Let $G$ be a connected semisimple algebraic group over the field of rational numbers $\mathbf{Q}$ or more generally over a global field $K$. The Tamagawa measure of $G$ is a canonical normalization of the Haar measure on $G$. (Recall that the latter is well-defined only up to scalar multiplication.)
The Tamagawa number of $G$ is essentially the Tamagawa measure of the quotient coset space $G(K)\backslash G(\mathbf{A}_K)$, where $\mathbf{A}$ is the ring of adeles of $K$.
This is conjectured and known in many cases to be a rational number, the quotient $Pic/Sha$ of the order of the Picard group divided by that of the Tate-Shafarevich group. (MO comment)
For $G= GL_1 = \mathbb{G}_m$ the multiplicative group then the quotient $G(\mathbf{Q})\backslash G(\mathbf{A})$ is the idele class group.
The further stacky quotient of $G(\mathbf{Q})\backslash G(\mathbf{A})$ by $G(\mathbb{A}_{\mathbb{Z}})$ is analogous, under the function field analogy and in view of the Weil uniformization theorem, with an incarnation of the moduli stack of G-principal bundles over a curve. A relation of the Tamagawa numbers to the properties of this stack is the content of the Weil conjecture on Tamagawa numbers.)
Wikipedia, Tamagawa numbers, Weil conjecture on Tamagawa numbers
Spencer Bloch, A note on height pairings, Tamagawa numbers, and the Birch and Swinnerton-Dyer conjecture, Invent. Math. 58, 65-76 (1980) (pdf)
Jacob Lurie, Tamagawa Numbers via Nonabelian Poincare Duality, Lecture notes (web)
Relation to Yang-Mills theory and the moduli space of connections:
Some other relation to quantum mechanics and maybe to the Witten genus (?) is claimed in
Last revised on June 24, 2018 at 13:17:34. See the history of this page for a list of all contributions to it.