Modular Forms A Classical Approach

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Modular Forms: A Classical Approach

Henri Cohen,Fredrik Strömberg
Modular Forms A Classical Approach Book PDF

Author : Henri Cohen,Fredrik Strömberg
Publisher : American Mathematical Soc.
Page : 700 pages
File Size : 53,7 Mb
Release : 2017-08-02
Category : Forms (Mathematics).
ISBN : 9780821849477

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Modular Forms: A Classical Approach by Henri Cohen,Fredrik Strömberg Pdf

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

Henri Cohen,Fredrik Strömberg
Modular Forms Book PDF

Author : Henri Cohen,Fredrik Strömberg
Publisher : Unknown
Page : 700 pages
File Size : 40,5 Mb
Release : 2017
Category : MATHEMATICS
ISBN : 1470440814

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Modular Forms by Henri Cohen,Fredrik Strömberg Pdf

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 a.

Modular Forms, a Computational Approach

William A. Stein
Modular Forms a Computational Approach Book PDF

Author : William A. Stein
Publisher : American Mathematical Soc.
Page : 290 pages
File Size : 53,6 Mb
Release : 2007-02-13
Category : Mathematics
ISBN : 9780821839607

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Modular Forms, a Computational Approach by William A. Stein Pdf

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.

Siegel Modular Forms

Ameya Pitale
Siegel Modular Forms Book PDF

Author : Ameya Pitale
Publisher : Springer
Page : 138 pages
File Size : 49,9 Mb
Release : 2019-05-07
Category : Mathematics
ISBN : 9783030156756

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Siegel Modular Forms by Ameya Pitale Pdf

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.

A First Course in Modular Forms

Fred Diamond,Jerry Shurman
A First Course in Modular Forms Book PDF

Author : Fred Diamond,Jerry Shurman
Publisher : Springer Science & Business Media
Page : 450 pages
File Size : 46,6 Mb
Release : 2006-03-30
Category : Mathematics
ISBN : 9780387272269

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A First Course in Modular Forms by Fred Diamond,Jerry Shurman Pdf

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.

Topics in Classical Automorphic Forms

Henryk Iwaniec
Topics in Classical Automorphic Forms Book PDF

Author : Henryk Iwaniec
Publisher : American Mathematical Soc.
Page : 274 pages
File Size : 48,7 Mb
Release : 1997
Category : Automorphic forms
ISBN : 9780821807774

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Topics in Classical Automorphic Forms by Henryk Iwaniec Pdf

This volume discusses various perspectives of the theory of automorphic forms drawn from the author's notes from a Rutgers University graduate course. In addition to detailed and often nonstandard treatment of familiar theoretical topics, the author also gives special attention to such subjects as theta- functions and representatives by quadratic forms. Annotation copyrighted by Book News, Inc., Portland, OR

The 1-2-3 of Modular Forms

Jan Hendrik Bruinier,Gerard van der Geer,Günter Harder,Don Zagier
The 1 2 3 of Modular Forms Book PDF

Author : Jan Hendrik Bruinier,Gerard van der Geer,Günter Harder,Don Zagier
Publisher : Springer Science & Business Media
Page : 266 pages
File Size : 54,5 Mb
Release : 2008-02-10
Category : Mathematics
ISBN : 9783540741190

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The 1-2-3 of Modular Forms by Jan Hendrik Bruinier,Gerard van der Geer,Günter Harder,Don Zagier Pdf

This book grew out of three series of lectures given at the summer school on "Modular Forms and their Applications" at the Sophus Lie Conference Center in Nordfjordeid in June 2004. The first series treats the classical one-variable theory of elliptic modular forms. The second series presents the theory of Hilbert modular forms in two variables and Hilbert modular surfaces. The third series gives an introduction to Siegel modular forms and discusses a conjecture by Harder. It also contains Harder's original manuscript with the conjecture. Each part treats a number of beautiful applications.

Modular Forms

Lloyd James Peter Kilford
Modular Forms Book PDF

Author : Lloyd James Peter Kilford
Publisher : Unknown
Page : 0 pages
File Size : 40,9 Mb
Release : 2015
Category : Algebraic spaces
ISBN : 1783265450

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Modular Forms by Lloyd James Peter Kilford Pdf

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.

Heads in Grammatical Theory

Greville G. Corbett,Norman M. Fraser,Scott McGlashan
Heads in Grammatical Theory Book PDF

Author : Greville G. Corbett,Norman M. Fraser,Scott McGlashan
Publisher : Cambridge University Press
Page : 364 pages
File Size : 52,5 Mb
Release : 1993-06-24
Category : Language Arts & Disciplines
ISBN : 052140245X

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Heads in Grammatical Theory by Greville G. Corbett,Norman M. Fraser,Scott McGlashan Pdf

A study of the idea of the 'head' or dominating element of a phrase.

Introduction to Modular Forms

Serge Lang
Introduction to Modular Forms Book PDF

Author : Serge Lang
Publisher : Springer Science & Business Media
Page : 267 pages
File Size : 49,9 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783642514470

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Introduction to Modular Forms by Serge Lang Pdf

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#

Introduction to Elliptic Curves and Modular Forms

Neal I. Koblitz
Introduction to Elliptic Curves and Modular Forms Book PDF

Author : Neal I. Koblitz
Publisher : Springer Science & Business Media
Page : 262 pages
File Size : 40,9 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461209096

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Introduction to Elliptic Curves and Modular Forms by Neal I. Koblitz Pdf

The theory of elliptic curves and modular forms provides a fruitful meeting ground for such diverse areas as number theory, complex analysis, algebraic geometry, and representation theory. This book starts out with a problem from elementary number theory and proceeds to lead its reader into the modern theory, covering such topics as the Hasse-Weil L-function and the conjecture of Birch and Swinnerton-Dyer. This new edition details the current state of knowledge of elliptic curves.

Harmonic Maass Forms and Mock Modular Forms: Theory and Applications

Kathrin Bringmann,Amanda Folsom,Ken Ono,Larry Rolen
Harmonic Maass Forms and Mock Modular Forms Theory and Applications Book PDF

Author : Kathrin Bringmann,Amanda Folsom,Ken Ono,Larry Rolen
Publisher : American Mathematical Soc.
Page : 391 pages
File Size : 45,5 Mb
Release : 2017-12-15
Category : Forms (Mathematics)
ISBN : 9781470419448

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Harmonic Maass Forms and Mock Modular Forms: Theory and Applications by Kathrin Bringmann,Amanda Folsom,Ken Ono,Larry Rolen Pdf

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.

Hilbert Modular Forms: mod $p$ and $p$-Adic Aspects

Fabrizio Andreatta,Eyal Zvi Goren
Hilbert Modular Forms mod p and p Adic Aspects Book PDF

Author : Fabrizio Andreatta,Eyal Zvi Goren
Publisher : American Mathematical Soc.
Page : 100 pages
File Size : 54,7 Mb
Release : 2005
Category : Mathematics
ISBN : 9780821836095

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Hilbert Modular Forms: mod $p$ and $p$-Adic Aspects by Fabrizio Andreatta,Eyal Zvi Goren Pdf

We study Hilbert modular forms in characteristic $p$ and over $p$-adic rings. In the characteristic $p$ theory we describe the kernel and image of the $q$-expansion map and prove the existence of filtration for Hilbert modular forms; we define operators $U$, $V$ and $\Theta_\chi$ and study the variation of the filtration under these operators. Our methods are geometric - comparing holomorphic Hilbert modular forms with rational functions on a moduli scheme with level-$p$ structure, whose poles are supported on the non-ordinary locus.In the $p$-adic theory we study congruences between Hilbert modular forms. This applies to the study of congruences between special values of zeta functions of totally real fields. It also allows us to define $p$-adic Hilbert modular forms 'a la Serre' as $p$-adic uniform limit of classical modular forms, and compare them with $p$-adic modular forms 'a la Katz' that are regular functions on a certain formal moduli scheme. We show that the two notions agree for cusp forms and for a suitable class of weights containing all the classical ones. We extend the operators $V$ and $\Theta_\chi$ to the $p$-adic setting.

Elementary Modular Iwasawa Theory

Haruzo Hida
Elementary Modular Iwasawa Theory Book PDF

Author : Haruzo Hida
Publisher : World Scientific
Page : 446 pages
File Size : 45,5 Mb
Release : 2021-10-04
Category : Mathematics
ISBN : 9789811241383

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Elementary Modular Iwasawa Theory by Haruzo Hida Pdf

This book is the first to provide a comprehensive and elementary account of the new Iwasawa theory innovated via the deformation theory of modular forms and Galois representations. The deformation theory of modular forms is developed by generalizing the cohomological approach discovered in the author's 2019 AMS Leroy P Steele Prize-winning article without using much algebraic geometry.Starting with a description of Iwasawa's classical results on his proof of the main conjecture under the Kummer-Vandiver conjecture (which proves cyclicity of his Iwasawa module more than just proving his main conjecture), we describe a generalization of the method proving cyclicity to the adjoint Selmer group of every ordinary deformation of a two-dimensional Artin Galois representation.The fundamentals in the first five chapters are as follows:Many open problems are presented to stimulate young researchers pursuing their field of study.

The Theory of Jacobi Forms

Martin Eichler,Don Zagier
The Theory of Jacobi Forms Book PDF

Author : Martin Eichler,Don Zagier
Publisher : Springer Science & Business Media
Page : 150 pages
File Size : 51,6 Mb
Release : 2013-12-14
Category : Mathematics
ISBN : 9781468491623

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The Theory of Jacobi Forms by Martin Eichler,Don Zagier Pdf

The functions studied in this monogra9h are a cross between elliptic functions and modular forms in one variable. Specifically, we define a Jacobi form on SL (~) to be a holomorphic function 2 (JC = upper half-plane) satisfying the t\-10 transformation eouations 2Tiimcz· k CT +d a-r +b z) (1) ((cT+d) e cp(T, z) cp CT +d ' CT +d (2) rjl(T, z+h+]l) and having a Four·ier expansion of the form 00 e2Tii(nT +rz) (3) cp(T, z) 2: c(n, r) 2:: rE~ n=O 2 r ~ 4nm Here k and m are natural numbers, called the weight and index of rp, respectively. Note that th e function cp (T, 0) is an ordinary modular formofweight k, whileforfixed T thefunction z-+rjl( -r, z) isa function of the type normally used to embed the elliptic curve ~/~T + ~ into a projective space. If m= 0, then cp is independent of z and the definition reduces to the usual notion of modular forms in one variable. We give three other examples of situations where functions satisfying (1)-(3) arise classically: 1. Theta series. Let Q: ~-+ ~ be a positive definite integer valued quadratic form and B the associated bilinear form.