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The volume of a homogeneous space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092060/t0920601.png" /> associated with the group of adèles (cf. [[Adèle|Adèle]]) of a connected [[Linear algebraic group|linear algebraic group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092060/t0920602.png" /> defined over a global field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092060/t0920603.png" /> with respect to the [[Tamagawa measure|Tamagawa measure]]. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092060/t0920604.png" /> is the subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092060/t0920605.png" /> consisting of those adèles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092060/t0920606.png" /> for which
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{{MSC|20G30|11E72,22E55}}
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{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092060/t0920607.png" /></td> </tr></table>
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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
  
for any character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092060/t0920608.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092060/t0920609.png" /> which is defined over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092060/t09206010.png" /> (the product is taken with respect to all valuations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092060/t09206011.png" /> in the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092060/t09206012.png" /> of normalized valuations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092060/t09206013.png" />). The finiteness of the Tamagawa number follows from reduction theory (see [[#References|[5]]]).
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$$\prod_{\nu \in V} | \chi(g_\nu)|_\nu = 1$$
  
When describing the values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092060/t09206014.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092060/t09206015.png" /> is an algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092060/t09206016.png" />-torus, then
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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 {{Cite|Pl}}).
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092060/t09206017.png" /></td> </tr></table>
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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
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092060/t09206018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092060/t09206019.png" /> are the order of the one-dimensional Galois cohomology group of the module of rational characters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092060/t09206020.png" /> of the torus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092060/t09206021.png" /> and the order of its Shafarevich–Tate group, respectively. On the basis of this formula an example was constructed of a torus for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092060/t09206022.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092060/t09206023.png" /> be a semi-simple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092060/t09206024.png" />-group, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092060/t09206025.png" /> be the universal covering which is defined over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092060/t09206026.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092060/t09206027.png" /> be the fundamental group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092060/t09206028.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092060/t09206029.png" /> be its character group; then
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$$\tau(T) = { { [H^1(K,\hat T)] }\over{[{\rm Shaf}(T)] } }$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092060/t09206030.png" /></td> </tr></table>
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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 {{Cite|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 {{Cite|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
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092060/t09206031.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092060/t09206032.png" /> is the order of the kernel of the canonical mapping
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$$\tau(G) = \tau(\hat G) { {h^0(\hat F) }\over{i^1(\hat F)}},$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092060/t09206033.png" /></td> </tr></table>
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where $h^0(\hat F) = [H^0(K,F)]$, and $i^1(\hat F)$ is the order of the kernel of the canonical mapping
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$$H^1(K,\hat F) \to \prod_{\nu\in V} H^1(K_\nu,\hat F).$$
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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 ({{Cite|We}}, {{Cite|We2}}, {{Cite|Ma}}), and also for Chevalley groups over number fields (see {{Cite|}}) and over global function fields {{Cite|Ha}}.
  
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]]].
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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 {{Cite|Ko}}.
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Weil's conjecture has been proved by R. Kottwitz {{Cite|Ko2}} for number fields, modulo the validity of the [[Hasse principle|Hasse principle]]. (The latter has also been established.)
  
 
====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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092060/t09206034.png" />"  ''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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{|
 
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|-
 
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|valign="top"|{{Ref|CaFr}}||valign="top"| J.W.S. Cassels (ed.)  A. Fröhlich (ed.), ''Algebraic number theory'', Acad. Press  (1967) {{MR|0215665}}  {{ZBL|0153.07403}}
 
+
|-
====Comments====
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|valign="top"|{{Ref|Ha}}||valign="top"|  G. Harder,   ''Chevalley groups over function fields and automorphic forms''  Ann. of Math., '''100'''  (1974)  pp. 249 306 {{MR|0563090}} {{ZBL|0309.14041}}
The Shafarevich–Tate group is also called Tate–Shafarevich group; cf. [[Galois cohomology|Galois cohomology]] for its definition.
+
|-
 
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|valign="top"|{{Ref|Ko}}||valign="top"| R.E. Kottwitz,  ''Stable trace formula: cuspidal tempered terms''  Duke Math. J., '''51'''  (1984)  pp. 611 650  {{MR|0757954}} {{ZBL|0576.22020}}
For the relation between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092060/t09206035.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092060/t09206036.png" /> see [[#References|[a1]]].
+
|-
 
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|valign="top"|{{Ref|Ko2}}||valign="top"| R.E. Kottwitz,  ''Tamagawa numbers''  Ann. of Math., '''127'''  (1988)  pp. 629 646 {{MR|0942522}} {{ZBL|0678.22012}}
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.)
+
|-
 
+
|valign="top"|{{Ref|Ma}}||valign="top"| J.G.M. Mars,  ''The Tamagawa number of ${}^2A_n$'' Ann. of Math., '''89'''  (1969)  pp. 557 574 {{MR|0263828}} {{ZBL|0193.21502}}
====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>
+
|valign="top"|{{Ref|On}}||valign="top"| T. Ono,  ''On the Tamagawa number of algebraic tori'' Ann. of Math., '''78'''  (1963)  pp. 47 73 {{MR|0156851}} {{ZBL|0122.39101}}
 +
|-
 +
|valign="top"|{{Ref|On2}}||valign="top"| T. Ono,  ''On the relative theory of Tamagawa numbers'' Ann. of Math., '''82'''  (1965)  pp. 88 111 {{MR|0177991}} {{ZBL|0135.08804}}
 +
|-
 +
|valign="top"|{{Ref|Pl}}||valign="top"|  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  {{MR|0659426}} {{ZBL|0513.20028}}
 +
|-
 +
|valign="top"|{{Ref|We}}||valign="top"| A. Weil,  ''Sur certains groupes d'opérateurs unitaires''  Acta Math., '''111'''  (1964)  pp. 143 211 {{MR|0165033}}  {{ZBL|0203.03305}}
 +
|-
 +
|valign="top"|{{Ref|We2}}||valign="top"| A. Weil,  ''Sur la formule de Siegel dans la théorie des groupes classiques''  Acta Math., '''113'''  (1965)  pp. 1 87 {{MR|0223373}}  {{ZBL|0161.02304}}
 +
|-
 +
|}

Latest revision as of 21:31, 4 March 2012

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

[CaFr] J.W.S. Cassels (ed.) A. Fröhlich (ed.), Algebraic number theory, Acad. Press (1967) MR0215665 Zbl 0153.07403
[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
How to Cite This Entry:
Tamagawa number. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tamagawa_number&oldid=13490
This article was adapted from an original article by A.S. Rapinchuk (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article