Note on Tamagawa numbers of tori attached to CM algebras

TL;DR

This paper proves that any integer power of 2 can be realized as the Tamagawa number of a torus attached to a CM algebra without relying on unproven conjectures, extending previous results.

Contribution

It removes the assumption of the generalized Landau conjecture in establishing Tamagawa numbers for tori attached to CM algebras.

Findings

Any integer power of 2 can be realized as a Tamagawa number of such tori.

The result holds unconditionally, removing previous conjecture dependencies.

Advances understanding of Tamagawa numbers in algebraic number theory.

Abstract

We prove that any integer power of $2$ can be realized as the Tamagawa number of a torus attached to a CM algebra considered by Guo Sheu Yu and Liang Yang Yu. Such a result is obtained by Liang Yang Yu under assuming "a generalized Landau conjecture". The main contribution of this paper is to remove the above assumption.