On Tamagawa numbers of CM tori

TL;DR

This paper computes Tamagawa numbers for CM tori, linking them to Galois cohomology and proving that all powers of 2 can occur as such numbers, thus confirming a conjecture of Ono.

Contribution

It systematically studies Tamagawa numbers of CM tori associated with Galois CM fields and proves that all powers of 2 are realizable as these numbers.

Findings

Tamagawa numbers of CM tori are computed explicitly.

Every power of 2 appears as a Tamagawa number of some CM torus.

The results confirm Ono's conjecture for CM tori.

Abstract

In this article 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--Gordon and of Guo--Sheu--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.