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Difference between revisions of "Tamagawa measure"

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A measure $\tau$ on the group $G_A$ of adèles (cf. [[Adèle|Adèle]])
 
A measure $\tau$ on the group $G_A$ of adèles (cf. [[Adèle|Adèle]])
 
of a connected [[Linear algebraic group|linear algebraic group]] $G$
 
of a connected [[Linear algebraic group|linear algebraic group]] $G$

Revision as of 13:50, 29 January 2012


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 [1] and [2]). 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 [1], [3].) 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

[1] A. Weil, "Sur certaines groupes d'opérateurs unitaires" Acta Math. , 111 (1964) pp. 143–211
[2] J.W.S. Cassels (ed.) A. Fröhlich (ed.) , Algebraic number theory , Acad. Press (1986)
[3] T. Ono, "On the Tamagawa number of algebraic tori" Ann. of Math. , 78 : 1 (1963) pp. 47–73


Comments

References

[a1] A. Weil, "Adèles and algebraic groups" , Birkhäuser (1982)
How to Cite This Entry:
Tamagawa measure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tamagawa_measure&oldid=19562
This article was adapted from an original article by A.S. Rapinchuk (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article