Tamagawa number

The volume of a homogeneous space associated with the group of adèles (cf. Adèle) of a connected linear algebraic group defined over a global field with respect to the Tamagawa measure. Here is the subgroup of consisting of those adèles for which

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

When describing the values of 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 is an algebraic -torus, then

where and are the order of the one-dimensional Galois cohomology group of the module of rational characters of the torus and the order of its Shafarevich–Tate group, respectively. On the basis of this formula an example was constructed of a torus for which 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 be a semi-simple -group, let be the universal covering which is defined over , let be the fundamental group of , and let be its character group; then

where , and is the order of the kernel of the canonical mapping

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 " 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 and 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