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On Tamagawa numbers of CM tori

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

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.

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.

Paper Structure

This paper contains 29 sections, 60 theorems, 157 equations.

Table of Contents

  1. Introduction
  2. Preliminaries, background and some known results
  3. Class numbers and Tamagawa numbers of algebraic tori
  4. Values of Tamagawa numbers
  5. Some known results for CM tori
  6. Transfer maps, corestriction maps and extensions
  7. Transfer maps
  8. Connection with corestriction maps
  9. Connection with class field theory
  10. Relative transfer maps
  11. Connection with class field theory
  12. Cohomology groups of algebraic tori
  13. $H^1(\Lambda^1)$ and $H^1(\Lambda)$
  14. $\Sha^2(\Lambda)$
  15. Computations of some product cases
  16. ...and 14 more sections

Key Result

Theorem 1.1

Let the notation be as above. (1) There is a canonical isomorphism $H^1(G,\Lambda^1)\simeq \bigoplus_{i} N_{i}^{\rm ab,\vee}$. (2) There is a canonical isomorphism where $\mathop{\rm Ver}\nolimits_{G,N_i}:G\to N_i^{\rm ab}$ is the transfer map. (3) Assume that $K_i/E_i$ is cyclic with Galois group $N_i$ for all $i$. Then $\Sha^2(\Lambda)\simeq H^2(\mathbb{Z})'/{\rm Im}(\delta)$ and (4) If we fur

Theorems & Definitions (122)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4: Ono's formula
  • proof
  • Remark 2.5
  • ...and 112 more