I‑AIMS Institute for Artificial Intelligence
and Mathematical Sciences
/

The locus of curves with prescribed automorphism group

Kay Magaard, Tanush Shaska, Sergey Shpectorov, Helmut Völklein

I-AIMS paper 2001-2 · Published in Sūrikaisekikenkyūsho Kōkyūroku (2002), 112–141 · 2002

Let G be a finite group, and g ≥ 2. We study the locus of genus g curves that admit a G-action of given type, and inclusions between such loci. We use this to study the locus of genus g curves with prescribed automorphism group G. We completely classify these loci for g = 3 (including equations for the corresponding curves), and for g ≤ 10 we classify those loci corresponding to “large” G.

MSC classes: 14H37

PDF

References

  1. Robert D. M. Accola, On the number of automorphisms of a closed Riemann surface, Trans. Amer. Math. Soc. 131 (1968), 398–408. ↑
  2. José Bertin, Compactification des schémas de Hurwitz, C. R. Acad. Sci. Paris Sér. I Math. 322 (1996), no. 11, 1063–1066. ↑
  3. Hans Ulrich Besche and Bettina Eick, The groups of order at most 1000 except 512 and 768, J. Symbolic Comput. 27 (1999), no. 4, 405–413. ↑
  4. Rolf Brandt, Uberd̈ie Automorphismengruppen von algebraischen Funktionenkörpern., Essen: Univ. (Gesamthochschule) Essen, Fachbereich Mathematik, 1988 (German). ↑
  5. Thomas Breuer, Characters and automorphism groups of compact Riemann surfaces, London Mathematical Society Lecture Note Series, vol. 280, Cambridge University Press, Cambridge, 2000. ↑
  6. Marston Conder, Hurwitz groups: a brief survey, Bull. Amer. Math. Soc. (N.S.) 23 (1990), no. 2, 359–370. ↑
  7. H. M. Farkas and I. Kra, Riemann surfaces, Second, Graduate Texts in Mathematics, vol. 71, Springer-Verlag, New York, 1992. ↑
  8. Michael D. Fried and Helmut Völklein, The inverse Galois problem and rational points on moduli spaces, Math. Ann. 290 (1991), no. 4, 771–800. ↑
  9. GAP, GAP – Groups, Algorithms, and Programming, Version 4.13.1, The GAP Group, 2001. ↑
  10. Masaaki Homma, Automorphisms of prime order of curves, Manuscripta Math. 33 (1980/81), no. 1, 99–109. ↑
  11. A. Hurwitz, On Riemann surfaces with monogenic transformations into themselves., Math. Ann. 41 (1893), 403–442 (German). ↑
  12. F. Klein, On the seventh order transformation of elliptic functions., Math. Ann. 14 (1879), 428–471 (German). ↑
  13. Ravi S. Kulkarni, Riemann surfaces admitting large automorphism groups, Extremal Riemann surfaces (San Francisco, CA, 1995), 1997, pp. 63–79. ↑
  14. Akikazu Kuribayashi and Hideyuki Kimura, Automorphism groups of compact Riemann surfaces of genus five, J. Algebra 134 (1990), no. 1, 80–103. ↑
  15. Akikazu Kuribayashi and Kaname Komiya, On Weierstrass points and automorphisms of curves of genus three, Algebraic geometry (Proc. Summer Meeting, Univ. Copenhagen, Copenhagen, 1978), 1979, pp. 253–299. ↑
  16. ———, On the structure of the automorphism group of a compact Riemann surface of genus 3, Bull. Fac. Sci. Engrg. Chuo Univ. 23 (1980), 1–34. ↑
  17. Akikazu Kuribayashi and Izumi Kuribayashi, On subgroups of automorphism groups of hyperelliptic Riemann surfaces of genus three, Bull. Fac. Sci. Engrg. Chuo Univ. 24 (1981), 21–37. ↑
  18. ———, On subgroups of automorphism groups of hyperelliptic Riemann surfaces of genus three. II, Bull. Fac. Sci. Engrg. Chuo Univ. 25 (1982), 5–43. ↑
  19. Silvio Levy (ed.), The eightfold way, Mathematical Sciences Research Institute Publications, vol. 35, Cambridge University Press, Cambridge, 1999. The beauty of Klein’s quartic curve. ↑
  20. A. M. Macbeath, On a curve of genus 7, Proc. London Math. Soc. (3) 15 (1965), 527–542. ↑
  21. C. Maclachlan, Abelian groups of automorphisms of compact Riemann surfaces, Proc. London Math. Soc. (3) 15 (1965), 699–712. ↑
  22. Kay Magaard, Sergey Shpectorov, and Helmut Völklein, A GAP package for braid orbit computation and applications, Experiment. Math. 12 (2003), no. 4, 385–393. ↑
  23. Gunter Malle, Genus zero translates of three point ramified Galois extensions, Manuscripta Math. 71 (1991), no. 1, 97–111. ↑
  24. Takanori Matsuno, Compact Riemann surfaces with large automorphism groups, J. Math. Soc. Japan 51 (1999), no. 2, 309–329. ↑
  25. John F. X. Ries, Subvarieties of moduli space determined by finite groups acting on surfaces, Trans. Amer. Math. Soc. 335 (1993), no. 1, 385–406. ↑
  26. Tanush Shaska and Helmut Völklein, Elliptic subfields and automorphisms of genus 2 function fields, Algebra, arithmetic and geometry with applications (West Lafayette, IN, 2000), 2004, pp. 703–723. ↑
  27. David Singerman, Finitely maximal Fuchsian groups, J. London Math. Soc. (2) 6 (1972), 29–38. ↑
  28. ———, Symmetries of Riemann surfaces with large automorphism group, Math. Ann. 210 (1974), 17–32. ↑
  29. H. Völklein, Moduli spaces for covers of the Riemann sphere, Israel J. Math. 85 (1994), no. 1-3, 407–430. ↑
  30. Helmut Völklein, Groups as Galois groups, Cambridge Studies in Advanced Mathematics, vol. 53, Cambridge University Press, Cambridge, 1996. An introduction. ↑
  31. Robert A. Wilson, The Monster is a Hurwitz group, J. Group Theory 4 (2001), no. 4, 367–374. ↑

All preprints