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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.
List of contents
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.
