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Source: Tamagawa number

In mathematics, the Tamagawa number

τ
(
G
)

{\displaystyle \tau (G)}

of a semisimple algebraic group defined over a global field k is the measure of

G
(

A

)

/

G
(
k
)

{\displaystyle G(\mathbb {A} )/G(k)}

, where

A

{\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

k

{\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.