The Ramanujan Journal

, Volume 47, Issue 2, pp 291–308 | Cite as

Equations and rational points of the modular curves \(X_0^+(p)\)

  • Pietro MercuriEmail author


Let p be an odd prime number and let \(X_0^+(p)\) be the quotient of the classical modular curve \(X_0(p)\) by the action of the Atkin–Lehner operator \(w_p\). In this paper, we show how to compute explicit equations for the canonical model of \(X_0^+(p)\). Then we show how to compute the modular parametrization, when it exists, from \(X_0^+(p)\) to an isogeny factor E of dimension 1 of its Jacobian \(J_0^+(p)\). Finally, we show how to use this map to determine the rational points on \(X_0^+(p)\) up to a large fixed height.


Modular curve Modular parametrization 

Mathematics Subject Classification

Primary: 11G18 Secondary: 14H25 


  1. 1.
    Atkin, A.O.L., Lehner, J.: Hecke operators on \(\Gamma _0(m)\). Math. Ann. 185, 134–160 (1970)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Baker, M.H., Gonzalez-Jimenez, E., Gonzalez, J., Poonen, B.: Finiteness results for modular curves of genus at least 2. Am. J. Math. 127(6), 1325–1387 (2005)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bilu, Y., Parent, P., Rebolledo, M.: Rational points on \(X^+_0(p^r)\) (2011). arXiv:1104.4641
  4. 4.
    Cremona, J.E.: Algorithms for Modular Elliptic Curves, 2nd edn. Cambridge University Press, Cambridge (1997)zbMATHGoogle Scholar
  5. 5.
    Diamond, F., Shurman, J.: A First Course in Modular Forms. Graduate Texts in Mathematics, vol. 228. Springer, New York (2005)Google Scholar
  6. 6.
    Galbraith, S.D.: Equations for modular curves. PhD Thesis (1996)Google Scholar
  7. 7.
    Galbraith, S.D.: Rational points on \(X_0^+(p)\). Exp. Math. 8(4), 311–318 (1999)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Griffiths, P., Harris, J.: Principles of Algebraic Geometry. Wiley, New York (1978)zbMATHGoogle Scholar
  9. 9.
    Hasegawa, Y., Hashimoto, K.: Hyperelliptic modular curves \(X_0^*(N)\) with square-free levels. Acta Arith. LXXVII(2), 179–193 (1996)CrossRefGoogle Scholar
  10. 10.
    Heegner, K.: Diophantische Analysis und Modulfunktionen. Math. Z. 56, 227–253 (1952)MathSciNetCrossRefGoogle Scholar
  11. 11.
    LMFDB: The database of L-functions, modular forms, and related objects.
  12. 12.
    Mazur, B.: Modular curves and the Eisenstein ideal. Publ. Math. IHES 47, 33–186 (1976)CrossRefGoogle Scholar
  13. 13.
    Mazur, B.: Rational points on modular curves, modular functions of one variable V. In: Proceedings of International Conference, Univ. Bonn, Bonn. Lecture Notes in Mathematics, vol. 601, pp. 107–148. Springer, Berlin (1977)Google Scholar
  14. 14.
    Mercuri, P.: Equations of Modular Curves Associated to Normalizers of non-split Cartan Subgroups. Preprint (2016)Google Scholar
  15. 15.
    Momose, F.: Rational points on the modular curves \(X_0^+(N)\). J. Math. Soc. Jpn. 39(2), 269–286 (1978)CrossRefGoogle Scholar
  16. 16.
    Ogg, A.P.: Rational points on certain elliptic modular curves, Analytic number theory. In: Proc. Sympos. Pure Math. 24, St. Louis, MO, 1972, pp. 221–231. American Mathematical Society, Providence (1973)Google Scholar
  17. 17.
    Saint-Donat, B.: On Petri’s analysis of the linear system of quadrics through a canonical curve. Math. Ann. 206, 157–175 (1973)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Shimura, M.: Defining equations of modular curves \(X_0(N)\). Tokyo J. Math. 18(2), 443–456 (1995)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Silverman, J.H.: The Arithmetic of Elliptic Curves. Graduate Texts in Mathematics, vol. 106. Springer, New York (1986)CrossRefGoogle Scholar
  20. 20.
    Silverman, J.H.: The difference between the Weil height and the canonical height on elliptic curves. Math. Comput. 55(192), 723–743 (1990)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Stark, H.M.: On complex quadratic fields with class number equal to one. Trans. Am. Math. Soc. 122, 112–119 (1966)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Stein, W.A.: The modular forms database.

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Sapienza UniversityRomeItaly

Personalised recommendations