Let $G$ be an algebraic reductive group defined over an algebraic function field $K$ in one variable with a finite field of constants $\mathbf{F}_q$. For any prime $p$ of $K$, unramified in the Galois splitting extension of $G$, let $K_p$ be the completion at $p$. If $T$ is the maximal central torus in $G \otimes_K K_p$, then the local Artin $L$-function, should be the inverse of the volume of the maximal compact subgroup of $T(K_p)$ (trivial in the semisimple case) w.r.t. to the local component of the Tamagawa measure. This would make the infinite product converge. In the number field case, this volume is $$ |k|^{-d} \cdot |\overline{T}(k)| =\det(1_d-h(F)/|k|) = L_p(1,\chi_T)^{-1}. $$ where $k$ is the residue field, $\overline{T}$ is the special fiber of the NR-finite type model of $T$, $d= \dim T$, $h(F)$ is the image of the Frobenius automorphism in the character module representation of the Galois group and $\chi_T$ is the character of the represnetation.

For some reason which I cannot understand, J. Oesterle in "Nombres de Tamagawa et groupes unipotents en caract\'eristique p": http://www.springerlink.com/content/r3r642748w7m4682/ section 2.5 has found that the volume w.r.t. the local component in the Tamagawa measure - in our function field case - is: $q^{-n-d} \cdot |\overline{G}(k)|$. It is not clear what is $n$. I saw that Kai Behrend and Ajneet Dhillon in their paper "The geometry of Tamagawa numbers of Chevalley groups": http://www.math.uwo.ca/~adhillon/papers/geotam.pdf are quoting in Proposition 6.1 Oesterle saying that $n$ is the degree of the prime $p$. According to that, the $L$-function should differ from the aforementioned one in the Number field case.
Could someone explain me how and why ?

Thank you, rony.

P.S. Sorry, but I have to ask you to close my previous question 75492.

  • To whoever voted to close as duplicate: This question is a much clearer question than the original question mathoverflow.net/questions/75492. Given that we have two questions open now, it seems a bit odd to vote to close the better-formulated question rather than the worse, no? – David Loeffler Sep 15 '11 at 15:15
  • @David. It was me. He should have edited the first one and not started a new one. He can also delete his own questions, to avoid duplicates. – Felipe Voloch Sep 15 '11 at 15:58
  • @Felipe: Rony asked his first question as a non-registered user, and therefore (?) could not edit or delete it... – Mikhail Borovoi Sep 15 '11 at 17:33

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