On finiteness conjectures for endomorphism algebras of abelian surfaces
Number Theory
2011-11-10 v3 Algebraic Geometry
Abstract
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.
Keywords
Cite
@article{arxiv.math/0312443,
title = {On finiteness conjectures for endomorphism algebras of abelian surfaces},
author = {Nils Bruin and E. Victor Flynn and Josep Gonzalez and Victor Rotger},
journal= {arXiv preprint arXiv:math/0312443},
year = {2011}
}
Comments
We have reorganized the article, correcting some misprints, improving some results and giving more detailed explanations and references