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The volume of a homogeneous space <math>
+
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
G_A^{(1)}/G_K
 
</math> associated with the group of adèles (cf. [[Adèle|Adèle]]) of a connected [[Linear algebraic group|linear algebraic group]] <math>
 
G
 
</math> defined over a global field <math>
 
K
 
</math> with respect to the [[Tamagawa measure|Tamagawa measure]]. Here <math>
 
G_A^{(1)}/G_K
 
</math> is the subgroup of <math>
 
G_A
 
</math> consisting of those adèles <math>
 
g = (g_\nu)_{\nu\in V} \in G_A
 
</math> for which
 
  
<center><math>
+
$$\prod_{\nu \in V} | \chi(g_\nu)|_\nu = 1$$
\prod_{\nu \in V} | \chi(g_\nu)|_\nu = 1
 
</math></center>
 
  
for any character <math>
+
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]]]).
\chi
 
</math> of <math>
 
G
 
</math> which is defined over <math>
 
K
 
</math> (the product is taken with respect to all valuations <math>
 
\nu
 
</math> in the set <math>
 
V
 
</math> of normalized valuations of <math>
 
K
 
</math>). The finiteness of the Tamagawa number follows from reduction theory (see [[#References|[5]]]).
 
  
When describing the values of <math>
+
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(G)
 
</math> 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 <math>
 
T
 
</math> is an algebraic <math>
 
K
 
</math>-torus, then
 
  
<center><math>
+
$$\tau(T) = { { [H^1(K,\hat T)] }\over{[{\rm Shaf}(T)] } }$$
\tau(T) = { { [H^1(K,\hat T)] }\over{[{\rm Shaf}(T)] } }
 
</math></center>
 
  
where <math>
+
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
[H^1(K,\hat T)]
 
</math> and <math>
 
[{\rm Shaf}(T)]
 
</math> are the order of the one-dimensional Galois cohomology group of the module of rational characters <math>
 
\hat T
 
</math> of the torus <math>
 
T
 
</math> and the order of its Shafarevich Tate group, respectively. On the basis of this formula an example was constructed of a torus for which <math>
 
\tau(T)
 
</math> 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 <math>
 
G
 
</math> be a semi-simple <math>
 
K
 
</math>-group, let <math>
 
\pi : \hat G \to G
 
</math> be the universal covering which is defined over <math>
 
K
 
</math>, let <math>
 
F = {\rm Ker}\; \pi
 
</math> be the fundamental group of <math>
 
G
 
</math>, and let <math>
 
\hat F
 
</math> be its character group; then
 
  
<center><math>
+
$$\tau(G) = \tau(\hat G) { {h^0(\hat F) }\over{i^1(\hat F)}},$$
\tau(G) = \tau(\hat G) { {h^0(\hat F) }\over{i^1(\hat F)}},
 
</math></center>
 
  
where <math>
+
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^0(\hat F) = [H^0(K,F)]
+
$$H^1(K,\hat F) \to \prod_{\nu\in V} H^1(K_\nu,\hat F).$$
</math>, and <math>
 
i^1(\hat F)
 
</math> is the order of the kernel of the canonical mapping
 
<center><math>
 
H^1(K,\hat F) \to \prod_{\nu\in V} H^1(K_\nu,\hat F).
 
</math></center>
 
 
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]]].
 
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====
 
====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 <math>
+
<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>
{}^2A_n
 
</math>"  ''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>
 
  
  
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The Shafarevich Tate group is also called Tate Shafarevich group; cf. [[Galois cohomology|Galois cohomology]] for its definition.
 
The Shafarevich Tate group is also called Tate Shafarevich group; cf. [[Galois cohomology|Galois cohomology]] for its definition.
  
For the relation between <math>
+
For the relation between $\tau(G)$ and $\tau(\hat G)$ see [[#References|[a1]]].
\tau(G)
 
</math> and <math>
 
\tau(\hat G)
 
</math> 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.)
 
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.)

Revision as of 21:39, 6 September 2011

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
How to Cite This Entry:
Tamagawa number. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tamagawa_number&oldid=19549
This article was adapted from an original article by A.S. Rapinchuk (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article