Kuga varieties are fiber varieties over symmetric spaces whose fibers are abelian varieties and have played an important role in the theory of Shimura varieties and number theory. This book is the first systematic exposition of these varieties and was written by their creators. It contains four chapters. Chapter 1 gives a detailed generalization to vector valued harmonic forms. These results are applied to construct Kuga varieties in Chapter 2 and to understand their cohomology groups. Chapter 3 studies Hecke operators which are the most basic operators in modular forms. All the previous results are applied in Chapter 4 to prove the modularity property of certain Kuga varieties. Note that the modularity property of elliptic curves is the key ingredient of Wiles’ proof of the Fermat’s Last Theorem. This book also contains a letter of Weil and a paper of Satake which fit well the topic of the book. Kuga簇是对称空间上的纤维簇，它的纤维是Abel簇。在Shimura簇和数论中，Kuga簇发挥了重要的作用。本书首次系统地阐述了这些簇。 本书共4章。第1章给出了到向量调和形式的详细推广。这些结果应用于第2章构造Kuga簇并理解其上的同调群。第3章研究模形式中最基本的Hecke算子。所有以前的结果应用在第4章证明Kuga簇的模块化性质。注意椭圆曲线的模块化性质是Wiles关于费马大定理证明的关键。 本书还包含了Weil的一封信和Satake的一篇文章，很切合这本书的主题。

前辅文
Volume I
  Chapter I Vector bundle valued harmonic forms
   1 An analogy of de Rham’s theorem
   2 Harmonic ρ-forms
   3 The type decomposition of harmonic ρ-forms
   4 Mountjoy’s abelian varieties
   5 Commutativity with ￠A
   6 Proof of commutativity theorems
   7 A wider frame: spherical functions
   8 An example: G = SL(2,R)
   9 Other examples, and discussions
  Chapter II Fibre variety over a symmetric space whose fibres are abelian varieties
   1 A fibre bundle V π ?→U
   2 Cohomology groups of V (Part I)
   3 Cohomology groups of V (Part II)
   4 Up-side-down operator θ, and the θ-invariant subspaces of H2(V )
   5 Fibre variety over a symmetric space whose fibres are abelian varieties
   6 Algebraic family of polarized abelian varieties
   7 Minimality of quotient varieties
  Appendix I A letter of AndréWeil
  Appendix II Holomorphic imbeddings of symmetric domains into a Siegel space
  References for volume I
Volume II
  Chapter III Hecke operators
   1 Goldman adelilzation
   2 Hecke operator operating on Hp (X,Γ,ρ) etc
   3 Hecke operator operating on -(X ×F)
   4 Hecke operators as algebraic correspondences
  Chapter IV Number theory of automorphic forms
   1 A fibre variety over an algebraic curveU = ΓX
   2 Harmonic forms on V , and the trace formulas
   3 Zeta-function of e V p
   4 Congruence Artin - L-functions
   5 Hecke polynomials as L-functions
  References for volume II