The Mathematical Institute, University of Oxford, Eprints Archive

On finiteness conjectures for modular quaternion algebras

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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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.

Item Type:Article
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 07:35
Last Modified:29 May 2015 18:29

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