內容大鋼
在本書中，著名數學家、Steele獎得主志村五郎以清晰易讀的風格，介紹了一個全新的數學領域。書中主題包括Witt定理和二次型上的Hasse 原理、Clifford代數的代數理論、自旋群和自旋表示。作者還給出了一些在其他地方不容易找到的基本結果。
本書的兩個重要主題是：（1）二次Diophantus方程，（2）正交群和Clifford群上的Euler積和Eisenstein級數。第一個主題的起點是Gauss的結果：一個整數作為三個平方和的本原表示的個數本質上是本原二元二次型的類數。本書給出了這一結果在代數數域中任意二次型上的推廣及其各種應用。對於第二個主題，作者證明了與Clifford群或正交群上的Hecke本征形式相關聯的Euler積存在亞純連續性。對於這樣的群上的Eisenstein級數，結論也是如此。
本書基本上是自封的，只需要讀者熟悉代數數論的相關知識。對於一些標準的事實，作者在敘述時給出了附有詳細證明的參考文獻。
目錄
Preface
Notation and Terminology
Introduction
Chapter I.Algebraic theory of quadratic forms, Clifford algebras, and spin groups
1.Quadratic forms and associative algebras
2.Clifford algebras
3.Clifford groups and spin groups
4.Parabolic subgroups
Chapter II.Quadratic forms, Clifford algebras, and spin groups over a local or global field
5.Orders and ideals in an algebra
6.Quadratic forms over a local field
7.Lower-dimensional cases and the Hasse principle
8.Part I.Clifford groups over a local field
8.Part II.Formal Hecke algebras and formal Euler factors
9.Orthogonal, Clifford, and spin groups over a global field
Chapter III.Quadratic Diophantine equations
10.Quadratic Diophantine equations over a local field
11.Quadratic Diophantine equations over a global field
12.The class number of an orthogonal group and sums of squares
13.Nonscalar quadratic Diophantine equations; Connection with the mass formula; A historical perspective
Chapter IV.Groups and symmetric spaces over R
14.Clifford and spin groups over R; The case of signature (1, m)
15.The case of signature (2, m)
16.Orthogonal groups over R and symmetric spaces
Chapter V.Euler products and Eisenstein series on or-thogonal groups
17.Automorphic forms and Euler products on an orthogonal group
18.Eisenstein series on Oω
19.Eisenstein series on Oη
20.Arithmetic description of the pullback of an Eisenstein series
21.Analytic continuation of Euler products and Eisenstein series
Chapter VI.Euler products and Eisenstein series on Clifford groups
22.Euler products on G+(V)
23.Eisenstein series on G(H, 2–1η)
24.Eisenstein series of general types on a Clifford group
25.Euler products for holomorphic forms on a Clifford group
26.Proof of the last main theorem
Appendix
A1.Differential operators on a semisimple Lie group
A2.Eigenvalues of integral operators
A3.Structure of Clifford algebras over R
A4.An embedding of G1(V) into a symplectic group
A5.Spin representations and Lie algebras
References
Frequently used symbols
Index
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