Bruin, N. and Flynn, E. V. and González, J. and Rotger, V. (2006) *On finiteness conjectures for modular quaternion algebras.* Mathematical Proceedings of the Cambridge Philosophical Society, 141 . pp. 383-408. ISSN 0305-0041

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## Abstract

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-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.

Item Type: | Article |
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Uncontrolled Keywords: | Shimura curves, Hilbert surfaces, Chabauty methods using elliptic curves, Heegner points. |

Subjects: | A - C > Algebraic geometry H - N > Number theory |

Research Groups: | Number Theory Group |

ID Code: | 798 |

Deposited By: | E. Victor Flynn |

Deposited On: | 01 Sep 2009 08:35 |

Last Modified: | 01 Sep 2009 08:35 |

### Available Versions of this Item

- On finiteness conjectures for modular quaternion algebras. (deposited 12 Jul 2006)
- On finiteness conjectures for modular quaternion algebras. (deposited 01 Sep 2009 08:35)
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- On finiteness conjectures for modular quaternion algebras. (deposited 01 Sep 2009 08:35)

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