The Web Of Modularity Arithmetic Of The Coefficients Of Modular Forms And Q Series
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The Web Of Modularity Arithmetic Of The Coefficients Of Modular Forms And Q Series
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Author : Ken Ono
language : en
Publisher: American Mathematical Soc.
Release Date : 2004
The Web Of Modularity Arithmetic Of The Coefficients Of Modular Forms And Q Series written by Ken Ono 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 2004 with Forms, Modular categories.
Chapter 1.
A First Course In Modular Forms
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Author : Fred Diamond
language : en
Publisher: Springer Science & Business Media
Release Date : 2006-03-30
A First Course In Modular Forms written by Fred Diamond 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 2006-03-30 with Mathematics categories.
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
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Author : L J P Kilford
language : en
Publisher: World Scientific Publishing Company
Release Date : 2015-03-12
Modular Forms written by L J P Kilford and has been published by World Scientific Publishing Company this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-03-12 with Mathematics categories.
Modular Forms is 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 various subjects, such as the theory of quadratic forms, the proof of Fermat's Last Theorem and the approximation of π. The text gives a balanced overview of both the theoretical and computational sides of its subject, allowing a variety of courses to be taught from it. This second edition has been revised and updated. New material on the future of modular forms as well as a chapter about longer-form projects for students has also been added.
Modular Forms A Classical Approach
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Author : Henri Cohen
language : en
Publisher: American Mathematical Soc.
Release Date : 2017-08-02
Modular Forms A Classical Approach written by Henri Cohen 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 2017-08-02 with Forms (Mathematics). categories.
The theory of modular forms is a fundamental tool used in many areas of mathematics and physics. It is also a very concrete and “fun” subject in itself and abounds with an amazing number of surprising identities. This comprehensive textbook, which includes numerous exercises, aims to give a complete picture of the classical aspects of the subject, with an emphasis on explicit formulas. After a number of motivating examples such as elliptic functions and theta functions, the modular group, its subgroups, and general aspects of holomorphic and nonholomorphic modular forms are explained, with an emphasis on explicit examples. The heart of the book is the classical theory developed by Hecke and continued up to the Atkin–Lehner–Li theory of newforms and including the theory of Eisenstein series, Rankin–Selberg theory, and a more general theory of theta series including the Weil representation. The final chapter explores in some detail more general types of modular forms such as half-integral weight, Hilbert, Jacobi, Maass, and Siegel modular forms. Some “gems” of the book are an immediately implementable trace formula for Hecke operators, generalizations of Haberland's formulas for the computation of Petersson inner products, W. Li's little-known theorem on the diagonalization of the full space of modular forms, and explicit algorithms due to the second author for computing Maass forms. This book is essentially self-contained, the necessary tools such as gamma and Bessel functions, Bernoulli numbers, and so on being given in a separate chapter.
Modular Forms
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Author : L J P Kilford
language : en
Publisher: World Scientific
Release Date : 2008-08-11
Modular Forms written by L J P 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 2008-08-11 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. Contents: Historical OverviewIntroduction to Modular FormsResults on Finite-DimensionalityThe Arithmetic of Modular FormsApplications of Modular FormsModular Forms in Characteristic pComputing with Modular FormsAppendices:MAGMA Code for Classical Modular FormsSAGE Code for Classical Modular FormsHints and Answers to Selected Exercises Readership: Academics, researchers and graduate students in number theory and computational mathematics. Keywords:Modular Forms;Computations;Modular Functions;Cusp Forms;Ramanujan Tau FunctionKey Features:Covers the computational side together with the theoryIncludes a wide variety of exercises, from short to research-project lengthContains historical asides and references to modular forms in mathematical culture, to help ground the subject and motivate student interestReviews: "This fascinating, contemporaneous, and even now unfolding story of current research in a historically brilliant part of mathematics is told with riveting attention to detail ... Almost all aspects one could wish for in the area of holomorphic modular forms are covered, as well as some selected topics about meromorphic modular functions." The Mathematical Intelligencer "The second and (perhaps) more interesting computational aspect conveyed in this book is the consistent use of explicit computations by hand. For example expressing modular forms in a given space in terms of Eisenstein series, Eta or Delta functions to verify and prove various statements and theorems. This aspect is further encouraged throughout the exercises, which by the way are numerous, relevant and well-written. This kind of very explicit computations are sadly missing in the literature although implicitly stated or used in many places. It is obviously well-known to experts but most students would never be exposed to these ideas unless actually playing around to prove theorems by themselves." Zentrallblatt MATH
Modular Forms
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Author : Claudia Alfes-Neumann
language : en
Publisher: Springer Nature
Release Date : 2021-10-11
Modular Forms written by Claudia Alfes-Neumann and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-10-11 with Mathematics categories.
In this essential, Claudia Alfes-Neumann discusses applications of the theory of modular forms and their importance as fundamental tools in mathematics. These functions - initially defined purely analytically - appear in many areas of mathematics: very prominently in number theory, but also in geometry, combinatorics, representation theory, and physics. After explaining necessary basics from complex analysis, the author defines modular forms and shows some applications in number theory. Furthermore, she takes up two important aspects of the theory surrounding modular forms: Hecke operators and L-functions of modular forms. The essentials conclude with an outlook on real-analytic generalizations of modular forms, which play an important role in current research. This Springer essential is a translation of the original German 1st edition essentials, Modulformen by Claudia Alfes-Neumann, published by Springer Fachmedien Wiesbaden GmbH, part of Springer Nature in 2020. The translation was done with the help of artificial intelligence (machine translation by the service DeepL.com). A subsequent human revision was done primarily in terms of content, so that the book will read stylistically differently from a conventional translation. Springer Nature works continuously to further the development of tools for the production of books and on the related technologies to support the authors.
Harmonic Maass Forms And Mock Modular Forms Theory And Applications
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Author : Kathrin Bringmann
language : en
Publisher: American Mathematical Soc.
Release Date : 2017-12-15
Harmonic Maass Forms And Mock Modular Forms Theory And Applications written by Kathrin Bringmann 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 2017-12-15 with Forms (Mathematics) categories.
Modular forms and Jacobi forms play a central role in many areas of mathematics. Over the last 10–15 years, this theory has been extended to certain non-holomorphic functions, the so-called “harmonic Maass forms”. The first glimpses of this theory appeared in Ramanujan's enigmatic last letter to G. H. Hardy written from his deathbed. Ramanujan discovered functions he called “mock theta functions” which over eighty years later were recognized as pieces of harmonic Maass forms. This book contains the essential features of the theory of harmonic Maass forms and mock modular forms, together with a wide variety of applications to algebraic number theory, combinatorics, elliptic curves, mathematical physics, quantum modular forms, and representation theory.
L Functions And Automorphic Forms
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Author : Jan Hendrik Bruinier
language : en
Publisher: Springer
Release Date : 2018-02-22
L Functions And Automorphic Forms written by Jan Hendrik Bruinier and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-02-22 with Mathematics categories.
This book presents a collection of carefully refereed research articles and lecture notes stemming from the Conference "Automorphic Forms and L-Functions", held at the University of Heidelberg in 2016. The theory of automorphic forms and their associated L-functions is one of the central research areas in modern number theory, linking number theory, arithmetic geometry, representation theory, and complex analysis in many profound ways. The 19 papers cover a wide range of topics within the scope of the conference, including automorphic L-functions and their special values, p-adic modular forms, Eisenstein series, Borcherds products, automorphic periods, and many more.
Lectures On Modular Forms
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Author : Joseph Lehner
language : en
Publisher:
Release Date : 1969
Lectures On Modular Forms written by Joseph Lehner and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1969 with Finite fields (Algebra) categories.
Number Theory Arithmetic In Shangri La
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Author : Shigeru Kanemitsu
language : en
Publisher: World Scientific
Release Date : 2013-02-20
Number Theory Arithmetic In Shangri La written by Shigeru Kanemitsu and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-02-20 with Mathematics categories.
This volume is based on the successful 6th China–Japan Seminar on number theory that was held in Shanghai Jiao Tong University in August 2011. It is a compilation of survey papers as well as original works by distinguished researchers in their respective fields. The topics range from traditional analytic number theory — additive problems, divisor problems, Diophantine equations — to elliptic curves and automorphic L-functions. It contains new developments in number theory and the topics complement the existing two volumes from the previous seminars which can be found in the same book series. Contents:On Jacobi Forms with Levels (Hiroki Aoki)Additive Representation in Thin Sequences, VIII: Diophantine Inequalities in Review (Jörg Brüdern, Koichi Kawada and Trevor D Wooley)Annexe to the Gallery: An Addendum to “Additive Representation in Thin Sequences, VIII: Diophantine Inequalities in Review” (Jörg Brüdern, Koichi Kawada and Trevor D Wooley)A Note on the Distribution of Primes in Arithmetic Progressions (Zhen Cui and Boqing Xue)Matrices of Finite Abelian Groups, Finite Fourier Transform and Codes (Shigeru Kanemitsu and Michel Waldschmidt)A Remark on a Result of Eichler (Yoshiyuki Kitaoka)On Weyl Sums over Primes in Short Intervals (Angel V Kumchev)On Congruences for Certain Binomial Coefficients of E Lehmer's Type (Takako Kuzumaki and Jerzy Urbanowicz)Sign Changes of the Coefficients of Automorphic L-Functions (Yuk-Kam Lau, Jianya Liu and Jie Wu)On Fourier Coefficients of Automorphic Forms (Guangshi Lü)The Twists of Hessian Elliptic Curves over Splitting Fields of Cubic Polynomials and the Related Elliptic 3-Folds (Katsuya Miyake)Asymptotic Voronoi's Summation Formulas and Their Duality for SL3(ℤ) (Xiumin Ren and Yangbo Ye)Jerzy Urbanowicz's Work in Pure Mathematics (Andrzej Schinzel)Conjectures Involving Arithmetical Sequences (Zhi-Wei Sun) Readership: Graduate students and researchers in number theory. Keywords:Diophantine Equation;Hessian Elliptic Curves;Automorphic L-functions;Jacobi Forms;Weyl Sums;Fourier Coefficients;Result of Eichler;Distribution of Primes in Arithmetic Progression