Let $G$ be a split semi-simple simply connected group over a global field $F$ and let $\omega$ be a top-degree differential form on $G$ without zeroes (defined over $F$). It is well known that $\omega$ defines a measure on the adele group $G(\mathbb A)$. The Tamagawa number formula states (if I understand correctly) that

1) If $F$ is a number field then the volume of $G(\mathbb A)/G(F)$ is 1

2) If $F$ is a functional field isomorphic to $\mathbb F_q(X)$ where $X$ is a projective curve over $\mathbb F_q$ then the above volume is equal to $q^{(g-1)\dim G}$ where $g$ is the genus of $X$.

My questions are the following:

a) Do I understand the statements correctly?

b) What is the reason why 1) and 2) look somewhat differently? Can one formulate the statement in a uniform way for all global fields?

Edit: In fact my understanding was wrong. In 1) one needs to multiply by the volume of $(\mathbb A/F)^{\dim G}$ which is equal exactly to $q^{(g-1)\dim G}$ in the functional case. I was confused by the case $F=\mathbb Q$ where the above factor is 1.

  • Side remark. Tamagawa number and susy: projecteuclid.org/euclid.cmp/1104201160 Quantum-mechanical calculations in the algebraic group theory M. A. Olshanetsky – Alexander Chervov Jan 31 '12 at 20:57
  • I think the Tamagawa number is still supposed to be 1 in the function field case. (Harder proved this for simply-connected Chevalley groups in "Chevalley groups over function fields and automorphic forms". See the last line in the paper.) Perhaps there are competing normalizations? – B R Jan 31 '12 at 22:30
  • Further, in Borel and Prasad's "Finiteness Theorems for Discrete Subgroups of Bounded Covolume in Semi-simple Groups" (1989), there is this line (at the end of Section 7.1): "The Tamagawa number of any simply connected group of inner type $A$ over an arbitrary global function field is known to be 1 (see [Weil's Adeles and Algebraic Groups]). However, whether this is the case in general over a global function field is not yet known." – B R Jan 31 '12 at 22:45
  • 1
    In the number field case this was proved using trace formula techniques by Kottwitz, following work of K.F.Lai and Langlands (see MR0942522). In the function field case, I believe the full result was recently (2011?) announced by Jacob Lurie. – fherzig Jan 31 '12 at 23:05
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    Actually, it's Gaitsgory-Lurie: web.me.com/teichner/Math/FRG-3.html – fherzig Jan 31 '12 at 23:09

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