Modular Forms: Basics and Beyond
This is an advanced book on modular forms. While there are many books published about modular forms, they are written at an elementary level, and not so interesting from the viewpoint of a reader who already knows the basics. This book offers somethi
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Goro Shimura
Modular Forms: Basics and Beyond
Goro Shimura Department of Mathematics Princeton University Princeton, NJ 08544-1000 USA [email protected]
ISSN 1439-7382 ISBN 978-1-4614-2124-5 e-ISBN 978-1-4614-2125-2 DOI 10.1007/978-1-4614-2125-2 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011941793 Mathematics Subject Classification (2010): 11F11, 11F27, 11F37, 11F67, 11M36, 11M41, 14K25
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PREFACE
It was forty years ago that my “Introduction to the arithmetic theory of automorphic functions” appeared. At present the terminology “modular form” can be counted among those most frequently heard in the conversations of mathematicians, and indeed, there are many textbooks on this topic. However, almost all of them are at the elementary level, and not so interesting from the viewpoint of the reader who already knows the basics. So, my intention in the present book is to offer something new that may satisfy the desire of such a reader. Therefore we naturally assume that the reader has at least rudimentary knowledge of modular forms of integral weight with respect to congruence subgroups of SL2 (Z), though we state every definition and some basic theorems on such forms. One of the principal new features of this book is the theory of modular forms of half-integral weight, another the discussion of theta functions and Eisenstein series of holomorphic and nonholomorphic types. Thus we have written the book so that the reader can learn such theories systematically. However, we present them with the following two themes as the ultimate aims: (I) The correspondence between the forms of half-integral weight and those of integral weight. (II) The arithmeticity of various Dirichlet series associated with modular forms of integral or half-integral weight. The correspondence of (I) associates a cusp form of weight k with a modular form of weight 2k − 1, where k is half an odd positive integer. I gave such a correspondence in my papers in 1973. In the present book I prove a stronger, perhaps the best possible, result with different methods. As for (II), a typical example is a
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