In mathematics, the Tamagawa number of a semisimple algebraic group defined over a global field is the measure of, where is the adele ring of. Tamagawa numbers were introduced by, and named after him by. Tsuneo Tamagawa's observation was that, starting from an invariant differential formω on, defined over, the measure involved was well-defined: while could be replaced by with a non-zero element of, the product formula for valuations in is reflected by the independence from of the measure of the quotient, for the product measure constructed from on each effective factor. The computation of Tamagawa numbers for semisimple groups contains important parts of classicalquadratic form theory.
Definition
Let be a global field, its ring of adeles, and a semisimple algebraic group defined over. Choose Haar measures on the completions such that has volume 1 for all but finitely many places. These then induce a Haar measure on, which we further assume is normalized so that has volume 1 with respect to the induced quotient measure. The Tamagawa measure on the adelic algebraic group is now defined as follows. Take a left-invariant -form on defined over, where is the dimension of. This, together with the above choices of Haar measure on the, induces Haar measures on for all places of. As is semisimple, the product of these measures yields a Haar measure on, called the Tamagawa measure. The Tamagawa measure does not depend on the choice of ω, nor on the choice of measures on the, because multiplying by an element of multiplies the Haar measure on by 1, using the product formula for valuations. The Tamagawa number is defined to be the Tamagawa measure of.
Weil's conjecture on Tamagawa numbers
Weil's conjecture on Tamagawa numbers states that the Tamagawa number of a simply connectedsimple algebraic group defined over a number field is 1. calculated the Tamagawa number in many cases of classical groups and observed that it is an integer in all considered cases and that it was equal to 1 in the cases when the group is simply connected. 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 and for the analogue over function fields over finite fields by Lurie and Gaitsgory in 2011.
