# 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

## 1 Answer

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

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Okay, I was interested more in the measure, which enters directly into dimension formulas of modular forms. But after all, your answer is correct and I actually was aware of Kottwitz's achievement. Thank you nevertheless. It seems that the normalisation enters by giving the maximal compact groups some non-unit normalization as in Gelbart-Jacquet. Thanks. –  Marc Palm Mar 14 '13 at 20:15
Oesterle's 1984 Inventiones paper on Tamagawa numbers (see Inv. Math. 78) has a very elegant discussion of the normalization issues to define the Tamagawa measure and Tamagawa number for rather general smooth connected affine groups over global fields. His normalizations are not governed by maximal compact subgroups (as they ought not be, since typically there are a lot of non-conjugate such subgroups!). He also cleaned up some points of possible confusion that arose in earlier work by others in the case of tori. –  user30379 Mar 17 '13 at 5:58