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 hopi...
This volume aims to present a straightforward and easily accessible survey of the analytic theory of quadratic forms. Written at an elementary level, the book provides a sound basis from which the reader can study advanced works and undertake original research. Roughly half a century ago C.L. Siegel discovered a new type of automorphic forms in several variables in connection with his famous work on the analytic theory of quadratic forms. Since then Siegel modular forms have been studied extensively because of their significance in both automorphic functions in several complex variables and number theory. The comprehensive theory of automorphic forms to subgroups of algebraic groups and the recent arithmetical theory of modular forms illustrate these two aspects in an illuminating manner. The text is based on the author's lectures given over a number of years and is intended for a one semester graduate course, although it can serve equally well for self study . The only prerequisites are a knowledge of algebra, number theory and complex analysis.
This book, now in its 2nd edition, is devoted to the arithmetical theory of Siegel modular forms and their L-functions. The central object are L-functions of classical Siegel modular forms whose special values are studied using the Rankin-Selberg method and the action of certain differential operators on modular forms which have nice arithmetical properties. A new method of p-adic interpolation of these critical values is presented. An important class of p-adic L-functions treated in the present book are p-adic L-functions of Siegel modular forms having logarithmic growth. The given construction of these p-adic L-functions uses precise algebraic properties of the arithmetical Shimura differential operator. The book will be very useful for postgraduate students and for non-experts looking for a quick approach to a rapidly developing domain of algebraic number theory. This new edition is substantially revised to account for the new explanations that have emerged in the past 10 years of the main formulas for special L-values in terms of arithmetical theory of nearly holomorphic modular forms.
Published for the Tata Institute of Fundamental Research
The Siegel moduli scheme classifies principally polarised abelian varieties and its compactification is an important result in arithmetic algebraic geometry. The main result of this monograph is to prove the existence of the toroidal compactification over Z (1/2). This result should have further applications and is presented here with sufficient background material to make the book suitable for seminar courses in algebraic geometry, algebraic number theory or automorphic forms.
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