The Tamagawa Number and the Volume of G(ℝ)/G(ℤ)

Abstract

In this chapter we present, following Weil [Wei], the definition and calculation of the Tamagawa number τ(G) of the algebraic ℚ-group G. Let $\mathbb{A}$ be the adele ring of ℚ. By definition,

$$\tau \left( G \right) = {\text{ vol}}\left( {G\left( \mathbb{A} \right)/G\left( \mathbb{Q} \right)} \right)$$

with respect to the Tamagawa measure, to be defined below. The main result, τ(G) = 1, is easily translated into the relation

$${\text{vol}}\left( {G\left( {\mathbb{R}} \right)/G\left( \mathbb{Z} \right)} \right) = {{\left( {\mathop{\Pi }\limits_{p} {\text{vol}}\left( {G\left( {{{\mathbb{Z}}_{p}}} \right)} \right)} \right)}^{{ - 1}}}$$

it is not hard to calculate the right-hand side, whereas the left-hand side is inaccessible in the direct way, which would consist in constructing a fundamental domain and integrating over it. (For SL _n , this can be done, see [Si 2].) The purpose of this chapter is to make the reader understand the miracle that τ(G) can be determined in spite of this.