\tau(G) = \tau(G_1)
Given a smooth, geometrically connected and projective curve C defined over a finite field k, let K=k(C) be the function field of rational functions on C. The Tamagawa number \tau(G) of a semisimple K-group G is defined as the covolume of the discrete group G(K) (embedded diagonally) in the adelic group G(A) with respect to the Tamagawa measure. The Weil conjecture, recently proved by Gaitsgory and Lurie, states that if G is simply-connected then \tau(G)=1.
Our aim is to prove, without relying on the Weil conjecture, the following fact:
Let G be a quasi-split inner form of a split semisimple and simply-connected K-group G_1. Then \tau(G)=\tau(G_1).
This Theorem can serve as an alternative proof to the Weil conjecture.
Meeting ID: 878 5613 2062
תאריך עדכון אחרון : 31/05/2022