For smooth linear group schemes over , we give a cohomological interpretation of the local Tamagawa measures as cohomological periods. This is in the spirit of the Tamagawa measures for motives defined by Bloch and Kato. We show that in the case of tori, the cohomological and the motivic Tamagawa measures coincide, which proves again the Bloch-Kato conjecture for motives associated to tori.
"A cohomological Tamagawa number formula." Nagoya Math. J. 202 45 - 75, June 2011. https://doi.org/10.1215/00277630-1260441