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A measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092050/t0920501.png" /> on the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092050/t0920502.png" /> 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/t092050/t0920503.png" /> defined over a global field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092050/t0920504.png" />. This measure is constructed as follows: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092050/t0920505.png" /> be a non-zero differential form on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092050/t0920506.png" /> of maximum degree which is defined over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092050/t0920507.png" />. For a valuation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092050/t0920508.png" /> in the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092050/t0920509.png" /> of equivalence classes of valuations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092050/t09205010.png" />, one denotes by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092050/t09205011.png" /> the Haar measure on the locally compact group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092050/t09205012.png" /> of points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092050/t09205013.png" /> over the completion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092050/t09205014.png" />, obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092050/t09205015.png" /> (see [[#References|[1]]] and [[#References|[2]]]). If the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092050/t09205016.png" /> taken over all non-Archimedean <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092050/t09205017.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092050/t09205018.png" /> is the group of integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092050/t09205019.png" />-adic points, is absolutely convergent (which is always the case for semi-simple and unipotent groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092050/t09205020.png" />), then one puts <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092050/t09205021.png" />. (Otherwise, to define <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092050/t09205022.png" /> in some non-canonical way, one introduces a system of numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092050/t09205023.png" />, called convergence factors, such that the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092050/t09205024.png" /> is absolutely convergent; then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092050/t09205025.png" />, see [[#References|[1]]], [[#References|[3]]].) The measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092050/t09205026.png" /> thus obtained does not depend on the initial choice of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092050/t09205027.png" />, and is the canonical Haar measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092050/t09205028.png" />. This allows one to speak about the volume of homogeneous spaces connected with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092050/t09205029.png" /> (see [[Tamagawa number|Tamagawa number]]).
+
A measure $\tau$ on the group $G_A$ of adèles (cf. [[Adèle|Adèle]])
 +
of a connected [[Linear algebraic group|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 [[#References|[1]]] and
 +
[[#References|[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 [[#References|[1]]],
 +
[[#References|[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|Tamagawa number]]).
  
 
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Revision as of 12:41, 11 September 2011

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=14742
This article was adapted from an original article by A.S. Rapinchuk (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article