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Modular Forms: Basics and Beyond

Author: Goro Shimura

Publisher: Springer Science & Business Media

Published: 2011-11-18

Total Pages: 183

ISBN-13: 146142125X

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 something new, which may satisfy the desire of such a reader. However, we state every definition and every essential fact concerning classical modular forms of one variable. One of the principal new features of this book is the theory of modular forms of half-integral weight, another being the discussion of theta functions and Eisenstein series of holomorphic and nonholomorphic types. Thus the book is presented so that the reader can learn such theories systematically.

Modular Forms

Author:

Publisher: Springer

Published: 2011-11-18

Total Pages: 188

ISBN-13: 9781461421269

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A First Course in Modular Forms

Author: Fred Diamond

Publisher: Springer Science & Business Media

Published: 2006-03-30

Total Pages: 450

ISBN-13: 0387272267

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This book introduces the theory of modular forms, from which all rational elliptic curves arise, with an eye toward the Modularity Theorem. Discussion covers elliptic curves as complex tori and as algebraic curves; modular curves as Riemann surfaces and as algebraic curves; Hecke operators and Atkin-Lehner theory; Hecke eigenforms and their arithmetic properties; the Jacobians of modular curves and the Abelian varieties associated to Hecke eigenforms. As it presents these ideas, the book states the Modularity Theorem in various forms, relating them to each other and touching on their applications to number theory. The authors assume no background in algebraic number theory and algebraic geometry. Exercises are included.

Modular Forms and Related Topics in Number Theory

Author: B. Ramakrishnan

Publisher: Springer Nature

Published: 2020-11-24

Total Pages: 240

ISBN-13: 9811587191

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This book collects the papers presented at the Conference on Number Theory, held at the Kerala School of Mathematics, Kozhikode, Kerala, India, from December 10–14, 2018. The conference aimed at bringing the active number theorists and researchers in automorphic forms and allied areas to demonstrate their current research works. This book benefits young research scholars, postdoctoral fellows, and young faculty members working in these areas of research.

Introduction to Siegel Modular Forms and Dirichlet Series

Author: Anatoli Andrianov

Publisher: Springer Science & Business Media

Published: 2008-12-22

Total Pages: 188

ISBN-13: 0387787526

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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.

Modular Forms, a Computational Approach

Author: William A. Stein

Publisher: American Mathematical Soc.

Published: 2007-02-13

Total Pages: 290

ISBN-13: 0821839608

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This marvellous and highly original book fills a significant gap in the extensive literature on classical modular forms. This is not just yet another introductory text to this theory, though it could certainly be used as such in conjunction with more traditional treatments. Its novelty lies in its computational emphasis throughout: Stein not only defines what modular forms are, but shows in illuminating detail how one can compute everything about them in practice. This is illustrated throughout the book with examples from his own (entirely free) software package SAGE, which really bring the subject to life while not detracting in any way from its theoretical beauty. The author is the leading expert in computations with modular forms, and what he says on this subject is all tried and tested and based on his extensive experience. As well as being an invaluable companion to those learning the theory in a more traditional way, this book will be a great help to those who wish to use modular forms in applications, such as in the explicit solution of Diophantine equations. There is also a useful Appendix by Gunnells on extensions to more general modular forms, which has enough in it to inspire many PhD theses for years to come. While the book's main readership will be graduate students in number theory, it will also be accessible to advanced undergraduates and useful to both specialists and non-specialists in number theory. --John E. Cremona, University of Nottingham William Stein is an associate professor of mathematics at the University of Washington at Seattle. He earned a PhD in mathematics from UC Berkeley and has held positions at Harvard University and UC San Diego. His current research interests lie in modular forms, elliptic curves, and computational mathematics.

Modular and Automorphic Forms & Beyond

Author: Hossein Movasati

Publisher: World Scientific Publishing Company

Published: 2021-10-12

Total Pages: 0

ISBN-13: 9789811238673

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The guiding principle in this monograph is to develop a new theory of modular forms which encompasses most of the available theory of modular forms in the literature, such as those for congruence groups, Siegel and Hilbert modular forms, many types of automorphic forms on Hermitian symmetric domains, Calabi-Yau modular forms, with its examples such as Yukawa couplings and topological string partition functions, and even go beyond all these cases. Its main ingredient is the so-called 'Gauss-Manin connection in disguise'.

Modular Forms with Integral and Half-Integral Weights

Author: Xueli Wang

Publisher: Springer Science & Business Media

Published: 2013-02-20

Total Pages: 436

ISBN-13: 3642293026

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"Modular Forms with Integral and Half-Integral Weights" focuses on the fundamental theory of modular forms of one variable with integral and half-integral weights. The main theme of the book is the theory of Eisenstein series. It is a fundamental problem to construct a basis of the orthogonal complement of the space of cusp forms; as is well known, this space is spanned by Eisenstein series for any weight greater than or equal to 2. The book proves that the conclusion holds true for weight 3/2 by explicitly constructing a basis of the orthogonal complement of the space of cusp forms. The problem for weight 1/2, which was solved by Serre and Stark, will also be discussed in this book. The book provides readers not only basic knowledge on this topic but also a general survey of modern investigation methods of modular forms with integral and half-integral weights. It will be of significant interest to researchers and practitioners in modular forms of mathematics. Dr. Xueli Wang is a Professor at South China Normal University, China. Dingyi Pei is a Professor at Guangzhou University, China.

Introduction to Modular Forms

Author: Serge Lang

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 267

ISBN-13: 3642514472

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From the reviews: "This book gives a thorough introduction to several theories that are fundamental to research on modular forms. Most of the material, despite its importance, had previously been unavailable in textbook form. Complete and readable proofs are given... In conclusion, this book is a welcome addition to the literature for the growing number of students and mathematicians in other fields who want to understand the recent developments in the theory of modular forms." #Mathematical Reviews# "This book will certainly be indispensable to all those wishing to get an up-to-date initiation to the theory of modular forms." #Publicationes Mathematicae#

Siegel Modular Forms

Author: Ameya Pitale

Publisher: Springer

Published: 2019-05-07

Total Pages: 138

ISBN-13: 3030156753

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This monograph introduces two approaches to studying Siegel modular forms: the classical approach as holomorphic functions on the Siegel upper half space, and the approach via representation theory on the symplectic group. By illustrating the interconnections shared by the two, this book fills an important gap in the existing literature on modular forms. It begins by establishing the basics of the classical theory of Siegel modular forms, and then details more advanced topics. After this, much of the basic local representation theory is presented. Exercises are featured heavily throughout the volume, the solutions of which are helpfully provided in an appendix. Other topics considered include Hecke theory, Fourier coefficients, cuspidal automorphic representations, Bessel models, and integral representation. Graduate students and young researchers will find this volume particularly useful. It will also appeal to researchers in the area as a reference volume. Some knowledge of GL(2) theory is recommended, but there are a number of appendices included if the reader is not already familiar.