Tamagawa measure

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A measure on the group of adèles (cf. Adèle) of a connected linear algebraic group defined over a global field . This measure is constructed as follows: Let be a non-zero differential form on of maximum degree which is defined over . For a valuation in the set of equivalence classes of valuations of , one denotes by the Haar measure on the locally compact group of points of over the completion , obtained from (see [1] and [2]). If the product taken over all non-Archimedean , where is the group of integral -adic points, is absolutely convergent (which is always the case for semi-simple and unipotent groups ), then one puts . (Otherwise, to define in some non-canonical way, one introduces a system of numbers , called convergence factors, such that the product is absolutely convergent; then , see [1], [3].) The measure thus obtained does not depend on the initial choice of the form , and is the canonical Haar measure on . This allows one to speak about the volume of homogeneous spaces connected with (see Tamagawa number).


[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



[a1] A. Weil, "Adèles and algebraic groups" , Birkhäuser (1982)
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Tamagawa measure. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.S. Rapinchuk (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article