This volume contains the original lecture notes presented by A. Weil in which the concept of adeles was first introduced, in conjunction with various aspects of C. L. Siegel s work on quadratic forms. These notes have been supplemented by an extended bibliography, and by Takashi Ono s brief survey of subsequent research.
Serving as an introduction to the subject, these notes may also provide stimulation for further research.
Inhaltsverzeichnis
I. Preliminaries on Adele-Geometry. - 1. 1. Adeles. - 1. 2. Adele-spaces attached to algebraic varieties. - 1. 3. Restriction of the basic field. - II. Tamagawa Measures. - 2. 1. Preliminaries. - 2. 2. The case of an algebraic variety: the local measure. - 2. 3. The global measure and the convergence factors. - 2. 4. Algebraic groups and Tamagawa numbers. - III. The Linear, Projective and Symplectic Groups. - 3. 1. The zeta-function of a central division algebra. - 3. 2. The projective group of a central division algebra. - 3. 3. Isogenies. - 3. 4. End of proof of Theorem 3. 3. 1. : central simple algebras. - 3. 5. The symplectic group. - 3. 6. Isogenies for products of linear groups. - 3. 7. Application to some orthogonal and hermitian groups. - 3. 8. The zeta-function of a central simple algebra. - IV. The other Classical Groups. - 4. 1. Classification and general theorems. - 4. 2. End of proof of Theorem 4. 1. 3 (types 01, L2(a), S2). - 4. 3. The local zeta-functions for a quadratic form. - 4. 4. The Tamagawa number (hermitian and quaternionic cases). - 4. 5. The Tamagawa number of the orthogonal group. - Appendix 2. (by T. Ono) A short survey of subsequent research on Tamagawa numbers.
