## Journal of the Mathematical Society of Japan

### Non-elliptic Shimura curves of genus one

#### Abstract

We present explicit models for non-elliptic genus one Shimura curves $X_0(D, N)$ with $\Gamma_0(N)$-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

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

Dates
First available in Project Euclid: 21 May 2007

https://projecteuclid.org/euclid.jmsj/1179759530

Digital Object Identifier
doi:10.2969/jmsj/1179759530

Mathematical Reviews number (MathSciNet)
MR2276174

Zentralblatt MATH identifier
1123.11019

#### Citation

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. https://projecteuclid.org/euclid.jmsj/1179759530

#### References

• 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. http://www.maths.nott.ac.uk/personal/jec/ftp/data/INDEX.html.
• 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 http://arxiv.org/math.NT/0409020.
• 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 http://magma.maths.usyd.edu.au/magma/
• 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.