Skip to main content
SpringerLink
Log in

Alternating groups and moduli space lifting invariants

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

The genus of a curve discretely separates decidely different algebraic relations in two variables to focus us on the connected moduli space M g . Yet, modern applications also require a data variable (function) on the curve. The resulting spaces are versions, depending on our needs for this data variable, of Hurwitz spaces. A Nielsen class (§1.1) consists of r >- 3 conjugacy classes C in the data variable monodromy G. It generalizes the genus.

Some Nielsen classes define connected spaces. To detect, however, the components of others requires further subtler invariants. We regard our Main Result (MR) as level 0 of Spin invariant information on moduli spaces.

In the MR, G = A n (the alternating group), r counts the data variable branch points and \(C = {C_{{3^r}}}\) is r repetitions of the 3-cycle conjugacy class. This Nielsen class defines two spaces called absolute and inner: \(H{({A_n},{C_{{3^r}}})^{abs}}\) of degree n, genus g = r − (n − 1) > 0 covers and \(H{({A_n},{C_{{3^r}}})^{in}}\) parametrizing Galois closures of such covers. The parity of a spin invariant precisely identifies the two components of each space. The inner result is the deeper.

We examine the effect of combining the MR, [ArP05] and ½-canonical classes on M g . First: §5.2 considers an analog of a famous conjecture of Shafarevich: With H the composite group of all Galois extensions K/Q with group some alternating group, does the canonical map G ℚ → H have pro-free kernel. Second: Thm. 6.15 produces nonzero automorphic (θ-null power) functions on the reduced Hurwitz spaces \({H_ + }{({A_n},{C_{{3^r}}})^{abs,rd}}\) (resp. \({H_ - }{({A_n},{C_{{3^r}}})^{abs,rd}}\) when r is even (resp. odd), for either g=1 or n≥12g+4.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. Artebani and P. Pirola, Algebraic functions with even monodromy, Proceedings of the American Mathematical Society 133 (2005), 331–341.

    Article  MATH  MathSciNet  Google Scholar 

  2. M. F. Atiyah, Riemann surfaces and spin structures, Annales scientifiques de l’École Normale Supérieure 4 (1971), 47–62.

    MATH  MathSciNet  Google Scholar 

  3. P. Bailey and M. Fried, Hurwitz monodromy, spin separation and higher levels of a Modular Tower, in Proceedings of Symposia in Pure Mathematics 70 (2002), 79–221 (M. Fried and Y. Ihara, eds.) 1999 von Neumann Symposium, August 16–27, 1999 MSRI.

  4. R. Biggers and M. Fried, Moduli spaces of covers and the Hurwitz monodromy group, Crelle’s Journal 335 (1982), 87–121.

    MATH  MathSciNet  Google Scholar 

  5. P. Dèbes and M. D. Fried, Nonrigid constructions in Galois theory, Pacific Journal of Mathematics 163 (1994), 81–122.

    MATH  MathSciNet  Google Scholar 

  6. H. Farkas and I. Kra, Theta Constants, Riemann Surfaces and the Modular Group, AMS Graduate Text Series 37, American Mathematical Society, Providence, RI, 2001.

    MATH  Google Scholar 

  7. J. Fay, Theta Functions on Riemann Surfaces, Lecture Notes in Mathematics 352, Springer-Verlag, Heidelberg, 1973.

    Google Scholar 

  8. M. Fried, Fields of definition of function fields and Hurwitz families and groups as Galois groups, Communications in Algebra 5 (1977), 17–82.

    Article  MATH  MathSciNet  Google Scholar 

  9. M. Fried, Combinatorial computation of moduli dimension of Nielsen classes of covers, Contemporary Mathematics 89 (1989), 61–79.

    MathSciNet  Google Scholar 

  10. M. Fried, Arithmetic of 3 and 4 branch point covers: A bridge provided by noncongruence subgroups of SL2(ℤ), Progress in Mathematics 81 (1990), 77–117.

  11. M. Fried, Enhanced review: Serre’s Topics in Galois Theory, Proceedings of the Recent Developments in the Inverse Galois Problem, AMS Contemporary Mathematics 186 (1995), 15–32.

  12. M. Fried, Introduction to Modular Towers: Generalizing the relation between dihedral groups and modular curves, Proceedings AMS-NSF Summer Conference, Vol. 186, Cont. Math series, Recent Developments in the Inverse Galois Problem, American Mathematical Society, Providence, RI, 1995, pp. 111–171.

    Google Scholar 

  13. M. Fried, Relating two genus 0 problems of John Thompson, Volume for John Thompson’s 70th birthday, Kluwer, New York, October 2004. At www.math.uci.edu/~mfried/#mt.

    Google Scholar 

  14. M. Fried, The Main Conjecture of Modular Towers and its higher rank generalization, in Groupes de Galois (Lum. 2004) (Bertrand and Debes, eds.), Congres 13 (2006), 165–233.

  15. M. Fried, Connectedness of families of covers of the sphere of An-type, At www.math.uci.edu/~mfried/paplist-mt/twoorbit.pdf.

  16. M. D. Fried, On-line Definitions related to Inverse Galois and Hurwitz spaces, http://www.math.uci.edu/~mfried/deflist-cov.html.

  17. M. Fried, Riemann’s Existence Theorem:An elementary approach to moduli, Four of its five chapters, and a synopsis of each at www.math.uci.edu/~mfried/#ret.

  18. M. Fried and M. Jarden, Diophantine properties of subfields of \({\bar Q}\), American Journal of Mathematics 100 (1978), 774–787.

    Article  MathSciNet  Google Scholar 

  19. M. Fried and M. Jarden, Field Arithmetic, Ergebnisse der Mathematik III, Vol. 11, Springer-Verlag, Heidelberg, 1986; 2nd Edition 2004.

    Google Scholar 

  20. M. Fried, E. Klassen and Y. Kopeliovic, Realizing alternating groups as monodromy groups of genus one covers, Proceedings of the American Mathematical Society 129 (2001), 111–119.

    Article  MATH  MathSciNet  Google Scholar 

  21. M. Fried and Y. Kopeliovic, Applying Modular Towers to the inverse Galois problem, Geometric Galois Actions II Dessins d’Enfants, Mapping Class Groups and Moduli, London Math. Soc. Lecture Notes, Vol. 243, Cambridge University Press, Cambridge, 1997, pp. 172–197.

    Google Scholar 

  22. M. Fried and H. Völklein, The inverse Galois problem and rational points on moduli spaces, Mathematische Annalen 290 (1991), 771–800.

    Article  MATH  MathSciNet  Google Scholar 

  23. M. Fried and H. Völklein, The embedding problem over an Hilbertian-PAC field, Annals of Math 135 (1992), 469–481.

    Article  Google Scholar 

  24. H. Grauert and H. Remmert, 3 papers in Comptes Rendus, Paris 245 (1957). 819–822, 822–825, 918–921.

    MATH  MathSciNet  Google Scholar 

  25. R. Guralnick and K. Magaard, On the minimal degree of a primitive permutation group, Journal of Algebra 207 (1998), 127–145.

    Article  MATH  MathSciNet  Google Scholar 

  26. R. Guralnick and M. Neubauer, Monodromy groups of branched coverings: the generic case, in Recent Developments in the Inverse Galois Problem (Seattle, Wa, 1993), Contemporary Mathematics 186, American Mathematical Society, Providence, RI, 1995, pp. 325–352.

    Google Scholar 

  27. R. Guralnick and J. Shareshian, Symmetric and Alternating Groups as Monodromy Groups of Riemann Surfaces I: Generic Covers and Covers with Many Branch Points, Memoirs of the American Mathematical Society 189 (2007), No. 886, 128 pp.

  28. R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics 52, Springer-Verlag, Berlin, 1977.

    Google Scholar 

  29. J. Igusa, Theta Functions, Berlin, Springer, 1972.

    MATH  Google Scholar 

  30. A. Krazer and F. Prym, Neue Grundlagen einer Theorie der Allgemeinen Thetafunctionen, Teubner, Leipzig, 1892.

    MATH  Google Scholar 

  31. J. Lewittes, Riemann surfaces and the theta functions, Acta Mathematica 111 (1964), 37–61.

    Article  MATH  MathSciNet  Google Scholar 

  32. F. Liu and B. Osserman, The Irreducibility of Certain Pure-cycle Hurwitz Spaces, American Journal of Mathematics 130 (2008), 1687–1708

    Article  MATH  MathSciNet  Google Scholar 

  33. K. Magaard, S. Shpectorov and H. Voelklein, A GAP package for braid orbit computation, and applications, to appear in Experimenatl Mathematics, kaym@math.wayne.edu/~kaym/research has the program (for variants of Unix).

  34. K. Magaard and H. Voelklein, The monodromy group of a function on a general curve, Israel Journal of Mathematics 141 (2004), 355–368.

    Article  MATH  MathSciNet  Google Scholar 

  35. G. Malle and B.H. Matzat, Inverse Galois Theory, Monographs in Mathematics, Springer, Berlin, 1999.

    MATH  Google Scholar 

  36. A. Mayer, On the Jacobi Inversion Theorem, Ph.D. thesis, Princeton, 1961.

  37. J.-F. Mestre, Extensions régulières de ℚ(t) de groupe de Galois à n, Journal of Algebra 131 (1990), 483–495.

    Article  MATH  MathSciNet  Google Scholar 

  38. D. Mumford, Introduction to Algebraic Geometry, Harvard University Notes, Cambridge, MA, 1966.

    Google Scholar 

  39. D. Mumford, Theta-characteristics of an algebraic curve, Annales scientifiques de l’École Normale Supérieure 4 (1971), 181–192.

    MATH  MathSciNet  Google Scholar 

  40. D. Mumford, Curves and their Jacobians, University of Michigan Press, Ann Arbor, 1976.

    Google Scholar 

  41. D. Mumford, Tata Lectures on Theta II, Progress in Mathematics 43, Birkhäuser, Boston, Basel, Stuttgart, 1984.

    MATH  Google Scholar 

  42. D. Mumford and J. Fogarty, Geometric Invariant Theory, 2nd. edn., Ergebnisse der Math. 34, Springer-Verlag, Berlin, 1984.

    Google Scholar 

  43. J. Neukirch, Ü ber die absoluten Galois gruppen algebriascher Zahlkörper, Jahresbericht der Deutschen Mathematiker-Vereinigung 76 (1974) 18–37.

  44. H. Poincaré, Remarques diverses sur les fonctions abéliennes, Jour. de Math. (5th series) I (1895).

  45. J.-P. Serre, L’invariant de Witt de la forme T(x 2), Comentarii Mathematici Helvetici 59 (1984), 651–676.

  46. J.-P. Serre, Relêvements dans à n, Comptes Rendus Mathématique. Académie des Sciences. Paris 311 (1990), 477–482.

    Google Scholar 

  47. J.-P. Serre, Revêtements a ramification impaire et thêta-caractéristiques, Comptes Rendus Mathématique. Académie des Sciences. Paris 311 (1990), 547–552.

    Google Scholar 

  48. J.-P. Serre, Topics in Galois Theory, notes taken by H. Darmon, Bartlett and Jones Publishers, Boston, MA, 1992.

    MATH  Google Scholar 

  49. J.-P. Serre, Galois Cohomology, Springer-Verlag, Berlin, 1997.

    MATH  Google Scholar 

  50. G. Shimura, Abelian Varieties with Complex Multiplication and Modular Functions, Princeton University Press, Princeton, 1998.

    MATH  Google Scholar 

  51. H. Völklein, Groups as Galois Groups, Cambridge Studies in Advanced Mathematics, 53, Cambridge University Press, Cambridge, 1996.

    Book  MATH  Google Scholar 

  52. H. Völklein, The Image of a Hurwitz space under the moduli map, in Progress in Galois Theory, (H. Völklein and T. Shaska, eds.), Springer Science, Berlin, 2005, pp. 135–150.

    Chapter  Google Scholar 

  53. S. Wewers, Construction of Hurwitz spaces, PhD thesis, I.E.M. - Essen, 1998.

Download references

Author information

Authors and Affiliations

  1. Emeritus, Mathematics Department, UC Irvine, Irvine, CA, 92697, USA

    Michael D. Fried

Authors
  1. Michael D. Fried
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michael D. Fried.

Additional information

Support from BiNational Science Foundation Grant-Israel #87-00038, Alexander von Humboldt Found., Institut für Experimentalle Math., July 1996, from NSF #DMS-99305590 and #DMS-0202259 and the Workshop on algebraic curves and monodromy, April 2004 in Milan.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fried, M.D. Alternating groups and moduli space lifting invariants. Isr. J. Math. 179, 57–125 (2010). https://doi.org/10.1007/s11856-010-0073-2

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-010-0073-2

Keywords

Search

Navigation