Physical description 
xiii, 271 pages ; 24 cm. 
Series 
Publications of the Mathematical Society of Japan ; 11. Kanō memorial lectures ; 1. 

Publications of the Mathematical Society of Japan ; 11.


Publications of the Mathematical Society of Japan. Kanō memorial lectures ; 1.

Notes 
Originally published: Tokyo : Iwanami Shoten ; Princeton, N.J. : Princeton University Press, 1971. 
Bibliography 
Includes bibliographical references (pages [262]264) and index. 
Contents 
Ch. 1. Fuchsian groups of the first kind  Ch. 2. Automorphic forms and functions  Ch. 3. Hecke operators and the zetafunctions associated with modular forms  Ch. 4. Elliptic curves  Ch. 5. Abelian extensions of imaginary quadratic fields and complex multiplication of elliptic curves  Ch. 6. Modular functions of higher level  Ch. 7. Zetafunctions of algebraic curves and abelian varieties  Ch. 8. The cohomology group associated with cusp forms  Ch. 9. Arithmetic Fuchsian groups. 
Summary 
The theory of automorphic forms is playing increasingly important roles in several branches of mathematics, even in physics, and is almost ubiquitous in number theory. This book introduces the reader to the subject and in particular to elliptic modular forms with emphasis on their numbertheoretical aspects. After two chapters geared toward elementary levels, there follows a detailed treatment of the theory of Hecke operators, which associate zeta functions to modular forms. At a more advanced level, complex multiplication of elliptic curves and abelian varieties is discussed. The main question is the construction of abelian extensions of certain algebraic number fields, which is traditionally called "Hilbert's twelfth problem." Another advanced topic is the determination of the zeta function of an algebraic curve uniformized by modular functions, which supplies an indispensable background for the recent proof of Fermat's last theorem by Wiles. 
Subject 
Automorphic functions.

ISBN 
0691080925 (paperback: acidfree) 
