Real Points on Shimura Curves

  • Andrew P. Ogg
Part of the Progress in Mathematics book series (PM, volume 35)


Let B be a quaternion algebra over Q, i.e., a central simple alegbra of dimension 4 over Q. We assume that, B is indefinite, and fix an identification of B = BR with M 2(R). The discriminant D of B is then the product of an even number of distinct primes. The completion B p = BQ p , is a skew field if p | D and is isomorphic to M 2(Q p ) if p × D, and D = 1 if and only if B is isomorphic to M 2(Q), i.e., B is not a skew field.


Hyperelliptic Curve Real Point Quaternion Algebra Single Orbit Hyperelliptic Involution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    A. O. L. Atkin and J. Lehner, Hecke operators on fo(m). Math. Ann. 185 (1970), 134–160.MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    B. Birch, W. Kuyk (ed.), Modular functins of one variable N. Lecture Notes in Mathematics 476. Berlin, Heidelberg, New York: Springer, 1975.Google Scholar
  3. [3]
    M. Eichler, Uber die Einheiten der Divisionsalgebren. Math. Ann. 114 (1937), 635–654.MathSciNetCrossRefGoogle Scholar
  4. [4]
    M. Eichler, Uber die ldealklassenzahl hyperkomplexer Systeme. Math. Z. 43 (1938), 481–494.MathSciNetCrossRefGoogle Scholar
  5. [5]
    M. Eichler, Zur Zahlentheorie der Quaternionen-Algebren. J. Reine Angew. Math. 195 (1955), 127–151.MathSciNetGoogle Scholar
  6. [6]
    M. Eichler, Uber die Darstellbarkeit von Modulformen durch Theta- reihen. J. Reine Angew. Math. 195 (1955), 156–171.MathSciNetMATHGoogle Scholar
  7. [7]
    Y. Ihara, Congruence relations and Shimura curves. Proc. Symposia Pure Math. 33(1979), part 2, 291–311.Google Scholar
  8. [8]
    B. Mazur, Rational isogenies of prime degree. Invent. Math. 44 (1978), 129–162.MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    B. Mazur and H. P. F. Swinnerton-Dyer, Arithmetic of Weil curves. Invent. Math. 25 (1974), 1–61.MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    J. -F. Miction, Courbes de Shimura hyperelliptiques. Bull. Soc. Math. France 109 (1981), 217–225.Google Scholar
  11. [11]
    A. Ogg, Rational points on certain elliptic modular curves. Proc. Symposia Pure Math. 24 (1973), 221–231.MathSciNetCrossRefGoogle Scholar
  12. [12]
    A. Ogg, Ilyperelliptic modular curves. Bull. Soc. Math. France 102 (1974), 449–462.MATHGoogle Scholar
  13. [13]
    K. Ribet, Sur les variétés abéliennes à multiplications réelles. C. R. Acad. Sci. Paris 291 (1980), Série A, 121–123.Google Scholar
  14. [14]
    G. Shimura, On the zeta-functions of the algebraic curves unifor- mized by certain automorphic functions. J. Math. Soc. Japan 13 (1961), 275–331.MathSciNetMATHCrossRefGoogle Scholar
  15. [15]
    G. Shimura, On the theory of automorphic functions. Ann. of Math. 70 (1959), 101–144.MathSciNetMATHCrossRefGoogle Scholar
  16. [16]
    G. Shimura, Construction of class fields and zeta functions of al- gebraic curves. Ann. of Math. 85 (1967), 58–159.MathSciNetMATHCrossRefGoogle Scholar
  17. [17]
    G. Shimura, On the real points of an arithmetic quotient of a bounded symmetric domain. Math. Ann. 215 (1975), 135–164.MathSciNetMATHCrossRefGoogle Scholar
  18. [18]
    M. F. Vignéras, Arithmétique des Algèbres de Quaternions. Lecture Notes in Mathematics 800. Berlin, Heidelberg, New York: Springer, 1980.Google Scholar

Copyright information

© Springer Science+Business Media New York 1983

Authors and Affiliations

  • Andrew P. Ogg
    • 1
  1. 1.Department of MathematicsUniversity of CaliforniaBerkeleyUSA

Personalised recommendations