Adèles and algebraic groups. (Appendix 1: The case of the group \(G_2\), by M. Demazure. Appendix 2: A short survey of subsequent research on Tamagawa numbers, by T. Ono).

*(English)*Zbl 0493.14028
Progress in Mathematics, Vol. 23. Boston-Basel-Stuttgart: Birkhäuser. v, 126 pp. DM 30.00 (1982).

This very influential work only existed up to now as mimeographed lecture notes, not
easily available from the Institute for Advanced Study, and it is a very welcome move
to have it reprinted in book form. After the introduction of idèles in number
theory by Chevalley, A. Weil had already seen in 1938 that, more generally, adèles
could also be used in that theory, and in 1957, T. Ono had made a first study of
adelized algebraic groups. Then, in 1959, Tamagawa discovered what are now called
Tamagawa measures and their relation to Siegel’s work on quadratic forms, but his
paper on this subject was only published in 1966 [T. Tamagawa, Proc. Sympos.
Pure Math. 9, 113–121 (1966; Zbl 0178.23801)]. A. Weil, who had seen Tamagawa’s
manuscript, at once realized its importance, and made it a subject of lectures he
gave in 1959–60 (and which are reproduced in this book), enriched by many original
viewpoints and results, giving the impetus to the modern theory of arithmetic groups.

Chapter I describes the process of adelization of algebraic varieties and algebraic groups. Chapter II introduces Tamagawa measures: if \(G\) is an algebraic linear group defined over a global field \(k\) and \(G_A\) the adelized group, \(G_k\) is a discrete subgroup of \(G_A\); if \(G_A/G_k\) has finite measure for any left Haar measure on \(G\), there is a privileged Haar measure on \(G_A\), called the Tamagawa measure, and the measure \(\tau(G)\) of \(G_A/G_k\) for that measure is the Tamagawa number.

Chapter III begins the computation of the Tamagawa numbers for the classical groups; one starts by computing \(\tau(G)\) when \(G\) is the projective group of a division algebra \(D\) of rank \(n^2\) over its center equal to \(k\); it is shown that \(\tau(G)=n\), by a method consisting in computing the residues of the zeta-function of \(D\) at its poles, since these residues can be expressed in terms of \(\tau(G)\). From then on one passes to the case in which \(G\) is the special linear group of \(D\), using general results on the relations between the Tamagawa numbers of two isogenous groups, which involve also the computation of Tamagawa numbers for tori. Owing to the well-known isomorphisms between the classical groups of low dimensions, one obtains the Tamagawa numbers of orthogonal and unitary groups for the lowest dimensions.

From then on, Chapter IV obtains the Tamagawa numbers for most of the classical groups by induction on the dimension, using again the zeta-function.

Appendix I by Demazure computes the Tamagawa number of the group of type \(G_2\) in the Cartan classification, using the algebra of Cayley numbers. – Appendix II by T. Ono reviews rapidly the results obtained on Tamagawa numbers since 1960: Weil conjectured that \(\tau(G)=1\) for all simply connected semisimple groups; this is not yet completely proved but has been verified in all cases in which the computation of \(\tau(G)\) has been made. T. Ono has studied the Tamagawa number of tori, but the results are not yet complete. It is a pity that the short passage of Tamagawa’s paper in which he established the relation between the expression \(\tau(G)=2\) for the orthogonal group and Siegel’s main theorem for quadratic forms has not been inserted in the book.

Chapter I describes the process of adelization of algebraic varieties and algebraic groups. Chapter II introduces Tamagawa measures: if \(G\) is an algebraic linear group defined over a global field \(k\) and \(G_A\) the adelized group, \(G_k\) is a discrete subgroup of \(G_A\); if \(G_A/G_k\) has finite measure for any left Haar measure on \(G\), there is a privileged Haar measure on \(G_A\), called the Tamagawa measure, and the measure \(\tau(G)\) of \(G_A/G_k\) for that measure is the Tamagawa number.

Chapter III begins the computation of the Tamagawa numbers for the classical groups; one starts by computing \(\tau(G)\) when \(G\) is the projective group of a division algebra \(D\) of rank \(n^2\) over its center equal to \(k\); it is shown that \(\tau(G)=n\), by a method consisting in computing the residues of the zeta-function of \(D\) at its poles, since these residues can be expressed in terms of \(\tau(G)\). From then on one passes to the case in which \(G\) is the special linear group of \(D\), using general results on the relations between the Tamagawa numbers of two isogenous groups, which involve also the computation of Tamagawa numbers for tori. Owing to the well-known isomorphisms between the classical groups of low dimensions, one obtains the Tamagawa numbers of orthogonal and unitary groups for the lowest dimensions.

From then on, Chapter IV obtains the Tamagawa numbers for most of the classical groups by induction on the dimension, using again the zeta-function.

Appendix I by Demazure computes the Tamagawa number of the group of type \(G_2\) in the Cartan classification, using the algebra of Cayley numbers. – Appendix II by T. Ono reviews rapidly the results obtained on Tamagawa numbers since 1960: Weil conjectured that \(\tau(G)=1\) for all simply connected semisimple groups; this is not yet completely proved but has been verified in all cases in which the computation of \(\tau(G)\) has been made. T. Ono has studied the Tamagawa number of tori, but the results are not yet complete. It is a pity that the short passage of Tamagawa’s paper in which he established the relation between the expression \(\tau(G)=2\) for the orthogonal group and Siegel’s main theorem for quadratic forms has not been inserted in the book.

Reviewer: J. Dieudonné

##### MSC:

14L35 | Classical groups (algebro-geometric aspects) |

20G35 | Linear algebraic groups over adèles and other rings and schemes |

14-02 | Research exposition (monographs, survey articles) pertaining to algebraic geometry |

11R56 | Adèle rings and groups |

43A70 | Analysis on specific locally compact and other abelian groups |

14L10 | Group varieties |

32M05 | Complex Lie groups, group actions on complex spaces |

11R52 | Quaternion and other division algebras: arithmetic, zeta functions |

20-02 | Research exposition (monographs, survey articles) pertaining to group theory |