# Volume of PGL(2,F) \ PGL(2, A)

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.

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This should be in Weil's Adeles and algebraic groups, chapter 3. – ACL Mar 14 '13 at 15:57
A natural measure gives that quotient measure essentially $\zeta_F(2)$. The renormalization as in Siegel, Weil and elsewhere will make it $1$ or a power of $2$, probably, as usual. For $PGL_n$, it's similarly $\zeta_F(2)\zeta_F(3)...\zeta_F(n)$. A classical argument over $\mathbb Q$ (which obviously generalizes), also for $Sp(n)$, is at math.umn.edu/~garrett/m/v/volumes.pdf – paul garrett Mar 14 '13 at 16:04
I believe the Tamagawa number is 1. books.google.com/… – Ian Agol Mar 14 '13 at 16:17
@Agol: The basic conjecture, which has been proved for all simple types, assumes a simply connected group here. Clozel has a useful Sem. Bourbaki expose 702, following the work by Kottwitz and others/ – Jim Humphreys Mar 14 '13 at 17:22
@Agol: The Tamagawa number is 1 for the special linear group. For the projective linear group, which is isogeneous to the special linear group, it is some rational number (whom I do not remember). – ACL Mar 14 '13 at 21:47

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