在本书中，著名数学家、Steele 奖得主志村五郎以清晰易读的风格，介绍了一个全新的数学领域。书中主题包括 Witt 定理和二次型上的 Hasse 原理、Clifford 代数的代数理论、自旋群和自旋表示。作者还给出了一些在其他地方不容易找到的基本结果。 本书的两个重要主题是：(1) 二次 Diophantus 方程，(2) 正交群和 Clifford 群上的 Euler 积和 Eisenstein 级数。第一个主题的起点是 Gauss 的结果：一个整数作为三个平方和的本原表示的个数本质上是本原二元二次型的类数。本书给出了这一结果在代数数域中任意二次型上的推广及其各种应用。对于第二个主题，作者证明了与 Clifford 群或正交群上的 Hecke 本征形式相关联的 Euler 积存在亚纯连续性。对于这样的群上的 Eisenstein 级数，结论也是如此。 本书基本上是自封的，只需要读者熟悉代数数论的相关知识。对于一些标准的事实，作者在叙述时给出了附有详细证明的参考文献。

前辅文
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 orthogonal 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