**Mathematical Surveys and Monographs**

Volume: 109;
2004;
275 pp;
Softcover

MSC: Primary 11; 15; 22;

Print ISBN: 978-1-4704-1562-4

Product Code: SURV/109.S

List Price: $80.00

AMS Member Price: $64.00

MAA Member Price: $72.00

**Electronic ISBN: 978-1-4704-1336-1
Product Code: SURV/109.E**

List Price: $80.00

AMS Member Price: $64.00

MAA Member Price: $72.00

#### Supplemental Materials

# Arithmetic and Analytic Theories of Quadratic Forms and Clifford Groups

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*Goro Shimura*

In this book, award-winning author Goro Shimura treats new areas and
presents relevant expository material in a clear and readable style. Topics
include Witt's theorem and the Hasse principle on quadratic forms, algebraic
theory of Clifford algebras, spin groups, and spin representations. He also
includes some basic results not readily found elsewhere.

The two principle themes are:

(1) Quadratic Diophantine equations;

(2) Euler products and Eisenstein series on orthogonal groups and Clifford
groups.

The starting point of the first theme is the result of Gauss that the number
of primitive representations of an integer as the sum of three squares is
essentially the class number of primitive binary quadratic forms. Presented are
a generalization of this fact for arbitrary quadratic forms over algebraic
number fields and various applications. For the second theme, the author proves
the existence of the meromorphic continuation of a Euler product associated
with a Hecke eigenform on a Clifford or an orthogonal group. The same is done
for an Eisenstein series on such a group.

Beyond familiarity with algebraic number theory, the book is mostly
self-contained. Several standard facts are stated with references for detailed
proofs.

Goro Shimura won the 1996 Steele Prize for Lifetime Achievement for
“his important and extensive work on arithmetical geometry and
automorphic forms”.

#### Readership

Graduate students and research mathematicians interested in number theory and algebraic groups.

#### Table of Contents

# Table of Contents

## Arithmetic and Analytic Theories of Quadratic Forms and Clifford Groups

- Table of Contents v6 free
- Preface vii8 free
- Notation and Terminology ix10 free
- Introduction 112 free
- Chapter I. Algebraic theory of quadratic forms, Clifford algebras, and spin groups 920 free
- Chapter II. Quadratic forms, Clifford algebras, and spin groups over a local or global field 3748
- 5. Orders and ideals in an algebra 3748
- 6. Quadratic forms over a local field 4556
- 7. Lower-dimensional cases and the Hasse principle 5263
- 8. Part I. Clifford groups over a local field 6273
- 8. Part II. Formal Hecke algebras and formal Euler factors 7283
- 9. Orthogonal, Clifford, and spin groups over a global field 8091

- Chapter III. Quadratic Diophantine equations 93104
- 10. Quadratic Diophantine equations over a local field 93104
- 11. Quadratic Diophantine equations over a global field 101112
- 12. The class number of an orthogonal group and sums of squares 113124
- 13. Nonscalar quadratic Diophantine equations; Connection with the mass formula; A historical perspective 126137

- Chapter IV. Groups and symmetric spaces over R 139150
- Chapter V. Euler products and Eisenstein series on orthogonal groups 163174
- 17. Automorphic forms and Euler products on an orthogonal group 163174
- 18. Eisenstein series on o[sup(w)] 173184
- 19. Eisenstein series on o[sup(η)] 181192
- 20. Arithmetic description of the pullback of an Eisenstein series 187198
- 21. Analytic continuation of Euler products and Eisenstein series 196207

- Chapter VI. Euler products and Eisenstein series on Clifford groups 205216
- Appendix 243254
- References 272283
- Frequently used symbols 274285
- Index 275286 free