Tamagawa measure
This category corresponds roughly to MSC {{{id}}} {{{title}}}; see {{{id}}} at MathSciNet and {{{id}}} at zbMATH.
The Tamagawa measure is
a measure $\tau$ on the group $G_A$ of adèles (cf. Adele group)
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 and
). 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 ,
.) 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] | J.W.S. Cassels (ed.) A. Fröhlich (ed.), Algebraic number theory, Acad. Press (1965) MR0215665 Template:ZBL |
| [2] | T. Ono, "On the Tamagawa number of algebraic tori" Ann. of Math., 78 : 1 (1963) pp. 47–73 MR0156851 Template:ZBL |
| [3] | A. Weil, "Sur certains groupes d'opérateurs unitaires" Acta Math., 111 (1964) pp. 143–211 MR0165033 Template:ZBL |
| [4] | A. Weil, "Adèles and algebraic groups", Birkhäuser (1982) MR0670072 Template:ZBL |
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