This is a fundamental book on modular forms, abelian varieties, etc. by one of the masters. Despite the remark in MathSciNet below that it is "written in a clear style", I find it hard to read.
You can check availability in the Harvard library.
As the author notes in the Preface, there are two major topics treated
in this volume: complex multiplication of elliptic or elliptic modular
functions and applications of the theory of Hecke operators to the zetafunctions
of algebraic curves and abelian varieties. Chapters 46 are mainly devoted to the first topic mentioned above. In Chapter 4, "Elliptic curves", the author reviews some facts about elliptic curves: general facts over an arbitrary field, curves over $ C$, points of finite order, isogenies and endomorphisms, properties of the modular invariant. Chapter 5, "Abelian extensions of imaginary quadratic fields and complex multiplication of elliptic curves", is devoted to the study of the behavior of an elliptic curve $E$ with complex multiplications under $\text{Gal}(K_{ab}/K)$, where $K$ is an imaginary quadratic field isomorphic to $\text{End}_{ Q}(E)$ and $K_{ab}$ is the maximal abelian extension of $K$. The main result (Theorem 5.4) is stated in the adelic language. As a consequence the author derives the classical result on the construction of $K_{ab}$ by means of special values of elliptic or elliptic modular functions. In Chapter 6, "Modular functions of higher level", the author gives another formulation of the reciprocitylaw which is closely connected with the structure of the field $\scr F$ of all modular functions of all levels whose Fourier coefficients belong to a cyclotomic field. It is shown that the group of all automorphisms of $\scr F$ is isomorphic to the adelization of $\text{GL}_2( Q)$ modulo rational scalar matrices and the archimedean part. Then the reciprocitylaw in the maximal abelian extension of an imaginary quadratic field is given as a certain commutativity of the action of the adeles with the specialization of the functions of $\scr F$. It gives a new point of view on this classical subject. In Chapter 7, "Zetafunctions of algebraic curves and abelian varieties", the author treats the second topic mentioned above. The conjecture of Hasse and Weil is verified for the algebraic curves uniformized by modular functions. Further, it is shown that if a cusp form of weight 2 is a common eigenfunction of the Hecke operators, then the product of several "conjugated" Dirichlet series associated with it coincides up to finitely many Euler factors with the zetafunction of a certain abelian variety which is specifically given. As an application of this result it is shown that some arithmetic of a real quadratic fieldits units, abelian extensions,...may be obtained from the modular forms of "Nebentype" in Hecke's sense ($§$ 7.7). This very interesting topic has received a wide development in the deep investigations of the author, Doi, Naganuma and Miyake. In Chapter 8, "The cohomology group associated with cusp forms", the author gives an account of his wellknown results on the cohomology group associated with cusp forms, the action of Hecke operators on the cohomology group and the complex tori associated with the spaces of cusp forms. In Chapter 9, "Arithmetic Fuchsian groups", the author shows that most of the above results can be generalized to arithmetic Fuchsian groups obtained from quaternion algebras. The appendix contains some elementary facts on algebraic varieties, especially on algebraic curves and abelian varieties. The book contains many new points of view and deep results. It is written in a clear style and may also be used as a course textbook. \{See also the articles reviewed below [\#\#3350, 3351].\} Reviewed by A. N. Andrianov
