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CLASS NUMBERS OF CM ALGEBRAIC TORI, CM ABELIAN VARIETIES AND COMPONENTS OF UNITARY SHIMURA VARIETIES

Part of: Algebraic number theory: global fields Abelian varieties and schemes

Published online by Cambridge University Press:  28 October 2020

JIA-WEI GUO ,
NAI-HENG SHEU  and
CHIA-FU YU
JIA-WEI GUO
Affiliation:
Department of Mathematics National Taiwan University No. 1, Roosevelt Road, Section 4 Taipei 10617, Taiwanjiaweiguo312@gmail.com
NAI-HENG SHEU
Affiliation:
Department of Mathematics Indiana University Rawles Hall, 831 East 3rd Street Bloomington, Indiana 47405, USAnaihsheu@iu.edu
CHIA-FU YU*
Affiliation:
Institute of Mathematics Academia Sinica and NCTS 6F Astronomy Mathematics Building, No. 1, Roosevelt Road, Section 4 Taipei 10617, Taiwan
*
chiafu@math.sinica.edu.tw
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Abstract

We give a formula for the class number of an arbitrary complex mutliplication (CM) algebraic torus over $\mathbb {Q}$ . This is proved based on results of Ono and Shyr. As applications, we give formulas for numbers of polarized CM abelian varieties, of connected components of unitary Shimura varieties and of certain polarized abelian varieties over finite fields. We also give a second proof of our main result.

MSC classification

Primary: 14K22: Complex multiplication
Secondary: 11R29: Class numbers, class groups, discriminants
Type
Article
Information
Nagoya Mathematical Journal , Volume 245 , March 2022 , pp. 74 - 99
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Copyright
© The Authors, 2020. Foundation Nagoya Mathematical is the exclusive licensee of this article

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