On Tamagawa numbers of CM tori

Abstract

In this article we investigate the problem of computing Tamagawa numbers of CM tori, which play an important role in counting polarized CM abelian varieties, polarized abelian varieties with commutative endomorphism algebras over finite fields, and components of unitary Shimura varieties. Using Ono's formula and extending the methods of Li-Rüd and Guo-Sheu-Yu, we make a systematic study on Galois cohomological groups in question and compute the Tamagawa numbers of CM tori associated to various Galois CM fields. We also show that every (positive or negative) power of 2 is the Tamagawa number of a CM tori provided that a generalized Landau conjecture holds true, and show an evidence of our result unconditionally.