Tamagawa measure

2020 Mathematics Subject Classification: Primary: 20G30 Secondary: 12A8520G35 [MSN][ZBL]

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 [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) MR0215665 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 certains groupes d'opérateurs unitaires" Acta Math., 111 (1964) pp. 143–211 MR0165033 Zbl 0203.03305
[We2]	A. Weil, "Adèles and algebraic groups", Birkhäuser (1982) MR0670072 Zbl 0493.14028