Introduction to Siegel Modular Forms and Dirichlet Series

This is intended as a graduate course on Siegel modular forms, Hecke operators, and related zeta functions. The aim is to attract researchers to this beautiful field by presenting a concise and self-contained introduction to a developing area of number theory.

Several years ago I was invited to an American university to give one-term graduate course on Siegel modular forms, Hecke operators, and related zeta functions. The idea to present in a concise but basically complete and self-contained form an int- duction to an important and developing area based partly on my own work attracted me. I accepted the invitation and started to prepare the course. Unfortunately, the visit was not realized. But the idea of such a course continued to be alive till after a number of years this book was ?nally completed. I hope that this short book will serve to attract young researchers to this beautiful ?eld, and that it will simplify and make more pleasant the initial steps. No special knowledge is presupposed for reading this book beyond standard courses in algebra and calculus (one and several variables), although some skill in working with mathematical texts would be helpful. The reader will judge whether the result was worth the effort. Dedications. The ideas of Goro Shimura exerted a deep in?uence on the number theory of the second half of the twentieth century in general and on the author's formation in particular. When Andre ` Weil was signing a copy of his Basic Number Theory to my son, he wrote in Russian, To Fedor Anatolievich hoping that he will become a number theoretist. Fedor has chosen computer science. Now I pass on the idea to Fedor's daughter, Alexandra Fedorovna.

Will become a standard reference on the subject Intended for graduate students and keeps prerequisites to a minimum Gives provocative examples in the simplest and most accessible terms The author is an expert who has originated many important ideas in the subject

Texte du rabat

Introduction to Siegel Modular Forms and Dirichlet Series gives a concise and self-contained introduction to the multiplicative theory of Siegel modular forms, Hecke operators, and zeta functions, including the classical case of modular forms in one variable. It serves to attract young researchers to this beautiful field and makes the initial steps more pleasant. It treats a number of questions that are rarely mentioned in other books. It is the first and only book so far on Siegel modular forms that introduces such important topics as analytic continuation and the functional equation of spinor zeta functions of Siegel modular forms of genus two.

Unique features include:

• New, simplified approaches and a fresh outlook on classical problems

• The abstract theory of HeckeâShimura rings for symplectic and related groups

• The action of Hecke operators on Siegel modular forms

• Applications of Hecke operators to a study of the multiplicative properties of Fourier coefficients of modular forms

• The proof of analytic continuation and the functional equation (under certain assumptions) for Euler products associated with modular forms of genus two

*Numerous exercises

Anatoli Andrianov is a leading researcher at the St. Petersburg branch of the Steklov Mathematical Institute of the Russian Academy of Sciences. He is well known for his works on the arithmetic theory of automorphic functions and quadratic forms, a topic on which he has lectured at many universities around the world.

Résumé

From the reviews:

"Introduction to Siegel Modular Forms and Dirichlet Series is a compact but masterful presentation of this important generalization of the classical theory, and a good deal more. ... beneficiaries of this wonderful book are the obvious candidates: students of number theory with their qualifying examinations behind them, or very gifted undergraduates who have already learned group theory and complex analysis, some topology, some rings and fields, and so on." (Michael Berg, The Mathematical Association of America, May, 2009) "Andrianov ... offers a comparatively concrete, lowbrow treatment of Siegel modular forms, a large and important but still special class of higher-dimensional modular forms. Readers familiar with standard accounts of the one-dimensional story will find the flow and organization here comfortingly familiar ... . Andrianov has suppressed entirely the vantage of algebraic geometry, presumably for simplicity. ... Summing Up: Highly recommended. Upper-division undergraduate through professional collections." (D. V. Feldman, Choice, Vol. 47 (4), December, 2009) "The book under review is a concise and self-contained introduction to the Hecke theory of Siegel modular forms and zeta functions and is suitable for beginners. ... Siegel modular form is introduced and its analytic properties are investigated." (Hidenori Katsurada, Mathematical Reviews, Issue 2010 f)

Contenu
Modular Forms.- Dirichlet Series of Modular Forms.- HeckeShimura Rings of Double Cosets.- Hecke Operators.- Euler Factorization of Radial Series.