View source for Tamagawa number

{{User:Boris Tsirelson/MSC|11F70|11E72,22E55}}

The volume of a homogeneous space $G_A^{(1)}/G_K$ associated with the group of adèles (cf. [[Adèle|Adèle]]) of a connected [[Linear algebraic group|linear algebraic group]] $G$ defined over a global field $K$ with respect to the [[Tamagawa measure|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 [[#References|[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 [[#References|[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 [[#References|[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 ([[#References|[3]]], [[#References|[4]]], [[#References|[7]]]), and also for Chevalley groups over number fields (see [[#References|[2]]]) and over global function fields [[#References|[6]]].

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



====Comments====
The Shafarevich Tate group is also called Tate Shafarevich group; cf. [[Galois cohomology|Galois cohomology]] for its definition.

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

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

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.E. Kottwitz,   "Stable trace formula: cuspidal tempered terms"  ''Duke Math. J.'' , '''51'''  (1984)  pp. 611 650</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R.E. Kottwitz,   "Tamagawa numbers"  ''Ann. of Math.'' , '''127'''  (1988)  pp. 629 646</TD></TR></table>