Introduction to the Arithmetic Theory of Automorphic Functions (Publications of the Mathematical Society of Japan, Vol. 11) - Softcover

Shimura, Goro

Softcover

ISBN 10: 0691080925 ISBN 13: 9780691080925

Publisher: Princeton University Press, 1971

Synopsis

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 number-theoretical 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.

About this title

About the Author

Goro Shimura is Professor of Mathematics at Princeton University.

From the Back Cover

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 number-theoretical 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.

Edition details

• Publisher: Princeton University Press
• Publication date: 1971
• Language: English
• ISBN 10: 0691080925
• ISBN 13: 9780691080925
• Binding: Paperback
• Edition number: 1
• Number of pages: 288