We investigate the problem of computing Tamagawa numbers of CM tori. This problem arises naturally from the problem of counting polarized abelian varieties with commutative endomorphism algebras over finite fields, and polarized CM abelian varieties and components of unitary Shimura varieties in the works of Achter, Altug, Garcia and Gordon and of Guo, Sheu and Yu, respectively. We make a systematic study on Galois cohomology groups in a more general setting and compute the Tamagawa numbers of CM tori associated to various Galois CM fields. Furthermore, we show that every (positive or negative) power of $2$ is the Tamagawa number of a CM tori, proving the analogous conjecture of Ono for CM tori.

我们研究了计算 CM tori 玉川数的问题。这个问题自然地产生于计算有限域上具有交换自同态代数的极化阿贝尔簇的问题，以及 Achter、Altug、Garcia 和 Gordon 以及Guo、Sheu 和 Yu 的著作中的极化 CM 阿贝尔簇和酉 Shimura 簇的分量，分别。我们在更一般的环境中对伽罗瓦上同调群进行了系统研究，并计算了与各种伽罗瓦 CM 域相关的 CM tori 的玉川数。此外，我们证明了每一个（正或负）的力量$2$是CM tori的玉川数，证明了小野对CM tori的类似猜想。