Introduction to the arithmetic theory of automorphic functions.
Kanô Memorial Lectures, No. 1.
Publications of the Mathematical Society of Japan, No. 11. Iwanami Shoten, Publishers, Tokyo; Princeton University Press, Princeton, N.J., 1971. xiv+267 pp. \$10.00.
10D10 (12-02 14K22)

References: 0 Reference Citations: 99 Review Citations: 30
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 zeta-functions of algebraic curves and abelian varieties.

The first three chapters form an introductory account of the theory of automorphic functions of one complex variable, along with fundamentals of Hecke operators. In Chapter 1, "Fuchsian groups of the first kind", the author treats properties of a group of linear fractional transformations acting on the upper half plane: the classification of the transformations, and topological and Riemann structures on the quotient spaces. He illustrates the general theory by studying the modular group $\text{SL}_2( Z)$ and the congruence subgroups of it. Chapter 2, "Automorphic forms and functions", contains the foundations of the theory of automorphic forms and functions for Fuchsian groups of the first kind, including the evaluation of the dimension of the spaces of cusp forms by means of the Riemann-Roch theorem. In Chapter 3, "Hecke operators and the zeta-functions associated with modular forms", the author discusses the theory of Hecke operators in a modern framework: the Hecke rings, their structure for $\text{SL}_n( Z)$ and for the congruence subgroups, the evaluation of the corresponding formal Dirichlet series, representations of the Hecke rings on the spaces of automorphic forms for $n=2$ and functional equation of the associated zeta-functions.

Chapters 4--6 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 reciprocity-law 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 reciprocity-law 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, "Zeta-functions 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 zeta-function 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 field---its units, abelian extensions,...---may be obtained from the modular forms of "Neben-type" 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 well-known 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