The Ramanujan Journal

, Volume 17, Issue 1, pp 107–121 | Cite as

Further properties of a function of Ogg and Ligozat



Certain identities of Ramanujan may be succinctly expressed in terms of the rational function \(\breve{g}_{\chi}=\breve{f}_{\chi}-\frac{1}{\breve {f}_{\chi}}\) on the modular curve X0(N), where \(\breve{f}_{\chi}=w_{N}f_{\chi}\) and fχ is a certain modular unit on the Nebentypus cover Xχ(N) introduced by Ogg and Ligozat for prime \(N\equiv1\allowbreak\mkern5mu(\mathrm{mod}~4)\) and wN is the Fricke involution. These correspond to levels N=5,13, where the genus gN of X0(N) is zero. In this paper we study slightly more general kind of relations for each \(\breve{g}_{\chi}\) such that X0(N) has genus gN=1,2, and also for each \(h_{\chi}=g_{\chi}+\breve{g}_{\chi}\) such that the Atkin–Lehner quotient X0+(N) has genus gN+=1,2. Finally we study the normal closure of the field of definition of the zeros of the latter.


Modular units Nebentypus cover 

Mathematics Subject Classification (2000)

11E16 20H05 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Berndt, B.: Ramanujan’s Notebooks Part III. Springer, New York (1991) MATHGoogle Scholar
  2. 2.
    Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system I: the user language. J. Symb. Comp. 24(3-4), 235–265 (1997). MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Cremona, J.E.: Elliptic curves of conductor ≤20,000.
  4. 4.
    Csirik, J.: The kernel of the Eisenstein ideal, U.C. Berkeley Ph.D. thesis (1999), v+49 Google Scholar
  5. 5.
    Darmon, H.: Stark-Heegner points over real quadratic fields. In: Number Theory (Tiruchirapalli, 1996). Contemporary Mathematics, vol. 210, pp. 41–69. Am. Math. Soc., Providence (1998) Google Scholar
  6. 6.
    Grayson, D.R., Stillman, M.E.: Macaulay 2, a software system for research in algebraic geometry. Available at
  7. 7.
    Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers, 4th edn. Oxford University Press, London (1960) MATHGoogle Scholar
  8. 8.
    Jenkins, M.: Proof of an arithmetical theorem leading, by means of Gauss’ fourth demonstration of Legendre’s law of reciprocity, to the extension of that law. Proc. Lond. Math. Soc. 2, 29–32 (1867) CrossRefGoogle Scholar
  9. 9.
    Kubert, D.S., Lang, S.: Units in the modular function field. II. A full set of units. Math. Ann. 218(2), 175–189 (1975) CrossRefMathSciNetGoogle Scholar
  10. 10.
    Kubert, D.S., Lang, S.: Modular Units. Grundlehren der Mathematischen Wissenschaften, vol. 244. Springer, New York (1981) MATHGoogle Scholar
  11. 11.
    Mazur, B.: Modular curves and the Eisenstein ideal. Publ. Math. IHES 47, 33–186 (1977) MATHMathSciNetGoogle Scholar
  12. 12.
    Ogg, A.P.: Rational points on certain elliptic modular curves. In: Analytic Number Theory, Proc. Sympos. Pure Math., vol. XXIV, St. Louis Univ., St. Louis, Mo., 1972, pp. 221–231. Am. Math. Soc., Providence (1973) Google Scholar
  13. 13.
    The PARI Group, Bordeaux, PARI/GP, Version 2.1.5.
  14. 14.
    Serre, J.P.: A Course in Arithmetic. Springer, New York/Heidelberg (1973) MATHGoogle Scholar
  15. 15.
    Shimura, G.: Introduction to the Arithmetic Theory of Automorphic Functions. Princeton University Press, Princeton (1994) MATHGoogle Scholar
  16. 16.
    Singmaster, D.: Problem 1654. Math. Mag. 75(4), 317 (2002) MathSciNetCrossRefGoogle Scholar
  17. 17.
    Sun, Z.: Products of binomial coefficients modulo p 2. Acta Arith. 97(1), 87–98 (2001) MATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Tate, J.T.: Les conjectures de Stark sur les fonctions L d’Artin en s=0. In: Progress in Mathematics, vol. 47. Birkhäuser, Boston (1984). Lecture notes edited by Dominique Bernardi and Norbert Schappacher Google Scholar
  19. 19.
    Zagier, D.: Modular points, modular curves, modular surfaces and modular forms. In: Workshop Bonn 1984. Lecture Notes in Mathematics, vol. 1111, pp. 225–248. Springer, Berlin (1985) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Department of Pure Mathematics and Mathematical StatisticsUniversity of CambridgeCambridgeUK

Personalised recommendations