Tamagawa number

In mathematics, the Tamagawa number ${\displaystyle \tau (G)}$ of a semisimple algebraic group defined over a global field k is the measure of ${\displaystyle G(\mathbb {A} )/G(k)}$, where ${\displaystyle \mathbb {A} }$ is the adele ring of k. Tamagawa numbers were introduced by Tamagawa (1966), and named after him by Weil (1959).

Tsuneo Tamagawa's observation was that, starting from an invariant differential form ω on G, defined over k, the measure involved was well-defined: while ω could be replaced by cω with c a non-zero element of ${\displaystyle k}$, the product formula for valuations in k is reflected by the independence from c of the measure of the quotient, for the product measure constructed from ω on each effective factor. The computation of Tamagawa numbers for semisimple groups contains important parts of classical quadratic form theory.