Journal of the Mathematical Society of Japan

Non-elliptic Shimura curves of genus one

Josep GONZÁLEZ and Victor ROTGER

Full-text: Open access


We present explicit models for non-elliptic genus one Shimura curves X 0 ( D , N ) with Γ 0 ( N ) -level structure arising from an indefinite quaternion algebra of reduced discriminant D , and Atkin-Lehner quotients of them. In addition, we discuss and extend Jordan's work [10, Ch. III] on points with complex multiplication on Shimura curves.

Article information

J. Math. Soc. Japan, Volume 58, Number 4 (2006), 927-948.

First available in Project Euclid: 21 May 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11G18: Arithmetic aspects of modular and Shimura varieties [See also 14G35]
Secondary: 14G35: Modular and Shimura varieties [See also 11F41, 11F46, 11G18]

Shimura curve curve of genus one elliptic curve complex multiplication points


GONZÁLEZ, Josep; ROTGER, Victor. Non-elliptic Shimura curves of genus one. J. Math. Soc. Japan 58 (2006), no. 4, 927--948. doi:10.2969/jmsj/1179759530.

Export citation


  • S. Baba and H. Granath, Genus 2 curves with quaternionic multiplication, Preprint 2005.
  • P. Bayer, Uniformization of certain Shimura curves, Differential Galois Theory, Banach Center Publications, 58, Polish Academy of Sciences, 2002.
  • M. Bertolini and H. Darmon, Heegner points on Mumford-Tate curves, Invent. Math., 126 (1996), 413–456.
  • N. Bruin, V. Flynn, J. Gonzàlez and V. Rotger, On finiteness conjectures for endomorphism algebras of abelian surfaces, to appear in Math. Proc. Camb. Phil. Soc.
  • J. E. Cremona, Elliptic curve data, 21 June, 2004.
  • J. E. Cremona, Classical invariants and 2-descent on elliptic curves, J. Symbolic Comput., 31 (2001), 71–87.
  • N. Elkies, Shimura curve computations, Lect. Notes Comp. Sci., 1423 (1998), 1–49.
  • N. Elkies, Shimura curves for level-3 subgroups of the (2,3,7) triangle group, and some other examples, available at
  • J. González and V. Rotger, Equations of Shimura curves of genus two, Int. Math. Res. Not., 14 (2004), 661–674.
  • B. W. Jordan, On the Diophantine arithmetic of Shimura curves, Harvard Ph. D. Thesis, 1981.
  • B. W. Jordan and R. Livné, Local diophantine properties of Shimura curves, Math. Ann., 270 (1985), 235–248.
  • A. Kurihara, On some examples of equations defining Shimura curves and the Mumford uniformization, J. Fac. Sci. Univ. Tokyo, Sec. IA, 25 (1979), 277–301.
  • A. Kurihara, On $p$-adic Poincaré series and Shimura curves, Intern. J. Math., 5 (1994), 747–763.
  • The Magma Computational Algebra System. Available at
  • A. P. Ogg, Real points on Shimura curves, Arithmetic and geometry, Progr. Math., 35, Birkhäuser Boston, Boston, MA, 1983, 277–307.
  • K. A. Ribet, Sur les varietés abéliennes à multiplications réelles, C. R. Acad. Sci. Paris, 291 (1980), 121–123.
  • D. P. Roberts, Shimura curves analogous to $X_0(N)$, Harvard Ph. D. Thesis, 1989.
  • V. Rotger, On the group of automorphisms of Shimura curves and applications, Compos. Math., 132 (2002), 229–241.
  • V. Rotger, Modular Shimura varieties and forgetful maps, Trans. Amer. Math. Soc., 356 (2004), 1535–1550.
  • V. Rotger, Shimura curves embedded in Igusa's threefold, Modular curves and abelian varieties, (eds. J. Cremona, J.-C. Lario, J. Quer and K. Ribet), Progr. Math., 224, Birkhäuser, 2004, 263–273.
  • V. Rotger, A. Skorobogatov and A. Yafaev, Failure of the Hasse principle for Atkin-Lehner quotients of Shimura curves over $\Q$, Moscow Math. J., 5:2 (2005), 463–476.
  • G. Shimura, Construction of class fields and zeta functions of algebraic curves, Ann. Math., 85 (1967), 58–159.
  • G. Shimura, On the real points of an arithmetic quotient of a bounded symmetric domain, Math. Ann., 215 (1975), 135–164.
  • G. Shimura, On canonical models of arithmetic quotients of bounded symmetric domains, Ann. Math., 91 (1970), 144–222.
  • M. Stoll and J. E. Cremona, Minimal models for 2-coverings of elliptic curves, LMS J. Comput. Math., 5 (2002), 220–243.
  • M. F. Vignéras, Arithmétique des algèbres de quaternions, Lecture Notes in Math., 800, Springer, 1980.