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On Tamagawa numbers of CM tori

Pei-Xin Liang, Yasuhiro Oki, Hsin-Yi Yang and Chia-Fu Yu

Appendix: Jianing Li and Chia-Fu Yu

Vol. 18 (2024), No. 3, 583–629

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.

CM algebraic tori, Tamagawa numbers
Mathematical Subject Classification
Primary: 14K22
Secondary: 11R29
Received: 30 September 2022
Revised: 5 March 2023
Accepted: 13 May 2023
Published: 16 February 2024
Pei-Xin Liang
National Tsing Hua University
Yasuhiro Oki
Hokkaido University
Hsin-Yi Yang
Utrecht University
Chia-Fu Yu
Institute of Mathematics
Academia Sinica and NCTS
Jianing Li
Shandong University
Qingdao Campus
Chia-Fu Yu
Institute of Mathematics
Academia Sinica and NCTS

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