Fields of definition of building blocks with quaternionic multiplication

摘要

This paper investigates the fields of definition up to isogeny of the abelianvarieties called building blocks. A result of Ribet characterizes the fields ofdefinition of these varieties together with their endomorphisms, in terms of aGalois cohomology class canonically attached to them. However, when thebuilding blocks have quaternionic multiplication, then the field of definitionof the varieties can be strictly smaller than the field of definition of theirendomorphisms. What we do is to give a characterization of the fields ofdefinition of the varieties in this case (also in terms of their associatedGalois cohomology class), by translating the problem into the language of groupextensions with non-abelian kernel. We also make the computations that areneeded in order to calculate in practice these fields from ourcharacterization.