Tamagawa number

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

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

The computation of Tamagawa numbers for semisimple groups contains important parts of classical quadratic form theory.

Let k be a global field, A its ring of adeles, and G a semisimple algebraic group defined over k. Choose Haar measures on the completions kv of k such that Ov has volume 1 for all but finitely many places v. These then induce a Haar measure on A, which we further assume is normalized so that A/k has volume 1 with respect to the induced quotient measure.

Ono (1963) found examples where the Tamagawa numbers are not integers, but the conjecture about the Tamagawa number of simply connected groups was proven in general by several works culminating in a paper by Kottwitz (1988) and for the analogue over function fields over finite fields by Gaitsgory & Lurie (2019).