesimal presentations of the Torelli groups, J. Amer. Math. Soc. 10 (1997), no. 3, 597–651. MR 1431828, 10.1090/S0894-0347-97-00235-X
  • 16. Richard Hain, Relative weight filtrations on completions of mapping class groups, Groups of diffeomorphisms, Adv. Stud. Pure Math., vol. 52, Math. Soc. Japan, Tokyo, 2008, pp. 309–368. MR 2509715
  • 17. R. Hain: Lectures on Moduli Spaces of Elliptic Curves, in Transformation Groups and Moduli Spaces of Curves, Advanced Lectures in Mathematics, edited by Lizhen Ji, Shing-Tung Yau, no. 16 (2010), pp. 95-166, Higher Education Press, Beijing. arXiv:0812.1803.
  • 18. R. Hain: Monodromy of codimension-one sub-families of universal curves, arXiv:1006.3785.
  • 19. R. Hain: Remarks on non-abelian cohomology of proalgebraic groups, preprint, 2010.
  • 20. Richard Hain and Makoto Matsumoto, Weighted completion of Galois groups and Galois actions on the fundamental group of ℙ¹-{0,1,∞}, Compositio Math. 139 (2003), no. 2, 119–167. MR 2025807, 10.1023/B:COMP.0000005077.42732.93
  • 21. Richard Hain and Makoto Matsumoto, Tannakian fundamental groups associated to Galois groups, Galois groups and fundamental groups, Math. Sci. Res. Inst. Publ., vol. 41, Cambridge Univ. Press, Cambridge, 2003, pp. 183–216. MR 2012217
  • 22. Richard Hain and Makoto Matsumoto, Galois actions on fundamental groups of curves and the cycle 𝐶-𝐶⁻, J. Inst. Math. Jussieu 4 (2005), no. 3, 363–403. MR 2197063, 10.1017/S1474748005000095
  • 23. R. Hain, M. Matsumoto: Weighted completion of arithmetic mapping class groups, in preparation.
  • 24. R. Hain, M. Matsumoto, G. Pearlstein, T. Terasoma: Tannakian fundamental groups of categories of variations of mixed Hodge structure, in preparation.
  • 25. John Hubbard, Sur la non-existence de sections analytiques à la courbe universelle de Teichmüller, C. R. Acad. Sci. Paris Sér. A-B 274 (1972), A978–A979 (French). MR 0294719
  • 26. Dennis Johnson, The structure of the Torelli group. III. The abelianization of 𝒯, Topology 24 (1985), no. 2, 127–144. MR 793179, 10.1016/0040-9383(85)90050-3
  • 27. Alexandre I. Kabanov, Stability of Schur functors, J. Algebra 195 (1997), no. 1, 233–240. MR 1468891, 10.1006/jabr.1997.7049
  • 28. Minhyong Kim, The motivic fundamental group of 𝐏¹\sbs{0,1,∞} and the theorem of Siegel, Invent. Math. 161 (2005), no. 3, 629–656. MR 2181717, 10.1007/s00222-004-0433-9
  • 29. Minhyong Kim, The unipotent Albanese map and Selmer varieties for curves, Publ. Res. Inst. Math. Sci. 45 (2009), no. 1, 89–133. MR 2512779, 10.2977/prims/1234361156
  • 30. Toshitake Kohno and Takayuki Oda, The lower central series of the pure braid group of an algebraic curve, Galois representations and arithmetic algebraic geometry (Kyoto, 1985/Tokyo, 1986) Adv. Stud. Pure Math., vol. 12, North-Holland, Amsterdam, 1987, pp. 201–219. MR 948244
  • 31. Finn F. Knudsen, The projectivity of the moduli space of stable curves. III. The line bundles on 𝑀_{𝑔,𝑛}, and a proof of the projectivity of \overline𝑀_{𝑔,𝑛} in characteristic 0, Math. Scand. 52 (1983), no. 2, 200–212. MR 702954
  • 32. John P. Labute, On the descending central series of groups with a single defining relation, J. Algebra 14 (1970), 16–23. MR 0251111
  • 33. D. Mumford, J. Fogarty, and F. Kirwan, Geometric invariant theory, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], vol. 34, Springer-Verlag, Berlin, 1994. MR 1304906
  • 34. Hiroaki Nakamura, Naotake Takao, and Ryoichi Ueno, Some stability properties of Teichmüller modular function fields with pro-𝑙 weight structures, Math. Ann. 302 (1995), no. 2, 197–213. MR 1336334, 10.1007/BF01444493
  • 35. Jürgen Neukirch, Alexander Schmidt, and Kay Wingberg, Cohomology of number fields, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 323, Springer-Verlag, Berlin, 2008. MR 2392026
  • 36. B. Noohi, Fundamental groups of algebraic stacks, J. Inst. Math. Jussieu 3 (2004), no. 1, 69–103. MR 2036598, 10.1017/S1474748004000039
  • 37. Takayuki Oda, Etale homotopy type of the moduli spaces of algebraic curves, Geometric Galois actions, 1, London Math. Soc. Lecture Note Ser., vol. 242, Cambridge Univ. Press, Cambridge, 1997, pp. 85–95. MR 1483111
  • 38. A. Putman: The second rational homology group of the moduli space of curves with level structures, preprint 2008, arXiv:0809.4477.
  • 39. Wilfried Schmid, Variation of Hodge structure: the singularities of the period mapping, Invent. Math. 22 (1973), 211–319. MR 0382272
  • 40. John Stallings, Homology and central series of groups, J. Algebra 2 (1965), 170–181. MR 0175956
  • 41. Steven Zucker, Variation of mixed Hodge structure. II, Invent. Math. 80 (1985), no. 3, 543–565. MR 791674, 10.1007/BF01388730
  • 42. Jakob Stix, A monodromy criterion for extending curves, Int. Math. Res. Not. 29 (2005), 1787–1802. MR 2172341, 10.1155/IMRN.2005.1787
  • 43. Dennis Sullivan, On the intersection ring of compact three manifolds, Topology 14 (1975), no. 3, 275–277. MR 0383415
  • 44. André Weil, L’arithmétique sur les courbes algébriques, Acta Math. 52 (1929), no. 1, 281–315 (French). MR 1555278, 10.1007/BF02547409
  • 45. Kirsten Wickelgren, Lower central series obstructions to homotopy sections of curves over number fields, ProQuest LLC, Ann Arbor, MI, 2009. Thesis (Ph.D.)–Stanford University. MR 2713908

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (2010): 14G05, 14G27, 14H10, 14H25, 11G30, 14G32

Retrieve articles in all journals with MSC (2010): 14G05, 14G27, 14H10, 14H25, 11G30, 14G32


Additional Information

Richard Hain
Affiliation: Department of Mathematics, Duke University, Durham, North Carolina 27708-0320
Email: hain@math.duke.edu

DOI: http://dx.doi.org/10.1090/S0894-0347-2011-00693-0
Received by editor(s): January 27, 2010
Received by editor(s) in revised form: September 19, 2010, and December 29, 2010
Published electronically: January 25, 2011
Additional Notes: The author was supported in part by grant DMS-0706955 from the National Science Foundation and by MSRI
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.