nLab Tamagawa number

Idea

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)

Remark

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.

Remark

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.)

Related concepts

• Weil conjecture on Tamagawa numbers

• Birch and Swinnerton-Dyer conjecture

References

• 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:

• Aravind Asok, Brent Doran, Frances Kirwan, Yang-Mills theory and Tamagawa numbers (arXiv:0801.4733)

Some other relation to quantum mechanics and maybe to the Witten genus (?) is claimed in

• M. A. Olshanetsky, Quantum-mechanical calculations in the algebraic group theory, Comm. Math. Phys. Volume 132, Number 2 (1990), 441-459. (Euclid)