TABLE OF CONTENTS Preface vii Notation and Terminology ix Introduction 1 Chapter I. Algebraic theory of quadratic forms, Clifford algebras, and spin groups 9 1. Quadratic forms and associative algebras 9 2. Clifford algebras 15 3. Clifford groups and spin groups 20 4. Parabolic subgroups 28 Chapter II. Quadratic forms, Clifford algebras, and spin groups over a local or global field 37 5. Orders and ideals in an algebra 37 6. Quadratic forms over a local field 45 7. Lower-dimensional cases and the Hasse principle 52 Part I. Clifford groups over a local field 62 Part II. Formal Hecke algebras and formal Euler factors 72 9. Orthogonal, Clifford, and spin groups over a global field 80 Chapter III. Quadratic Diophantine equations 93 10. Quadratic Diophantine equations over a local field 93 11. Quadratic Diophantine equations over a global field 101 12. The class number of an orthogonal group and sums of squares 113 13. Nonscalar quadratic Diophantine equations Connection with the mass formula A historical perspective 126 Chapter IV. Groups and symmetric spaces over R 139 14. Clifford and spin groups over R The case of signature (1, m) 139 15. The case of signature (2, m) 146 16. Orthogonal groups over R and symmetric spaces 154 Chapter V. Euler products and Eisenstein series on or- thogonal groups 163 17. Automorphic forms and Euler products on an orthogonal group 163
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