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On finiteness conjectures for endomorphism algebras of abelian surfaces

Nils Bruin, E. Victor Flynn, Josep Gonzalez, Victor Rotger

Published 2003-12-24, updated 2007-07-20Version 3

It is conjectured that there exist finitely many isomorphism classes of simple endomorphism algebras of abelian varieties of GL_2-type over \Q of bounded dimension. We explore this conjecture when particularized to quaternion endomorphism algebras of abelian surfaces 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.

Comments: We have reorganized the article, correcting some misprints, improving some results and giving more detailed explanations and references
Journal: Math. Proc. Camb. Phil. Soc. 141:3 (2006), 383-408
Categories: math.NT, math.AG
Subjects: 11G18, 14G35
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