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El. knyga: Arithmetic and Analytic Theories of Quadratic Forms and Clifford Groups

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Arithmetic and Analytic Theories of Quadratic Forms and Clifford GroupsDidesnis paveikslėlis
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The two main themes of the book are (1) quadratic Diophantine equations; (2) Euler products and Eisenstein series on orthogonal groups and Clifford groups. Whereas the latest chapters of the book contain new results, a substantial portion of it is devoted to expository material related to these themes, such as Witt's theorem and the Hasse principle on quadratic forms, algebraic theory of Clifford algebras, spin groups, and spin representations.

The starting point of the first main 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. A generalization of this fact for arbitrary quadratic forms over algebraic number fields, as well as various applications are presented. As for the second theme, the existence of the meromorphic continuation of an Euler product associated with a Hecke eigenform on a Clifford or an orthogonal group is proved. The same is done for an Eisenstein series on such a group.

The book is practically self-contained, except that familiarity with algebraic number theory is assumed and several standard facts are stated without detailed proof, but with precise references.
Preface vii
Notation and Terminology ix
Introduction 1(8)
Chapter I Algebraic theory of quadratic forms, Clifford algebras, and spin groups
9(28)
1 Quadratic forms and associative algebras
9(6)
2 Clifford algebras
15(5)
3 Clifford groups and spin groups
20(8)
4 Parabolic subgroups
28(9)
Chapter II Quadratic forms, Clifford algebras, and spin groups over a local or global field
37(56)
5 Orders and ideals in an algebra
37(8)
6 Quadratic forms over a local field
45(7)
7 Lower-dimensional cases and the Hasse principle
52(10)
8 Part I. Clifford groups over a local field
62(10)
8 Part II. Formal Hecke algebras and formal Euler factors
72(8)
9 Orthogonal, Clifford, and spin groups over a global field
80(13)
Chapter III Quadratic Diophantine equations
93(46)
10 Quadratic Diophantine equations over a local field
93(8)
11 Quadratic Diophantine equations over a global field
101(12)
12 The class number of an orthogonal group and sums of squares
113(13)
13 Nonscalar quadratic Diophantine equations; Connection with the mass formula; A historical perspective
126(13)
Chapter IV Groups and symmetric spaces over R
139(24)
14 Clifford and spin groups over R; The case of signature (1, m)
139(7)
15 The case of signature (2, m)
146(8)
16 Orthogonal groups over R and symmetric spaces
154(9)
Chapter V Euler products and Eisenstein series on orthogonal groups
163(42)
17 Automorphic forms and Euler products on an orthogonal group
163(10)
18 Eisenstein series on Oω
173(8)
19 Eisenstein series Oη
181(6)
20 Arithmetic description of the pullback of an Eisenstein series
187(9)
21 Analytic continuation of Euler products and Eisenstein series
196(9)
Chapter VI Euler products and Eisenstein series on Clifford groups
205(38)
22 Euler products on G+(V)
205(7)
23 Eisenstein series on G(H, 2-1η)
212(6)
24 Eisenstein series of general types on a Clifford group
218(8)
25 Euler products for holomorphic forms on a Clifford group
226(8)
26 Proof of the last main theorem
234(9)
Appendix
243(29)
A1 Differential operators on a semisimple Lie group
243(7)
A2 Eigenvalues of integral operators
250(11)
A3 Structure of Clifford algebras over R
261(4)
A4 An embedding of G1(V) into a symplectic group
265(3)
A5 Spin representations and Lie algebras
268(4)
References 272(2)
Frequently used symbols 274(1)
Index 275
Goro Shimura, Princeton University, NJ, USA.
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