A Numerical Survey of the Reduction of Modular Curve Genus by Fricke’s Involutions

  • Harvey Cohn
Conference paper


The modular curve of order N, relating j(z) and j(z/N), can be simplified by a system of involutions first due to Fricke. A computation is presented for finding the genus of the simplified curves by using the method of isometric circles due to Poincaré. This is an internal construction (in memory) for the fundamental domain. A survey of the results for N ≤ 310 confirms the 64 values of N (largest 119) for which the genus reduces to 0 and 65 values of N (largest 288) for which it reduces to 1. It also reveals 55 values of N (largest 390) for which the genus reduces to 2. While other surveys have been made, (e.g., by A. Ogg and by A.O.L. Atkin), the present one has the advantage of using minimal computational tools and theoretical techniques. The elliptic cases are interesting because in some instances e. g., N = 40,48,63,65,75, the involution process does not produce a genus reduction for the torus, and Poincaré’s method shows how this works geometrically. Also, in many cases the elliptic curves can be parametrised analytically using discriminant functions in a manner originally introduced by Fricke.


Elliptic Curve Fundamental Domain Modular Function Modular Curve Modular Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    A.O.L. ATKIN, Table 5, in “Modular Functions of One Variable IV, Lecture Notes 476,” Springer-Verlag, Berlin and New York, 1975.Google Scholar
  2. 2.
    A.O.L. ATKIN & J. LEHNER, Hecke operators on Γ0(m), Math. Annalen 185 (1970), 134–160.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    H. COHN, “Introduction to the Construction of Class Fields,” Cambridge University Press, 1985.zbMATHGoogle Scholar
  4. 4.
    H. COHN, Fricke’s two-valued Modular Equation, Math. of Comput. 51 (1988), 787–807.zbMATHGoogle Scholar
  5. 5.
    H. COHN, Computation of singular moduli by multivalued modular equations, Proc. Colloq. Comput. Number Theory Debrecen 1989 (to appear).Google Scholar
  6. 6.
    R. FRICKE, “Lehrbuch der Algebra III (Algebraische Zahlen),” Vieweg, Braunschweig, 1928.zbMATHGoogle Scholar
  7. 7.
    R. FRICKE, Über die Berechnung der Klasseninvarianten, Acta Arithmetica 52 (1929), 257–279.MathSciNetzbMATHGoogle Scholar
  8. 8.
    T.Y. FUNG, Fundamental domains of modular subgroups using isometric circles, Dissertation CUNY (1990).Google Scholar
  9. 9.
    H. HELLING, Note über das Geschlecht gewisser arithmetischer Gruppen, Math. Ann. 207 (1973), 173–179.MathSciNetCrossRefGoogle Scholar
  10. 10.
    M.A. KENKU, Atkin-Lehner involutions and class number residuality, Acta Arithmetica 33 (1977), 1–9.MathSciNetzbMATHGoogle Scholar
  11. 11.
    M.A. KENKU, The modular curves X 0(65) and X 0(91) and rational isogeny, Math. Proc. Cambridge Philos. Soc 87 (1980), 15–20.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    P.G. KLUIT, On the normalizer of Γ0(N), in “Modular Functions of one Variable V, Lecture Notes 601,” Springer-Verlag, Berlin and New York, 1977, pp. 239–247.CrossRefGoogle Scholar
  13. 13.
    M. I. KNOPP, “Modular Functions in Analytic Number Theory,” Markham, Chicago, Ill, 1970.zbMATHGoogle Scholar
  14. 14.
    J. LEHNER, “Discontinuous Groups and Automorphic Functions,” American Mathematical Society, Providence, Rhode Island, 1964.CrossRefGoogle Scholar
  15. 15.
    G. LIGOZAT, Courbes Modulaires de genre 1, Bull, de la Soc. Math. de France 43 Supplement (1975), 1–80.MathSciNetGoogle Scholar
  16. 16.
    M. NEWMAN, Construction and application of a class of modular functions, Proc. London Math. Soc. 7 (1957), 334–350.zbMATHGoogle Scholar
  17. 17.
    A.P. OGG, Hyperelliptic modular curves, Bull. Soc. Math. France 102 (1974), 107–148.MathSciNetGoogle Scholar
  18. 18.
    A.P. OGG, Modular functions, in “Proc. Sympos. Pure Math. 37,” Amer. Math. Soc, Providence, Rhode Island, 1980.Google Scholar

Copyright information

© Springer Science+Business Media New York 1991

Authors and Affiliations

  • Harvey Cohn
    • 1
  1. 1.Mathematics DepartmentCity College (City University of New York)New YorkUSA

Personalised recommendations