V l l l PREFACE Since Clifford groups are closely related to quadratic forms, there is nothing special about presenting these two themes in the same volume, but traditionally Clifford groups have been discussed merely as a tool for investigating quadratic forms. In the present book, however, we put the groups on center stage, and investigate them for their own sake, as this approach gives a better perspective in the theory of Euler products. Though my chief intention is to present some new ideas on these two topics, I have also included expository material concerning arithmetic of quadratic forms such as the Hasse principle and the algebraic theory of Clifford algebras and spin groups, often with new methods, so that those portions of the book may serve as an introduction to some such subjects. Princeton May 2003 Goro Shimura
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