Skip to main content
SpringerLink
Log in

Relating Two Genus 0 Problems of John Thompson

  • Conference paper
Progress in Galois Theory

Part of the book series: Developments in Mathematics ((DEVM,volume 12))

Abstract

Excluding a precise list of groups like alternating, symmetric, cyclic and dihedral, from 1st year algebra (§7.2.3), we expect there are only finitely many monodromy groups of primitive genus 0 covers. Denote this nearly proven genus 0 problem as Problem g=00 2. We call the exceptional groups 0-sporadic. Example: Finitely many Chevalley groups are 0-sporadic. A proven result: Among polynomial 0-sporadic groups, precisely three produce covers falling in nontrivial reduced families. Each (miraculously) defines one natural genus 0 ℚ cover of the j-line. The latest Nielsen class techniques apply to these dessins d’enfant to see their subtle arithmetic and interesting cusps.

John Thompson earlier considered another genus 0 problem: To find θ-functions uniformizing certain genus 0 (near) modular curves. We call this Problem g=00 1. We pose uniformization problems for j-line covers in two cases. First: From the three 0-sporadic examples of Problem g=00 2. Second: From finite collections of genus 0 curves with aspects of Problem g=00 1.

Thanks to NSF Grant #DMS-0202259, support for the Thompson Semester at the University of Florida

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 89.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 119.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. W. Aitken, M. Fried and L. Holt, Davenport Pairs over finite fields, in proof, PJM, Dec. 2003.

    Google Scholar 

  2. M. Aschbacher, On conjectures of Guralnick and Thompson, J. Algebra 1990 #2, 277–343.

    Article  MATH  MathSciNet  Google Scholar 

  3. M. Aschbacher and L. Scott, Maximal subgroups of finite groups, J. Alg. 92 (1985), 44–80.

    Article  MATH  MathSciNet  Google Scholar 

  4. E. Artin, Geometric Algebra, Interscience tracts in pure and applied math. 3, 1957.

    Google Scholar 

  5. 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) editors M. Fried and Y. Ihara, 1999 von Neumann Conference on Arithmetic Fundamental Groups and Noncommutative Algebra, August 16–27, 1999 MSRI, 79–221.

    Google Scholar 

  6. J. Browne, Charles Darwin: The Power of Place, Knopf, 2003.

    Google Scholar 

  7. R.D. Carmichael, Introduction to the theory groups of finite order, Dover Pub. 1956.

    Google Scholar 

  8. J.-M. Couveignes and P. Cassou-Noguès, Factorisations explicites de g(y)-h(z), Acta Arith. 87 (1999), no. 4, 291–317.

    MATH  MathSciNet  Google Scholar 

  9. J.-M. Couveignes, Tools for the computation of families of covers, in Aspects of Galois Theory, Ed: H. Völklein, Camb. Univ. Press, LMS Lecture Notes 256 (1999), 38–65. The author informs me that due to a mistake of the editors, three pages have been cyclically permuted. The correct order of pages can be found following the numbering of formulae.

    Google Scholar 

  10. C.W. Curtis, W.M. Kantor and G.M. Seitz, The 2-transitive permutation representations of the finite Chevalley groups, TAMS 218 (1976), 1–59.

    Article  MATH  MathSciNet  Google Scholar 

  11. H. Davenport, D.J. Lewis and A. Schinzel, Equations of the form f(x) = g(y), Quart. J. Math. Oxford 12 (1961), 304–312.

    MATH  MathSciNet  Google Scholar 

  12. H. Davenport, and A. Schinzel, Two problems concerning polynomials, Crelle’s J. 214 (1964), 386–391.

    MathSciNet  Google Scholar 

  13. P. Debes and M. Emsalem, Harbater-Mumford components and Towers of Moduli Spaces, Presentation by M. Emsalem at Graz, July 2003, preprint Jan. 2004.

    Google Scholar 

  14. P. Debes and M.D. Fried, Nonrigid situations in constructive Galois theory, Pacific Journal 163 #1 (1994), 81–122.

    MATH  MathSciNet  Google Scholar 

  15. P. Debes and M.D. Fried, Integral specializations of families of rational functions, PJM 190 (1999), 45–85.

    MATH  MathSciNet  Google Scholar 

  16. H. Farkas and I. Kra, Theta Constants, Riemann Surfaces and the Modular Group, AMS graduate text series 37, 2001.

    Google Scholar 

  17. W. Feit, On symmetric balanced incomplete block designs with doubly transitive automorphism groups, J. of Comb.; Series A bf (1973), 221–247.

    Google Scholar 

  18. W. Feit, Some consequences of the classification of finite simple groups, Proceedings of Symposia in Pure Math: Santa Cruz Conference on Finite Groups, A.M.S. Publications 37 (1980), 175–181.

    MATH  MathSciNet  Google Scholar 

  19. W. Feit, E-mail to Peter Müller, Jan. 28, 1992.

    Google Scholar 

  20. M.D. Fried, On a conjecture of Schur, Mich. Math. J. 17 (1970), 41–45.

    Article  MATH  MathSciNet  Google Scholar 

  21. M.D. Fried, The field of definition of function fields and a problem in the reducibility of polynomials in two variables, Ill. J. of Math. 17 (1973), 128–146.

    MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  23. M. Fried, Galois groups and Complex Multiplication, T.A.M.S. 235 (1978) 141–162.

    Article  MATH  MathSciNet  Google Scholar 

  24. M.D. Fried, Exposition on an Arithmetic-Group Theoretic Connection via Riemann’s Existence Theorem, Proceedings of Symposia in Pure Math: Santa Cruz Conference on Finite Groups, A.M.S. Publications 37 (1980), 571–601.

    MATH  MathSciNet  Google Scholar 

  25. M. Fried, Extension of Constants, Rigidity, and the Chowla-Zassenhaus Conjecture, Finite Fields and their applications, Carlitz volume 1 (1995), 326–359.

    Article  MATH  MathSciNet  Google Scholar 

  26. M. D. Fried, Modular towers: Generalizing the relation between dihedral groups and modular curves, Proceedings AMS-NSF Summer Conference, 186, 1995, Cont. Math series, Recent Developments in the Inverse Galois Problem, 111–171.

    MATH  MathSciNet  Google Scholar 

  27. M.D. Fried, Separated variables polynomials and moduli spaces, Number Theory in Progress (Berlin-New York) (ed. J. Urbanowicz K. Gyory, H. Iwaniec, ed.), Walter de Gruyter, 1999, Proceedings of the Schinzel Festschrift, Summer 1997: Available from www.math.uci.edu/~mfried/#math, 169–228.

    Google Scholar 

  28. M.D. Fried, Prelude: Arithmetic fundamental groups and noncommutative algebra, Proceedings of Symposia in Pure Mathematics, 70 (2002) editors M. Fried and Y. Ihara, 1999 von Neumann Conference on Arithmetic Fundamental Groups and Noncommutative Algebra, August 16–27, 1999 MSRI, vii–xxx.

    Google Scholar 

  29. M.D. Fried, Extension of constants series and towers of exceptional covers, preprint available in list at www.math.uci.edu/~mfried/#math or www.math.uci.edu/~mfried/psfiles/exctow.html.

    Google Scholar 

  30. M.D. Fried, Riemann’s existence theorem: An elementary approach to moduli, Chaps. 1–4 available at www.math.uci.edu/~mfried/#ret.

    Google Scholar 

  31. M.D. Fried, R. Guralnick and J. Saxl, Schur covers and Carlitz’s conjecture, Israel J. 82 (1993), 157–225.

    MATH  MathSciNet  Google Scholar 

  32. M.D. Fried and D. Semmen, Schur multiplier types and Shimura-like systems of varieties, preprint available in the list at www.math.uci.edu/~mfried/#mt or www.math.uci.edu/~mfried/psfiles/schurtype.html.

    Google Scholar 

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

    Article  MATH  Google Scholar 

  34. D. Frohardt and K. Magaard, Composition Factors of Monodromy Groups, Annals of Math. 154 (2001), 1–19

    MathSciNet  Google Scholar 

  35. D. Frohardt, R.M. Guralnick and K. Magaard, Genus 0 actions of groups of Lie rank 1, in Proceedings of Symposia in Pure Mathematics 70 (2002) editors M. Fried and Y. Ihara, 1999 von Neumann Conference on Arithmetic Fundamental Groups and Noncommutative Algebra, August 16–27, 1999 MSRI, 449–483.

    Google Scholar 

  36. L. Gerritzen, F. Herrlich, and M. van der Put, Stable n-pointed trees of projective lines, Ind. Math. 50 (1988), 131–163.

    MATH  Google Scholar 

  37. R.M. Guralnick, The genus of a permutation group, in Groups, Combinatorics and Geometry, Ed: M. Liebeck and J. Saxl, LMS Lecture Note Series 165, CUP, Longdon, 1992.

    Google Scholar 

  38. R. Guralnick, P. Müller and J. Saxl, The rational function analoque of a question of Schur and exceptionality of permutations representations, Memoirs of the AMS 162 773 (2003), ISBN 0065-9266.

    Google Scholar 

  39. R.M. Guralnick and M.G. Neubauer, Monodromy groups of branched coverings: the generic case, Proceedings AMS-NSF Summer Conference, 186, 1995, Cont. Math series, Ed: M. Fried Recent Developments in the Inverse Galois Problem, 325–352.

    Google Scholar 

  40. R.M. Guralnick and J. Shareshian, Symmetric and Alternating Groups as Monodromy Groups of Riemann Surfaces I, preprint.

    Google Scholar 

  41. R.M. Guralnick and J.G. Thompson, Finite groups of genus 0, J. Algebra 131 (1990), 303–341.

    Article  MATH  MathSciNet  Google Scholar 

  42. M. Liebeck and J. Saxl, Minimal degrees of primitive permutation groups, with an application to monodromy groups of covers of Riemann surfaces, PLMS (3) 63 (1991), 266–314.

    MATH  MathSciNet  Google Scholar 

  43. M. Liebeck and A. Shalev, Simple groups, permutation groups, and probability, 497–520.

    Google Scholar 

  44. G. Malle and B.H. Matzat, Inverse Galois Theory, ISBN 3-540-62890-8, Monographs in Mathematics, Springer, 1999.

    Google Scholar 

  45. P. Müller, Kronecker conjugacy of polynomials, TAMS 350 (1998), 1823–1850.

    Article  MATH  Google Scholar 

  46. P. Müller, Primitive monodromy groups of polynomials, Recent developments in the inverse Galois problem AMS, Cont. Math. Series Editor: Michael (1995), 385–401.

    Google Scholar 

  47. K. Parshall and A. Rice, Mathematics unbound: The evolution of an international mathematical research community, 1800–1945, History of Math. vol. 23, AMS/LMS, Prov. RI, 2002.

    Google Scholar 

  48. A. Pizer, A Note on a Conjecture of Hecke, PJM 79 (1978), 541–548.

    MathSciNet  Google Scholar 

  49. D. Rowe, Review of [PaR02], BAMS 40 #4 (2003), 535–542.

    Google Scholar 

  50. J.-P. Serre, Abelian ℓ-adic representations and elliptic curves, 1st ed., McGill University Lecture Notes, Benjamin, New York • Amsterdam, 1968, in collaboration with Willem Kuyk and John Labute.

    Google Scholar 

  51. J.-P. Serre, Revêtements a ramification impaire et thêta-caractéristiques, C. R. Acad. Sci. Paris 311 (1990), 547–552.

    MATH  MathSciNet  Google Scholar 

  52. T. Shih, A note on groups of genus zero, Comm. Alg. 19 (1991), 2813–2826.

    MATH  MathSciNet  Google Scholar 

  53. C.L. Siegel, Analytic Zahlentheorie II Vorlesungen, gehalten im Wintersemester 1963/64 an der Universität Göttingen, mimeographed notes.

    Google Scholar 

  54. J.G. Thompson, Finite groups and modular functions, RIMS, 11(3) (1979), 347–351.

    MATH  Google Scholar 

  55. J.G. Thompson, Some Numerology between the Fischer-Griess Monster and the elliptic modular function, BLMS 11(3) (1979), 340–346.

    MATH  Google Scholar 

  56. U. Ray, Generalized Kac-Moody algebras and some related topics, BAMS 38 #1, 1–42.

    Google Scholar 

  57. R. Solomon, A brief history of the classification of finite simple groups, BAMS 38 #3 (2001), 315–352.

    Article  MATH  Google Scholar 

  58. H. Völklein, Groups as Galois Groups 53, Cambridge Studies in Advanced Mathematics, Camb. U. Press, Camb. England, 1996.

    MATH  Google Scholar 

  59. S. Wewers, Deformation of tame admissible covers of curves, in Aspects of Galois Theory, Ed: H. Völklein, Camb. Univ. Press, LMS Lecture Notes 256 (1999), 239–282.

    Google Scholar 

  60. K. Wohlfahrt, An extension of F. Klein’s level concept, Ill. J. Math. 8 (1964), 529–535.

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. 3135 Rosemont Way, Billings, MT, 59101

    Michael D. Fried

Authors
  1. Michael D. Fried
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. University of Florida, USA

    Helmut Voelklein

  2. University of Idaho, USA

    Tanush Shaska

Rights and permissions

Reprints and permissions

Copyright information

© 2005 Springer Science+Business Media, Inc.

About this paper

Cite this paper

Fried, M.D. (2005). Relating Two Genus 0 Problems of John Thompson. In: Voelklein, H., Shaska, T. (eds) Progress in Galois Theory. Developments in Mathematics, vol 12. Springer, Boston, MA. https://doi.org/10.1007/0-387-23534-5_4

Download citation

Publish with us

Policies and ethics

Search

Navigation