Skip to main content
SpringerLink
Log in

On Riemann surfaces of genus g with 4g–4 automorphisms

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

Abstract

In this article we study compact Riemann surfaces with a non-large group of automorphisms of maximal order, namely, compact Riemann surfaces of genus g with a group of automorphisms of order 4g–4. Under the assumption that g–1 is prime, we provide a complete classification of them and determine isogeny decompositions of the corresponding Jacobian varieties.

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. R. Accola, On the number of automorphisms of a closed Riemann surface, Transactions of the American Mathematical Society 131 (1968), 398–408.

    Article  MathSciNet  Google Scholar 

  2. G. Bartolini, A. F. Costa and M. Izquierdo, On the orbifold structure of the moduli space of Riemann surfaces of genera four and five, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matematicas 108 (2014), 769–793.

    Article  MathSciNet  Google Scholar 

  3. M. V. Belolipetsky and G. A. Jones, Automorphism groups of Riemann surfaces of genus p + 1, where p is prime. Glasgow Mathematical Journal 47 (2005), 379–393.

    Article  MathSciNet  Google Scholar 

  4. Ch. Birkenhake and H. Lange, Complex Abelian Varieties, Grundlehren der Mathema-tischen Wissenschaften, Vol. 302, Springer, Berlin, 2004.

  5. O. V. Bogopol'skiĭ, Classifying the actions of finite groups on orientable surfaces of genus 4, Siberian Advances in Mathematics 7 (1997), 9–38.

    MathSciNet  Google Scholar 

  6. S. A. Broughton, The equisymmetric stratification of the moduli space and the Krull dimension of mapping class groups, Topology and its Applications 37 (1990), 101–113.

    Article  MathSciNet  Google Scholar 

  7. S. A. Broughton, Finite groups actions on surfaces of low genus, Journal of Pure and Applied Algebra 69 (1991), 233–270.

    Article  MathSciNet  Google Scholar 

  8. E. Bujalance, A. F. Costa and M. Izquierdo, On Riemann surfaces of genus g with 4g automorphisms, Topology and its Applications 218 (2017), 1–18.

    Article  MathSciNet  Google Scholar 

  9. A. Carocca, S. Recillas and R. E. Rodríguez, Dihedral groups acting on Jacobians, in Complex Manifolds and Hyperbolic Geometry (Guanajuato, 2001), Contemporary Mathematics, Vol. 311, American Mathematical Society, Providence, RI, 2002, pp. 41–77.

    Article  Google Scholar 

  10. A. Carocca and R. E. Rodríguez, Jacobians with group actions and rational idempotents. Journal of Algebra 306 (2006), 322–343.

    Article  MathSciNet  Google Scholar 

  11. M. Conder, www.math.auckland.ac.nz/~conder/OrientableProperHypermaps101.txt.

  12. M. Conder, www.math.auckland.ac.nz/~conder/BigSurfaceActions-Genus2to101-ByType.txt.

  13. M. Conder and R. Kulkarni, Infinite families of automorphism groups of Riemann surfaces, in Discrete Groups and Geometry (Birmingham, 1991), London Mathematical Society Lecture Note Series, Vol. 173, Cambridge University Press, Cambridge, 1992, pp. 47–56.

    MATH  Google Scholar 

  14. A. F. Costa and M. Izquierdo, Equisymmetric strata of the singular locus of the moduli space of Riemann surfaces of genus 4, in Geometry of Riemann Surfaces, London Mathematical Society Lecture Note Series, Vol. 368, Cambridge University Press, Cambridge, 2010, pp. 120–138.

    MATH  Google Scholar 

  15. A. F. Costa and M. Izquierdo, One-dimensional families of Riemann surfaces of genus g with 4g + 4 automorphims, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matematicas 112 (2018), 623–631.

    Article  MathSciNet  Google Scholar 

  16. O. Debarre, Tores et variétés abéliennes complexes, Cours Spécialisés, Vol. 6, Societe Mathematique de France, Paris; EDP Sciences, Les Ulis, 1999.

  17. H. Farkas and I. Kra, Riemann Surfaces, Graduate Texts in Mathsematics, Vol. 71, Springer, New York-Berlin, 1980.

  18. J. Harvey, Cyclic groups of automorphisms of a compact Riemann surface, Quarterly Journal of Mathematics 17 (1966), 86–97.

    Article  MathSciNet  Google Scholar 

  19. J. Harvey, On branch loci in Teichmüller space, Transactions of the American Mathematical Society 153 (1971), 387–399.

    MathSciNet  MATH  Google Scholar 

  20. R. A. Hidalgo, L. Jiménez, S. Quispe and S. Reyes-Carocca, Quasiplatonic curves with symmetry group are definable over ℚ, Bulletin of the London Mathematical Society 49 (2017), 165–183.

    Article  MathSciNet  Google Scholar 

  21. M. Izquierdo, L. Jiménez and A. Rojas, Decomposition of Jacobian varieties of curves with dihedral actions via equisymmetric stratification, Revista Matemática Iberoameri-cana 35 (2019), 1259–1279.

    Article  MathSciNet  Google Scholar 

  22. M. Izquierdo and S. Reyes-Carocca, A note on large automorphism groups of compact Riemann surfaces, Journal of Algebra 547 (2020), 1–21.

    Article  MathSciNet  Google Scholar 

  23. E. Kani and M. Rosen, Idempotent relations and factors of Jacobians, Mathematische Annalen 284 (1989), 307–327.

    Article  MathSciNet  Google Scholar 

  24. C. Kassel and V. Turaev, Braid Groups, Graduate Texts in Mathematics, Vol. 247, Springer, New York, 2008.

  25. R. S. Kulkarni, A note on Wiman and Accola-Maclachlan surfaces. Annales Academiae Scientiarum Fennicae. Series A I. Mathematica 16 (1991), 83–94.

    Article  MathSciNet  Google Scholar 

  26. R. S. Kulkarni, Infinite families of surface symmetries, Israel Journal of Mathematics 76 (1991), 337–343.

    Article  MathSciNet  Google Scholar 

  27. R. S. Kulkarni, Riemann surfaces admitting large automorphism groups, in Extremal Riemann Surfaces (San Francisco, CA, 1995), Contemporary Mathematics, Vol. 201, American Mathematical Society, Providence, RI, 1997, pp. 63–79.

    Article  Google Scholar 

  28. A. Kuribayashi and I. Kuribayashi, Automorphism groups of compact Riemann surfaces of genus three and four, Journal of Pure and Applied Algebra 65 (1990), 277–292.

    Article  MathSciNet  Google Scholar 

  29. H. Lange and S. Recillas, Abelian varieties with group actions, Journal fur die Reine und Angewandte Mathematik 575 (2004), 135–155.

    MathSciNet  MATH  Google Scholar 

  30. C. Lomuto, Riemann surfaces with a large abelian group of automorphisms, Collectanea Mathematica 57 (2006), 309–318.

    MathSciNet  MATH  Google Scholar 

  31. A. MacBeath, The classification of non-euclidean crystallographic groups, Canadian Journal of Mathematics 19 (1967), 1192–1205.

    Article  MathSciNet  Google Scholar 

  32. C. Maclachlan, A bound for the number of automorphisms of a compact Riemann surface, Journal of the London Mathematical Society 44 (1969), 265–272.

    Article  MathSciNet  Google Scholar 

  33. K. Magaard, T. Shaska, S. Shpectorov and H. Völklein, The locus of curves with prescribed automorphism group, Sūrikaisekikenkyūsho Kōkyūroku 1267 (2002), 112–141.

    MathSciNet  Google Scholar 

  34. T. Matsuno, Compact Riemann surfaces with large automorphism groups, Journal of the Mathematical Society of Japan 51 (1999), 309–329.

    Article  MathSciNet  Google Scholar 

  35. J. Paulhus and A. M. Rojas, Completely decomposable Jacobian varieties in new genera, Experimental Mathematics 26 (2017), 430–445.

    Article  MathSciNet  Google Scholar 

  36. S. Recillas and R. E. Rodríguez, Jacobians and representations of S3, in Workshop on Abelian Varieties and Theta Functions (Morelia, 1996), Aportaciones Matematicas: Investigacion, Vol. 13, Sociedad Matemática Mexicana, México, 1998, pp. 117–140.

    Google Scholar 

  37. S. Reyes-Carocca, On the one-dimensional family of Riemann surfaces of genus q with 4q automorphisms, Journal of Pure and Applied Algebra 223 (2019), 2123–2144.

    Article  MathSciNet  Google Scholar 

  38. S. Reyes-Carocca and R. E. Rodríguez, A generalisation of Kani-Rosen decomposition theorem for Jacobian varieties, Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie V 19 (2019), 705–722.

    MathSciNet  MATH  Google Scholar 

  39. S. Reyes-Carocca and R. E. Rodríguez, On Jacobians with group action and coverings, Mathematische Zeitschrift 294 (2020), 209–227.

    Article  MathSciNet  Google Scholar 

  40. A. M. Rojas, Group actions on Jacobian varieties, Revista Matemática Iberoamerican 23 (2007), 397–420.

    Article  MathSciNet  Google Scholar 

  41. F. Schottky and H. Jung, Neue Sätze über Symmetralfunctionen und due Abel'schen Functionen der Riemann'schen Theorie, Sitzungsberichte der Preussischen Akademie der Wissenschaften, Physikalisch-Mathematische Klasse 1 (1909), 282–297.

    Google Scholar 

  42. J.-P. Serre, Linear Representations of Finite Groups, Graduate Texts in Mathematics, Vol. 42, Springer, New York-Heidelberg, 1977.

  43. D. Singerman, Finitely maximal Fuchsian groups, Journal of the London Mathematical Society 6 (1972), 29–38.

    Article  MathSciNet  Google Scholar 

  44. D. Singerman, Symmetries of Riemann surfaces with large automorphism group, Math-ematische Annalen 210 (1974), 17–32.

    Article  MathSciNet  Google Scholar 

  45. D. Swinarski, Equations of Riemann surfaces with automorphisms, in Mathematical Physics and Arithmetic Geometry, Contemporary Mathematics, Vol. 703, American Mathematical Society, Providence, RI, 2018, pp. 33–46.

    Article  MathSciNet  Google Scholar 

  46. A. Wiman, Über die hyperelliptischen Curven und diejenigen von Geschlechte p - Jwelche eindeutige Tiansformationen in sich zulassen, Bihang till Kungliga Svenska Vetenskapsakademiens Handlingar 21 (1895-6), 1–28.

    Google Scholar 

  47. W. Wirtinger, Untersuchungen über Theta Funktionen, Teubner, Berlin, 1895.

    MATH  Google Scholar 

Download references

Acknowledgment

The author is grateful to his colleague Angel Carocca who generously told him how to construct Example 2 in Section 7.

Author information

Authors and Affiliations

  1. Departamento de Matemática y Estadística, Universidad de La Frontera, Avenida Francisco Salazar, 01145, Temuco, Chile

    Sebastián Reyes-Carocca

Authors
  1. Sebastián Reyes-Carocca
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sebastián Reyes-Carocca.

Additional information

Partially supported by Fondecyt Grant 11180024, Redes Etapa Inicial Grant 2017-170071 and Anillo ACT1415 PIA-CONICYT Grant.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Reyes-Carocca, S. On Riemann surfaces of genus g with 4g–4 automorphisms. Isr. J. Math. 237, 415–436 (2020). https://doi.org/10.1007/s11856-020-2010-3

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-020-2010-3

Search

Navigation