On Tamagawa numbers of CM tori

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

doi:10.2140/ant.2024.18.583

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

ISSN 1944-7833 (online)

ISSN 1937-0652 (print)

Author index

To appear

Other MSP journals

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

DOI: 10.2140/ant.2024.18.583

Abstract

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.

Keywords

CM algebraic tori, Tamagawa numbers

Mathematical Subject Classification

Primary: 14K22

Secondary: 11R29

Milestones

Received: 30 September 2022

Revised: 5 March 2023

Accepted: 13 May 2023

Published: 16 February 2024

Authors

Pei-Xin Liang
National Tsing Hua University Taipei Taiwan

Yasuhiro Oki
Hokkaido University Sapporo Japan

Hsin-Yi Yang
Utrecht University Utrecht Netherlands

Chia-Fu Yu
Institute of Mathematics Academia Sinica and NCTS Taipei Taiwan

Jianing Li
Shandong University Qingdao Campus Qingdao China

Chia-Fu Yu
Institute of Mathematics Academia Sinica and NCTS Taipei Taiwan

Open Access made possible by participating institutions via Subscribe to Open.

Vol. 18, No. 3, 2024