TITLE: How to compute class number of a torus
QUESTION [7 upvotes]: Let $T$ be an algebraic torus over a number field $K$. 
Following notations in Ono's The Arithmetic of Tori, 
http://www.jstor.org/discover/10.2307/1970307?sid=21105671135711&uid=3739888&uid=2&uid=3739256&uid=4
we define 
$T_A$ the adele points, $T_K$ the $K$-rational point of $T$ and 
$$T_{A,S_{\infty}}=\prod_{v \in S_{\infty}}T(K_v)\times \prod_{v \notin S_{\infty}}T_v^c $$
The class number of $T$ is defined as 
$$ h_T := [{T_A}:{T_{A,S_{\infty}}T_K}] $$
My question is how do we compute $h_T$ in pratice. If $T=\mathbb{G}_m$, $h_T$ is just the class number of $K$ and we compute it using Minkowski's bounds. But I don't know how to compute $h_T$ in general. 
For an example, if $T$ is the torus $Spec(\mathbb{Q}[x,y]/(x^2+y^2-1))$, what is $h_T$ ?
Thank you very much.

REPLY [2 votes]: You can use relationships between different tori. For instance, your torus lives in a long exact sequence
$T\to \operatorname {Res} _{\mathbb Q(i) / \mathbb Q } \mathbb G_m  \to \mathbb G_m $
This gives you a cohomology long exact sequence:
$ H^0( \mathbb Z, \operatorname {Res} _{\mathbb Q(i) / \mathbb Q } \mathbb G_m) \to H^0( \mathbb Z, \mathbb G_m) \to H^1(\mathbb Z, T) \to H^1 ( \mathbb Z,  \operatorname {Res} _{\mathbb Q(i) / \mathbb Q }\mathbb G_m ) \to H^1(\mathbb Z, \mathbb G_m)$
Equivalently:
$ H^0( \mathbb Z[i], \mathbb G_m) \to H^0( \mathbb Z, \mathbb G_m) \to H^1(\mathbb Z, T) \to H^1 ( \mathbb Z[i], \mathbb G_m ) \to H^1(\mathbb Z, \mathbb G_m)$
Evaluating the $H^0$s as unit groups and $H^1$s as class groups:
$ \mu_4 \to \mu_2 \to H^1(\mathbb Z, T) \to 0 \to 1$
The map $\mu_4 \to \mu_2$ is the norm map, which is trivial, hence $H^1(\mathbb Z, T) = \mu_2$, and the class number is $2$.
You can probably do this in general but you need a spectral sequence.