Weil Conjecture On Tamagawa Numbers - Tamagawa Measure and Tamagawa Numbers

Tamagawa Measure and Tamagawa Numbers

Let k be a global field, A its ring of adeles, and G an algebraic group defined over k.

The Tamagawa measure on the adelic algebraic group G(A) is defined as follows. Take a left-invariant n-form ω on G(k) defined over k, where n is the dimension of G. This induces Haar measures on G(k_s) for all places of s, and hence a Haar measure on G(A), if the product over all places converges. This Haar measure on G(A) does not depend on the choice of ω, because multiplying ω by an element of k* multiplies the Haar measure on G(A) by 1, using the product formula for valuations.

The Tamagawa number τ(G) is the Tamagawa measure of G(A)/G(k).