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On finiteness conjectures for modular quaternion algebras

By N. Bruin, E. V. Flynn, J. González and V. Rotger


It is conjectured that there exist only finitely many isomorphism classes of endomorphism algebras of abelian varieties of bounded dimension over a number field of bounded degree. We explore this conjecture when restricted to quaternion endomorphism algebras of abelian surfaces of GL$_2$-type over Q by giving a moduli interpretation which translates the question into the diophantine arithmetic of Shimura curves embedded in Hilbert surfaces. We address the resulting problems on these curves by local and global methods, including Chabauty techniques on explicit equations of Shimura curves

Topics: Algebraic geometry, Number theory
Year: 2006
OAI identifier:

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  1. a program for 2-Selmer groups of elliptic curves over number fields, available at
  2. (2005). Abelian surfaces of GL2-type as Jacobians of curves,
  3. (2004). Abelian varieties over Q and modular forms, in Modular curves and abelian varieties,
  4. (2004). Abelian varieties over Q with large endomorphism algebras and their simple components over Q in Modular curves and abelian
  5. (1992). Abelian varieties with extra twist, cusp forms and elliptic curves over imaginary quadratic fields,
  6. (1970). Algebraic Number Theory, Graduate Texts in Mathematics 110,
  7. (1992). Algorithms for modular elliptic curves,
  8. (1980). Arithme´tique des alge`bres de quaternions,
  9. (2004). Arithmetic of Q-curves, in Modular curves and abelian varieties,
  10. (1999). Chabauty methods and covering techniques applied to generalized Fermat equations, Dissertation,
  11. (2003). Chabauty methods using elliptic curves,
  12. (1967). Construction of class fields and zeta functions of algebraic curves, doi
  13. (1968). Corps Locaux, Hermann,
  14. (2001). Covering collections and a challenge problem of Serre.
  15. (2004). Equations of Shimura curves of genus two,
  16. (2005). Failure of the Hasse principle for Atkin-Lehner quotients of Shimura curves over Q,
  17. (1999). Finding rational points on bielliptic genus 2 curves,
  18. Local and global points on moduli spaces of abelian surfaces with potential quaternionic multiplication,
  19. (1985). Local diophantine properties of Shimura curves,
  20. (1983). Mauvaise re´duction des courbes de Shimura, Se´minaire de the´orie des nombres,
  21. (1998). Models of Shimura varieties in mixed characteristic,
  22. (2004). Modular Shimura varieties and forgetful maps,
  23. Non-elliptic Shimura curves of genus one, submitted for publication.
  24. (2005). On abelian surfaces with potential quaternionic multiplication,
  25. (2003). On abelian varieties with quaternionic multiplication and their moduli, Universitat de
  26. (1994). On p-adic Poincare´ series and Shimura curves,
  27. (2001). On Q-derived polynomials,
  28. On QM-abelian surfaces with a model of GL2-type over
  29. (1979). On some examples of equations defining Shimura curves and the Mumford uniformization,
  30. (1996). On some examples of modular QM-abelian surfaces,
  31. (2001). On the Andre´-Oort for Hilbert modular surfaces,
  32. (1981). On the Diophantine arithmetic of Shimura curves, Harvard PhD. Thesis
  33. (2002). On the group of automorphisms of Shimura curves and applications,
  34. (1975). On the real points of an arithmetic quotient of a bounded symmetric domain, doi
  35. (1999). On the Sato-Tate conjecture for QM-curves of genus two,
  36. (2003). Quaternions, polarizations and class numbers,
  37. (1978). Rational isogenies of prime degree,
  38. Seminar at Boston University and Letter to Frans Oort,
  39. (2004). Shimura curves embedded in Igusa’s threefold, in Modular curves and abelian varieties,
  40. (1941). Sur les points rationnels des varie´te´s alge´briques dont l’irre´gularite´ est supe´rieure a` la dimension,
  41. (1987). Sur les repre´sentations modulaires de degre´
  42. (1986). The arithmetic of elliptic curves, doi
  43. (2004). The arithmetic of QM-abelian surfaces through their Galois representations,
  44. (1999). The Diophantine equations
  45. (1956). The field of definition of a variety, doi
  46. (2004). The field of moduli of quaternionic multiplication on abelian varieties, doi
  47. The Magma Computational Algebra System. Available from:
  48. (2001). Towers of 2-covers of hyperelliptic curves.
  49. (1991). Uniformisation p-adique des courbes de Shimura: les the´ore`mes de Cerednik et de Drinfeld. Aste´risque,

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