We study K3 surfaces with complex multiplication following the classical work of Shimura on CM abelian varieties. After we translate the problem in terms of the arithmetic of the CM field and its idèles, we proceed to study some abelian extensions that arise naturally in this context. We then make use of our computations to determine the fields of moduli of K3 surfaces with CM and to classify their Brauer groups. More specifically, we provide an algorithm that given a number field K and a CM number field E, returns a finite list of groups which contains $\text{Br}{(\bar{X})}^{G_K}$ for any K3 surface $X/K$ that has CM by the ring of integers of E. We run our algorithm when E is a quadratic imaginary field (a condition that translates into X having maximal Picard rank) generalizing similar computations already appearing in the literature.

在Shimura对CM阿贝尔变种进行经典工作之后，我们研究了具有复数乘法的K3曲面。在根据CM字段及其偶数的算法对问题进行了翻译之后，我们将继续研究在这种情况下自然产生的一些阿贝尔扩展。然后，我们利用我们的计算确定具有CM的K3曲面的模场，并对它们的Brauer组进行分类。更具体地说，我们提供了一种算法，该算法给定一个数字字段K和一个CM数字字段E，返回包含以下内容的组的有限列表：$\text{溴}{（\stackrel{〜}{X}）}^{G_{ķ}}$ 适用于任何K3表面 $X/ķ$通过CM的E的整数环得到CM 。当E是一个二次虚域（转化为具有最大皮卡德秩的X的条件）时，我们运行我们的算法，以概括文献中已经出现的类似计算。