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Difference between revisions of "Tamagawa measure"

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{{MSC|10C30|12A85,20G35}}
 
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of equivalence classes of valuations of $K$, one denotes by $\omega_\nu$ the
 
of equivalence classes of valuations of $K$, one denotes by $\omega_\nu$ the
 
Haar measure on the locally compact group $G_{K_v}$ of points of $G$ over
 
Haar measure on the locally compact group $G_{K_v}$ of points of $G$ over
the completion $K_\nu$, obtained from $\omega$ (see [[#References|[1]]] and
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the completion $K_\nu$, obtained from $\omega$ (see {{Cite|We}} and
[[#References|[2]]]). If the product $\prod\omega_\nu(G_{O_\nu})$ taken over all
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{{Cite|CaFr}}). If the product $\prod\omega_\nu(G_{O_\nu})$ taken over all
 
non-Archimedean $\nu$, where $G_{O_\nu}$ is the group of integral $\nu$-adic
 
non-Archimedean $\nu$, where $G_{O_\nu}$ is the group of integral $\nu$-adic
 
points, is absolutely convergent (which is always the case for
 
points, is absolutely convergent (which is always the case for
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to define $\tau$ in some non-canonical way, one introduces a system of
 
to define $\tau$ in some non-canonical way, one introduces a system of
 
numbers $(\lambda_\nu)_{\nu\in V}$, called convergence factors, such that the product $\prod_{\nu\in V} \lambda_\nu \omega_\nu (G_{O_\nu})$ is
 
numbers $(\lambda_\nu)_{\nu\in V}$, called convergence factors, such that the product $\prod_{\nu\in V} \lambda_\nu \omega_\nu (G_{O_\nu})$ is
absolutely convergent; then $\tau = \prod_{\nu\in V} \lambda_\nu \omega_\nu$, see [[#References|[1]]],
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absolutely convergent; then $\tau = \prod_{\nu\in V} \lambda_\nu \omega_\nu$, see {{Cite|We}},
[[#References|[3]]].) The measure $\tau$ thus obtained does not depend on
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{{Cite|On}}.) The measure $\tau$ thus obtained does not depend on
 
the initial choice of the form $\omega$, and is the canonical Haar measure
 
the initial choice of the form $\omega$, and is the canonical Haar measure
 
on $G_A$. This allows one to speak about the volume of homogeneous
 
on $G_A$. This allows one to speak about the volume of homogeneous
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====References====
 
====References====
<table><TR><TD valign="top">[1]</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">[2]</TD> <TD valign="top"> J.W.S. Cassels (ed.)  A. Fröhlich (ed.) , ''Algebraic number theory'' , Acad. Press  (1986)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> T. Ono,  "On the Tamagawa number of algebraic tori"  ''Ann. of Math.'' , '''78''' :  1  (1963)  pp. 47–73</TD></TR></table>
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|valign="top"|{{Ref|CaFr}}||valign="top"| J.W.S. Cassels (ed.)  A. Fröhlich (ed.), ''Algebraic number theory'', Acad. Press  (1965)   {{MR|0911121}} {{MR|0255512}} {{MR|0215665}}  {{ZBL|0645.12001}} {{ZBL|0153.07403}}
 
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====Comments====
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|valign="top"|{{Ref|On}}||valign="top"| T. Ono,  "On the Tamagawa number of algebraic tori"  ''Ann. of Math.'', '''78''' :  1  (1963)  pp. 47–73 {{MR|0156851}}  {{ZBL|0122.39101}}   
 
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|-
 
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|valign="top"|{{Ref|We}}||valign="top"|  A. Weil,  "Sur certaines groupes d'opérateurs unitaires"  ''Acta Math.'', '''111'''  (1964)  pp. 143–211     
====References====
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> A. Weil,  "Adèles and algebraic groups" , Birkhäuser  (1982)</TD></TR></table>
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|valign="top"|{{Ref|We2}}||valign="top"| A. Weil,  "Adèles and algebraic groups", Birkhäuser  (1982) {{MR|0670072}}  {{ZBL|0493.14028}}   
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Revision as of 20:02, 5 March 2012

2020 Mathematics Subject Classification: Primary: 10C30 Secondary: 12A8520G35 [MSN][ZBL]

A measure $\tau$ on the group $G_A$ of adèles (cf. Adèle) of a connected linear algebraic group $G$ defined over a global field $K$. This measure is constructed as follows: Let $\omega$ be a non-zero differential form on $G$ of maximum degree which is defined over $K$. For a valuation $\nu$ in the set $V$ of equivalence classes of valuations of $K$, one denotes by $\omega_\nu$ the Haar measure on the locally compact group $G_{K_v}$ of points of $G$ over the completion $K_\nu$, obtained from $\omega$ (see [We] and [CaFr]). If the product $\prod\omega_\nu(G_{O_\nu})$ taken over all non-Archimedean $\nu$, where $G_{O_\nu}$ is the group of integral $\nu$-adic points, is absolutely convergent (which is always the case for semi-simple and unipotent groups $G$), then one puts $\tau=\prod_{\nu\in V} \omega_\nu$. (Otherwise, to define $\tau$ in some non-canonical way, one introduces a system of numbers $(\lambda_\nu)_{\nu\in V}$, called convergence factors, such that the product $\prod_{\nu\in V} \lambda_\nu \omega_\nu (G_{O_\nu})$ is absolutely convergent; then $\tau = \prod_{\nu\in V} \lambda_\nu \omega_\nu$, see [We], [On].) The measure $\tau$ thus obtained does not depend on the initial choice of the form $\omega$, and is the canonical Haar measure on $G_A$. This allows one to speak about the volume of homogeneous spaces connected with $G_A$ (see Tamagawa number).

References

[CaFr] J.W.S. Cassels (ed.) A. Fröhlich (ed.), Algebraic number theory, Acad. Press (1965) MR0911121 MR0255512 MR0215665 Zbl 0645.12001 Zbl 0153.07403
[On] T. Ono, "On the Tamagawa number of algebraic tori" Ann. of Math., 78 : 1 (1963) pp. 47–73 MR0156851 Zbl 0122.39101
[We] A. Weil, "Sur certaines groupes d'opérateurs unitaires" Acta Math., 111 (1964) pp. 143–211
[We2] A. Weil, "Adèles and algebraic groups", Birkhäuser (1982) MR0670072 Zbl 0493.14028
How to Cite This Entry:
Tamagawa measure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tamagawa_measure&oldid=20755
This article was adapted from an original article by A.S. Rapinchuk (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article