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Tamagawa number

Contents

Idea

Let GG be a connected semisimple algebraic group over the field of rational numbers Q\mathbf{Q} or more generally over a global field KK. The Tamagawa measure of GG is a canonical normalization of the Haar measure on GG. (Recall that the latter is well-defined only up to scalar multiplication.)

The Tamagawa number of GG is essentially the Tamagawa measure of the quotient coset space G(K)\G(A K)G(K)\backslash G(\mathbf{A}_K), where A\mathbf{A} is the ring of adeles of KK.

This is conjectured and known in many cases to be a rational number, the quotient Pic/ShaPic/Sha of the order of the Picard group divided by that of the Tate-Shafarevich group. (MO comment)

Remark

For G=GL 1=𝔾 mG= GL_1 = \mathbb{G}_m the multiplicative group then the quotient G(Q)\G(A)G(\mathbf{Q})\backslash G(\mathbf{A}) is the idele class group.

Remark

The further stacky quotient of G(Q)\G(A)G(\mathbf{Q})\backslash G(\mathbf{A}) by G(𝔸 )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.)

References

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)

Last revised on June 24, 2018 at 13:17:34. See the history of this page for a list of all contributions to it.