Tamagawa number

The volume of a homogeneous space $G_A^{(1)}/G_K$ associated with the group of adèles (cf. Adèle) of a connected linear algebraic group $G$ defined over a global field $K$ with respect to the Tamagawa measure. Here $G_A^{(1)}/G_K$ is the subgroup of $G_A$ consisting of those adèles $g = (g_\nu)_{\nu\in V} \in G_A$ for which

$$\prod_{\nu \in V} | \chi(g_\nu)|_\nu = 1$$

for any character $\chi$ of $G$ which is defined over $K$ (the product is taken with respect to all valuations $\nu$ in the set $V$ of normalized valuations of $K$). The finiteness of the Tamagawa number follows from reduction theory (see [5]).

When describing the values of $\tau(G)$ it is convenient to distinguish the cases of unipotent groups, algebraic tori and semi-simple groups. For unipotent groups the Tamagawa number is always equal to 1. If $T$ is an algebraic $K$-torus, then

$$\tau(T) = { { [H^1(K,\hat T)] }\over{[{\rm Shaf}(T)] } }$$

where $ [H^1(K,\hat T)]$ and $[{\rm Shaf}(T)]$ are the order of the one-dimensional Galois cohomology group of the module of rational characters $\hat T$ of the torus $T$ and the order of its Shafarevich Tate group, respectively. On the basis of this formula an example was constructed of a torus for which $\tau(T)$ is not an integer [8]. The determination of the Tamagawa number of a semi-simple group over a number field can be reduced to the case of a simply-connected group [9]: Let $G$ be a semi-simple $K$-group, let $\pi : \hat G \to G$ be the universal covering which is defined over $K$, let $F = {\rm Ker}\; \pi$ be the fundamental group of $G$, and let $\hat F$ be its character group; then

$$\tau(G) = \tau(\hat G) { {h^0(\hat F) }\over{i^1(\hat F)}},$$

where $h^0(\hat F) = [H^0(K,F)]$, and $i^1(\hat F)$ is the order of the kernel of the canonical mapping $$H^1(K,\hat F) \to \prod_{\nu\in V} H^1(K_\nu,\hat F).$$ It is the conjectured that for all simply-connected groups the Tamagawa number is equal to 1 (the Weil conjecture). This was proved for most types of simple groups over number fields ([3], [4], [7]), and also for Chevalley groups over number fields (see [2]) and over global function fields [6].

References

[1]	J.W.S. Cassels (ed.) A. Fröhlich (ed.) , Algebraic number theory , Acad. Press (1986)
[2]	, Arithmetic groups and automorphic functions , Moscow (1969) (In Russian; translated from English and French)
[3]	A. Weil, "Sur certaines groupes d'opérateurs unitaires" Acta Math. , 111 (1964) pp. 143 211
[4]	A. Weil, "Sur la formule de Siegel dans la théorie des groupes classiques" Acta Math. , 113 (1965) pp. 1 87
[5]	V.P. Platonov, "The arithmetic theory of algebraic groups" Russian Math. Surveys , 37 : 3 (1982) pp. 1 62 Uspekhi Mat. Nauk , 37 : 3 (1982) pp. 3 54
[6]	G. Harder, "Chevalley groups over function fields and automorphic forms" Ann. of Math. , 100 (1974) pp. 249 306
[7]	J.G.M. Mars, "The Tamagawa number of ${}^2A_n$" Ann. of Math. , 89 (1969) pp. 557 574
[8]	T. Ono, "On the Tamagawa number of algebraic tori" Ann. of Math. , 78 (1963) pp. 47 73
[9]	T. Ono, "On the relative theory of Tamagawa numbers" Ann. of Math. , 82 (1965) pp. 88 111

Comments

The Shafarevich Tate group is also called Tate Shafarevich group; cf. Galois cohomology for its definition.

For the relation between $\tau(G)$ and $\tau(\hat G)$ see [a1].

Weil's conjecture has been proved by R. Kottwitz [a2] for number fields, modulo the validity of the Hasse principle. (The latter has also been established.)

References

[a1]	R.E. Kottwitz, "Stable trace formula: cuspidal tempered terms" Duke Math. J. , 51 (1984) pp. 611 650
[a2]	R.E. Kottwitz, "Tamagawa numbers" Ann. of Math. , 127 (1988) pp. 629 646