The arithmetic of QM-abelian surfaces through their Galois representations

Luis V. Dieulefait, V. Rotger

Published 2003-03-13, updated 2004-06-30Version 2

This note provides an insight to the diophantine properties of abelian surfaces with quaternionic multiplication over number fields. We study the fields of definition of the endomorphisms on these abelian varieties and the images of the Galois representations on their Tate modules. We illustrate our results with several explicit examples.