View source for Tamagawa measure

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]]).

====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>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Weil,   "Adèles and algebraic groups" , Birkhäuser  (1982)</TD></TR></table>