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On finiteness conjectures for endomorphism algebras of abelian surfaces

Published online by Cambridge University Press:  01 December 2006

NILS BRIUN
Affiliation:
Department of Mathematics, Simon Fraser University, Burnaby, BC, Canada V5A 1S6. e-mail: bruin@member.ams.org
E. VICTOR FLYNN
Affiliation:
Mathematical Institute, University of Oxford, Oxford OX1 3LB, United Kingdom. e-mail: flynn@maths.ox.ac.uk
JOSEP GONZALEZ
Affiliation:
Universitat Politècnica de Catalunya, Departament de Matemàtica Aplicada IV (EUPVG), Av. Victor Balaguer s/n, 08800 Vilanova i la Geltrú, Spain. e-mail: josepg@mat.upc.es, vrotger@mat.upc.es
VICTOR ROTGER
Affiliation:
Universitat Politècnica de Catalunya, Departament de Matemàtica Aplicada IV (EUPVG), Av. Victor Balaguer s/n, 08800 Vilanova i la Geltrú, Spain. e-mail: josepg@mat.upc.es, vrotger@mat.upc.es

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 $\mathrm{GL}_2$-type over $\mathbb{Q}$ 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.

Type
Research Article
Copyright
© 2006 Cambridge Philosophical Society

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