On Tamagawa numbers of 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 is the Tamagawa number of a CM tori, proving the analogous conjecture of Ono for CM tori.
Cite
@article{arxiv.2109.04121,
title = {On Tamagawa numbers of CM tori},
author = {Pei-Xin Liang and Yasuhiro Oki and Hsin-Yi Yang and Chia-Fu Yu},
journal= {arXiv preprint arXiv:2109.04121},
year = {2024}
}
Comments
Four authors. Add Section 8 which is merged from results of arXiv:2206.15230 and Section 9, 38 pages