A cohomological Tamagawa number formula

TL;DR

This paper provides a cohomological interpretation of local Tamagawa measures for smooth linear group schemes over integers, linking them to cohomological periods and confirming the Bloch-Kato conjecture for tori.

Contribution

It introduces a cohomological framework for Tamagawa measures and proves their equivalence to motivic measures for tori, confirming a key conjecture.

Findings

Cohomological interpretation of Tamagawa measures established

Equivalence of cohomological and motivic measures for tori proved

Reproves the Bloch-Kato conjecture for motives associated to tori

Abstract

For smooth linear group schemes over $\bbZ$ 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 reproves the Bloch-Kato conjecture for motives associated to tori.