This paper investigates the fields of definition up to isogeny of the abelian varieties called building blocks. A result of Ribet characterizes the fields of definition of these varieties together with their endomorphisms, in terms of a Galois cohomology class canonically attached to them. However, when the building blocks have quaternionic multiplication, then the field of definition of the varieties can be strictly smaller than the field of definition of their endomorphisms. What we do is to give a characterization of the fields of definition of the varieties in this case (also in terms of their associated Galois cohomology class), by translating the problem into the language of group extensions with non-abelian kernel. We also make the computations that are needed in order to calculate in practice these fields from our characterization.
Fields of definition of K3 surfaces with complex multiplication
设$X/ \mathbb{C}$是一个与CM域$E$的整数环进行复乘的K3曲面.我们发现，$X$总是可以定义在阿贝尔扩展$K/E$明确确定的判别形式的格$\mathrm{NS}(X)$。然后，我们通过伽罗瓦下降构造了一个$X$ over $K$的模型，我们研究了它的一些基本性质，特别是我们明确地确定了它的伽罗瓦表示。最后，我们应用我们的结果给出了$X$的最小定义域的上、下界，它是由$E$的类数和$\mathrm{NS}(X)$的判别式给出的。
