Modular Forms And Hecke Operators

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Modular Forms And Hecke Operators

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Author : A. N. Andrianov
language : en
Publisher: American Mathematical Soc.
Release Date : 2016-01-29

Modular Forms And Hecke Operators written by A. N. Andrianov and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-01-29 with categories.

he concept of Hecke operators was so simple and natural that, soon after Hecke's work, scholars made the attempt to develop a Hecke theory for modular forms, such as Siegel modular forms. As this theory developed, the Hecke operators on spaces of modular forms in several variables were found to have arithmetic meaning. Specifically, the theory provided a framework for discovering certain multiplicative properties of the number of integer representations of quadratic forms by quadratic forms. Now that the theory has matured, the time is right for this detailed and systematic exposition of its fundamental methods and results. Features: The book starts with the basics and ends with the latest results, explaining the current status of the theory of Hecke operators on spaces of holomorphic modular forms of integer and half-integer weight congruence-subgroups of integral symplectic groups.Hecke operators are considered principally as an instrument for studying the multiplicative properties of the Fourier coefficients of modular forms. It is the authors' intent that Modular Forms and Hecke Operators help attract young researchers to this beautiful and mysterious realm of number theory.

Modular Forms

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Author : Lloyd James Peter Kilford
language : en
Publisher: World Scientific
Release Date : 2008

Modular Forms written by Lloyd James Peter Kilford and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008 with Mathematics categories.

This book presents a graduate student-level introduction to the classical theory of modular forms and computations involving modular forms, including modular functions and the theory of Hecke operators. It also includes applications of modular forms to such diverse subjects as the theory of quadratic forms, the proof of Fermat's last theorem and the approximation of pi. It provides a balanced overview of both the theoretical and computational sides of the subject, allowing a variety of courses to be taught from it.

Modular Forms And Hecke Operators

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Author : Anatoliĭ Nikolaevich Andrianov
language : en
Publisher:
Release Date : 1995

Modular Forms And Hecke Operators written by Anatoliĭ Nikolaevich Andrianov and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1995 with Forms, Modular categories.

The concept of Hecke operators was so simple and natural that, soon after Hecke's work, scholars made the attempt to develop a Hecke theory for modular forms, such as Siegel modular forms. As this theory developed, the Hecke operators on spaces of modular forms in several variables were found to have arithmetic meaning. Specifically, the theory provided a framework for discovering certain multiplicative properties of the number of integer representations of quadratic forms by quadratic forms. Now that the theory has matured, the time is right for this detailed and systematic exposition of its.

Modular Forms And Hecke Operators

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Author : Robert Alexander Rankin
language : en
Publisher:
Release Date : 1999*

Modular Forms And Hecke Operators written by Robert Alexander Rankin and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1999* with Hecke operators categories.

Modular Forms And Hecke Operators

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Author : A. N. Andrianov V. G. Zhuravlev
language : en
Publisher: American Mathematical Soc.
Release Date : 1995-08-28

Modular Forms And Hecke Operators written by A. N. Andrianov V. G. Zhuravlev and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 1995-08-28 with Mathematics categories.

The concept of Hecke operators was so simple and natural that, soon after Hecke's work, scholars made the attempt to develop a Hecke theory for modular forms, such as Siegel modular forms. As this theory developed, the Hecke operators on spaces of modular forms in several variables were found to have arithmetic meaning. Specifically, the theory provided a framework for discovering certain multiplicative properties of the number of integer representations of quadratic forms by quadratic forms. Now that the theory has matured, the time is right for this detailed and systematic exposition of its fundamental methods and results. Features: The book starts with the basics and ends with the latest results, explaining the current status of the theory of Hecke operators on spaces of holomorphic modular forms of integer and half-integer weight congruence-subgroups of integral symplectic groups. Hecke operators are considered principally as an instrument for studying the multiplicative properties of the Fourier coefficients of modular forms. It is the authors' intent that Modular Forms and Hecke Operators help attract young researchers to this beautiful and mysterious realm of number theory.

Modular Forms

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Author : Toshitsune Miyake
language : en
Publisher: Springer Science & Business Media
Release Date : 2006-02-17

Modular Forms written by Toshitsune Miyake and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-02-17 with Mathematics categories.

For the most part, this book is the translation from Japanese of the earlier book written jointly by Koji Doi and the author who revised it substantially for the English edition. It sets out to provide the reader with the basic knowledge of elliptic modular forms necessary to understand the recent developments in number theory. The first part gives the general theory of modular groups, modular forms and Hecke operators, with emphasis on the Hecke-Weil theory of the relation between modular forms and Dirichlet series. The second part is on the unit groups of quaternion algebras, which are seldom dealt with in books. The so-called Eichler-Selberg trace formula of Hecke operators follows next and the explicit computable formula is given. In the last chapter, written for the English edition, Eisenstein series with parameter are discussed following the recent work of Shimura: Eisenstein series are likely to play a very important role in the future progress of number theory, and thischapter provides a good introduction to the topic.

Quadratic Forms And Hecke Operators

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Author : Anatolij N. Andrianov
language : en
Publisher: Springer
Release Date : 1987

Quadratic Forms And Hecke Operators written by Anatolij N. Andrianov and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 1987 with Mathematics categories.

The numerous explicit formulae of the classical theory of quadratic forms revealed remarkable multiplicative properties of the numbers of integral representations of integers by positive definite integral quadratic forms. These properties were explained by the original theory of Hecke operators. As regards the integral representations of quadratic forms in more than one variable by quadratic forms, no multiplicative properties were known at that time, and so there was nothing to explain. However, the idea of Hecke operators was so natural and attractive that soon attempts were made to cultivate it in the neighbouring field of modular forms of several variables. The approach has proved to be fruitful; in particular, a number of multiplicative properties of integral representations of quadratic forms by quadratic forms were eventually discovered. By now the theory has reached a certain maturity, and the time has come to give an up-to-date report in a concise form, in order to provide a solid ground for further development. The purpose of this book is to present in the form of a self-contained text-book the contemporary state of the theory of Hecke operators on the spaces of hoi om orphic modular forms of integral weight (the Siegel modular forms) for congruence subgroups of integral symplectic groups. The book can also be used for an initial study of modular forms of one or several variables and theta-series of positive definite integral quadratic forms.

Problems In The Theory Of Modular Forms

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Author : M. Ram Murty
language : en
Publisher: Springer
Release Date : 2016-11-25

Problems In The Theory Of Modular Forms written by M. Ram Murty and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-11-25 with Mathematics categories.

This book introduces the reader to the fascinating world of modular forms through a problem-solving approach. As such, besides researchers, the book can be used by the undergraduate and graduate students for self-instruction. The topics covered include q-series, the modular group, the upper half-plane, modular forms of level one and higher level, the Ramanujan τ-function, the Petersson inner product, Hecke operators, Dirichlet series attached to modular forms and further special topics. It can be viewed as a gentle introduction for a deeper study of the subject. Thus, it is ideal for non-experts seeking an entry into the field.

Quadratic Forms And Hecke Operators

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Author : Anatoliĭ Nikolaevich Andrianov
language : en
Publisher: Springer Verlag
Release Date : 1987

Quadratic Forms And Hecke Operators written by Anatoliĭ Nikolaevich Andrianov and has been published by Springer Verlag this book supported file pdf, txt, epub, kindle and other format this book has been release on 1987 with Mathematics categories.

Hecke Operators On Modular Forms And Integrals

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Author : Reine Younis Al-Housseini
language : en
Publisher:
Release Date : 2009

Hecke Operators On Modular Forms And Integrals written by Reine Younis Al-Housseini and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009 with categories.

We define modular forms of nonnegative weight k on the full modular group Gamma and give some of their properties. We proceed to show the behavior of Hecke ope rators on modular forms. Then we discuss the multiplicative property of Hecke op erators. We notice that differentiating a modular form of negative weight -k, k + 1 times turns out to be a modular form of weight k + 2. However, integrating a modular form of weight k + 2 does not give a modular form but it gives what we call an Eichler integral. So we study Eichler integrals of negative weight and s ee the action of Hecke operators on them. Finally we introduce entire modular in tegrals and study the behavior of Hecke operators when they act on them.