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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

Topics:
Algebraic geometry, Number theory

Year: 2006

OAI identifier:
oai:generic.eprints.org:798/core69

Provided by:
Mathematical Institute Eprints Archive

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