Tamagawa number

2020 Mathematics Subject Classification: Primary: 20G30 Secondary: 11E7222E55 [MSN][ZBL]

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 [Pl]).

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 [On]. 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 [On2]: 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 ([We], [We2], [Ma]), and also for Chevalley groups over number fields (see ) and over global function fields [Ha].

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 [Ko]. Weil's conjecture has been proved by R. Kottwitz [Ko2] for number fields, modulo the validity of the Hasse principle. (The latter has also been established.)

References

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[Ha]	G. Harder, Chevalley groups over function fields and automorphic forms Ann. of Math., 100 (1974) pp. 249 306 MR0563090 Zbl 0309.14041
[Ko]	R.E. Kottwitz, Stable trace formula: cuspidal tempered terms Duke Math. J., 51 (1984) pp. 611 650 MR0757954 Zbl 0576.22020
[Ko2]	R.E. Kottwitz, Tamagawa numbers Ann. of Math., 127 (1988) pp. 629 646 MR0942522 Zbl 0678.22012
[Ma]	J.G.M. Mars, The Tamagawa number of ${}^2A_n$ Ann. of Math., 89 (1969) pp. 557 574 MR0263828 Zbl 0193.21502
[On]	T. Ono, On the Tamagawa number of algebraic tori Ann. of Math., 78 (1963) pp. 47 73 MR0156851 Zbl 0122.39101
[On2]	T. Ono, On the relative theory of Tamagawa numbers Ann. of Math., 82 (1965) pp. 88 111 MR0177991 Zbl 0135.08804
[Pl]	V.P. Platonov, Arithmetic theory of algebraic groups Russian Math. Surveys, 37 : 3 (1982) pp. 1 62 Uspekhi Mat. Nauk, 37 : 3 (1982) pp. 3 54 MR0659426 Zbl 0513.20028
[We]	A. Weil, Sur certains groupes d'opérateurs unitaires Acta Math., 111 (1964) pp. 143 211 MR0165033 Zbl 0203.03305
[We2]	A. Weil, Sur la formule de Siegel dans la théorie des groupes classiques Acta Math., 113 (1965) pp. 1 87 MR0223373 Zbl 0161.02304