Canonical models of arithmetic (1; e)-curves

Abstract

The Fuchsian groups of signature (1; e) are the simplest class of Fuchsian groups for which the calculation of the corresponding quotient of the upper half plane presents a challenge. This paper considers the finite list of arithmetic (1; e)-groups. We define canonical models for the associated quotients by relating these to genus 1 Shimura curves. These models are then calculated by applying results on the \({\mathfrak{p}}\)-adic uniformization of Shimura curves and Hilbert modular forms.

This is a preview of subscription content, access via your institution.

References

  1. 1

    Boutot, J.-F., Carayol, H.: Uniformisation p-adique des courbes de Shimura: les théorèmes de Čerednik et de Drinfel’d. Astérisque 196–197, 7 (1991); 45–158 (1992)

  2. 2

    Bosma W., Cannon J., Playoust C.: The Magma algebra system. I. The user language. J. Symbolic Comput 24(3–4), 235–265 (1997)

    MathSciNet  MATH  Article  Google Scholar 

  3. 3

    Boutot, J.-F., Zink, T.: The p-adic Uniformization of Shimura Curves. Preprint. http://www.mathematik.uni-bielefeld.de/~zink/p-adicuni.ps

  4. 4

    Carayol H.: Sur la mauvaise réduction des courbes de Shimura. Compos. Math. 59(2), 151–230 (1986)

    MathSciNet  MATH  Google Scholar 

  5. 5

    Dieulefait L., Dimitrov M.: Explicit determination of images of Galois representations attached to Hilbert modular forms. J. Number Theory 117(2), 397–405 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  6. 6

    Dembélé, L., Donnelly, S.: Computing Hilbert modular forms over fields with nontrivial class group. Algorithmic number theory. Lecture Notes in Computer Science, vol. 5011, pp. 371–386. Springer, Berlin (2008)

  7. 7

    Doi, K., Naganuma, H.: On the algebraic curves uniformized by arithmetical automorphic functions. Ann. Math. (2) 86, 449–460 (1967)

    Google Scholar 

  8. 8

    Eichler M.: Zur Zahlentheorie der Quaternionen-Algebren. J. Reine Angew. Math 195(1955), 127–151 (1956)

    MathSciNet  Google Scholar 

  9. 9

    Elkies, N.D.: Shimura curve computations. Algorithmic Number Theory (Portland, OR, 1998). Lecture Notes in Computer Science, vol. 1423, pp. 1–47. Springer, Berlin (1998)

  10. 10

    González J., Rotger V.: Non-elliptic Shimura curves of genus one. J. Math. Soc. Jpn 58(4), 927–948 (2006)

    MATH  Article  Google Scholar 

  11. 11

    Hallouin E.: Computation of a cover of Shimura curves using a Hurwitz space. J. Algebra 321(2), 558–566 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  12. 12

    Hida H.: On abelian varieties with complex multiplication as factors of the Jacobians of Shimura curves. Am. J. Math 103(4), 727–776 (1981)

    MathSciNet  MATH  Article  Google Scholar 

  13. 13

    Hijikata H., Pizer A., Shemanske T.: Orders in quaternion algebras. J. Reine Angew. Math 394, 59–106 (1989)

    MathSciNet  MATH  Google Scholar 

  14. 14

    Kurihara A.: On some examples of equations defining Shimura curves and the Mumford uniformization. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 25(3), 277–300 (1979)

    MathSciNet  Google Scholar 

  15. 15

    Kirschmer M., Voight J.: Algorithmic enumeration of ideal classes for quaternion orders. SIAM J. Comput. 39(5), 1714–1747 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  16. 16

    Livné, R.: Cubic exponential sums and Galois representations. Current Trends in Arithmetical Algebraic Geometry (Arcata, Calif., 1985). Contemp. Math., vol. 67, pp. 247–261. Amer. Math. Soc., Providence (1987)

  17. 17

    Milne, J.S.: Introduction to Shimura Varieties. http://www.jmilne.org/math/xnotes/svi.html

  18. 18

    Molina, S.: Equations of hyperelliptic Shimura curves. Preprint. http://arxiv.org/abs/1004.3675 (2010)

  19. 19

    Neukirch, J.: Algebraic number theory. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 322. Springer, Berlin (1999)

  20. 20

    Ogg, A.P.: Real points on Shimura curves. Arithmetic and Geometry, vol. I. Progr. Math., vol. 35, pp. 277–307. Birkhäuser, Boston (1983)

  21. 21

    Ribet K.A.: On modular representations of \({{{\rm Gal}}(\overline {\bf Q}/{\bf Q})}\) arising from modular forms. Invent. Math 100(2), 431–476 (1990)

    MathSciNet  MATH  Article  Google Scholar 

  22. 22

    Shimura, G.: On canonical models of arithmetic quotients of bounded symmetric domains. Ann. Math. (2) 91, 144–222 (1970)

    Google Scholar 

  23. 23

    Sijsling, J.: Equations for arithmetic pointed tori. Ph.D. thesis, Universiteit Utrecht (2010)

  24. 24

    Sijsling, J.: Magma programs for arithmetic pointed tori. http://sites.google.com/site/sijsling/programs (2010)

  25. 25

    Sijsling, J.: Arithmetic (1;e)-curves and Belyĭ maps. Math.Comp. (2011, to appear)

  26. 26

    Skinner C.M., Wiles A.J.: Residually reducible representations and modular forms. Inst. Hautes Études Sci. Publ. Math. 89, 5–126 (1999)

    MathSciNet  MATH  Google Scholar 

  27. 27

    Socrates J., Whitehouse D.: Unramified Hilbert modular forms, with examples relating to elliptic curves. Pac. J. Math 219(2), 333–364 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  28. 28

    Takeuchi K.: A characterization of arithmetic Fuchsian groups. J. Math. Soc. Jpn 27(4), 600–612 (1975)

    MATH  Article  Google Scholar 

  29. 29

    Takeuchi K.: Arithmetic Fuchsian groups with signature (1; e). J. Math. Soc. Jpn. 35(3), 381–407 (1983)

    MATH  Article  Google Scholar 

  30. 30

    Varshavsky Y.: p-adic uniformization of unitary Shimura varieties. Inst. Hautes Études Sci. Publ. Math 87, 57–119 (1998)

    MathSciNet  MATH  Article  Google Scholar 

  31. 31

    Vignéras, Marie-France: Arithmétique des algèbres de quaternions. Lecture Notes in Mathematics, vol. 800. Springer, Berlin (1980)

  32. 32

    Voight J.: Computing fundamental domains for Fuchsian groups. J. Théor. Nombres Bordeaux 21(2), 469–491 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  33. 33

    Voight J.: Shimura curves of genus at most two. Math. Comput 78(266), 1155–1172 (2009)

    MathSciNet  MATH  Google Scholar 

  34. 34

    Voight, J.: Computing automorphic forms on Shimura curves over fields with arbitrary class number. Algorithmic number theory. Lecture Notes in Computer Science, vol. 6197, pp. 357–371. Springer, Berlin (2010)

  35. 35

    Zhang, S.: Heights of Heegner points on Shimura curves. Ann. Math. (2) 153(1), 27–147 (2001)

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Jeroen Sijsling.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Sijsling, J. Canonical models of arithmetic (1; e)-curves. Math. Z. 273, 173–210 (2013). https://doi.org/10.1007/s00209-012-1000-5

Download citation

Keywords

  • Shimura curves
  • Arithmetic groups
  • Uniformization
  • Explicit methods

Mathematics Subject Classification (2010)

  • Primary 11G18
  • Secondary 14G35
  • 14Q05