Let $F$ be a global field. What is the measure of $PGL\_2(F) \backslash PGL\_2(\mathbb{A})$?


This depends of course on the normalizations of the Haar measures on $PGL\_2(F)$ and $PGL\_2(\mathbb{A})$. Probably its most likely available in the literature for $PGL\_2(F)$ admits the discrete measure and $PGL\_2(\mathbb{A})$ the Tamagawa measure, but I couldn't find!?


I remember that there was a question about the measure of $SL\_n(\mathbb{Z}) \backslash SL\_n(\mathbb{R})$ here in the past, but couldn't find it.


It should be related to special values of the Dedekind zeta function.



Volume of PGL(2,F) \ PGL(2, A)
If you take the Tamagawa measure, the measure is 2. It is equal to the index of the universal covering, so for a simply connected group $G$ the volue of $G({\mathbb A})/G(F)$ is one. This is Kottwitz's Theorem, formerly known as the Tamagawa number conjecture. A proof is in
<http://www.jstor.org/discover/10.2307/2007007?uid=3737864&uid=2129&uid=2&uid=70&uid=4&sid=21101828946301>


