Computations With Modular Forms

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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 : 49,5 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.

Modular Forms

L J P Kilford
Modular Forms Book PDF

Author : L J P Kilford
Publisher : World Scientific Publishing Company
Page : 252 pages
File Size : 52,7 Mb
Release : 2015-03-12
Category : Mathematics
ISBN : 9781783265473

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Modular Forms by L J P 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.

Computational Aspects of Modular Forms and Galois Representations

Bas Edixhoven,Jean-Marc Couveignes
Computational Aspects of Modular Forms and Galois Representations Book PDF

Author : Bas Edixhoven,Jean-Marc Couveignes
Publisher : Princeton University Press
Page : 438 pages
File Size : 47,6 Mb
Release : 2011-05-31
Category : Mathematics
ISBN : 9781400839001

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Computational Aspects of Modular Forms and Galois Representations by Bas Edixhoven,Jean-Marc Couveignes Pdf

Modular forms are tremendously important in various areas of mathematics, from number theory and algebraic geometry to combinatorics and lattices. Their Fourier coefficients, with Ramanujan's tau-function as a typical example, have deep arithmetic significance. Prior to this book, the fastest known algorithms for computing these Fourier coefficients took exponential time, except in some special cases. The case of elliptic curves (Schoof's algorithm) was at the birth of elliptic curve cryptography around 1985. This book gives an algorithm for computing coefficients of modular forms of level one in polynomial time. For example, Ramanujan's tau of a prime number p can be computed in time bounded by a fixed power of the logarithm of p. Such fast computation of Fourier coefficients is itself based on the main result of the book: the computation, in polynomial time, of Galois representations over finite fields attached to modular forms by the Langlands program. Because these Galois representations typically have a nonsolvable image, this result is a major step forward from explicit class field theory, and it could be described as the start of the explicit Langlands program. The computation of the Galois representations uses their realization, following Shimura and Deligne, in the torsion subgroup of Jacobian varieties of modular curves. The main challenge is then to perform the necessary computations in time polynomial in the dimension of these highly nonlinear algebraic varieties. Exact computations involving systems of polynomial equations in many variables take exponential time. This is avoided by numerical approximations with a precision that suffices to derive exact results from them. Bounds for the required precision--in other words, bounds for the height of the rational numbers that describe the Galois representation to be computed--are obtained from Arakelov theory. Two types of approximations are treated: one using complex uniformization and another one using geometry over finite fields. The book begins with a concise and concrete introduction that makes its accessible to readers without an extensive background in arithmetic geometry. And the book includes a chapter that describes actual computations.

Computations with Modular Forms

Gebhard Böckle,Gabor Wiese
Computations with Modular Forms Book PDF

Author : Gebhard Böckle,Gabor Wiese
Publisher : Springer Science & Business Media
Page : 376 pages
File Size : 51,7 Mb
Release : 2014-01-23
Category : Mathematics
ISBN : 9783319038476

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Computations with Modular Forms by Gebhard Böckle,Gabor Wiese Pdf

This volume contains original research articles, survey articles and lecture notes related to the Computations with Modular Forms 2011 Summer School and Conference, held at the University of Heidelberg. A key theme of the Conference and Summer School was the interplay between theory, algorithms and experiment. The 14 papers offer readers both, instructional courses on the latest algorithms for computing modular and automorphic forms, as well as original research articles reporting on the latest developments in the field. The three Summer School lectures provide an introduction to modern algorithms together with some theoretical background for computations of and with modular forms, including computing cohomology of arithmetic groups, algebraic automorphic forms, and overconvergent modular symbols. The 11 Conference papers cover a wide range of themes related to computations with modular forms, including lattice methods for algebraic modular forms on classical groups, a generalization of the Maeda conjecture, an efficient algorithm for special values of p-adic Rankin triple product L-functions, arithmetic aspects and experimental data of Bianchi groups, a theoretical study of the real Jacobian of modular curves, results on computing weight one modular forms, and more.

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 : 273 pages
File Size : 44,5 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: 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 : 52,5 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.

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 : 43,7 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.

Elliptic Curves, Hilbert Modular Forms and Galois Deformations

Laurent Berger,Gebhard Böckle,Lassina Dembélé,Mladen Dimitrov,Tim Dokchitser,John Voight
Elliptic Curves Hilbert Modular Forms and Galois Deformations Book PDF

Author : Laurent Berger,Gebhard Böckle,Lassina Dembélé,Mladen Dimitrov,Tim Dokchitser,John Voight
Publisher : Springer Science & Business Media
Page : 257 pages
File Size : 41,6 Mb
Release : 2013-06-13
Category : Mathematics
ISBN : 9783034806183

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Elliptic Curves, Hilbert Modular Forms and Galois Deformations by Laurent Berger,Gebhard Böckle,Lassina Dembélé,Mladen Dimitrov,Tim Dokchitser,John Voight Pdf

The notes in this volume correspond to advanced courses held at the Centre de Recerca Matemàtica as part of the research program in Arithmetic Geometry in the 2009-2010 academic year. The notes by Laurent Berger provide an introduction to p-adic Galois representations and Fontaine rings, which are especially useful for describing many local deformation rings at p that arise naturally in Galois deformation theory. The notes by Gebhard Böckle offer a comprehensive course on Galois deformation theory, starting from the foundational results of Mazur and discussing in detail the theory of pseudo-representations and their deformations, local deformations at places l ≠ p and local deformations at p which are flat. In the last section,the results of Böckle and Kisin on presentations of global deformation rings over local ones are discussed. The notes by Mladen Dimitrov present the basics of the arithmetic theory of Hilbert modular forms and varieties, with an emphasis on the study of the images of the attached Galois representations, on modularity lifting theorems over totally real number fields, and on the cohomology of Hilbert modular varieties with integral coefficients. The notes by Lassina Dembélé and John Voight describe methods for performing explicit computations in spaces of Hilbert modular forms. These methods depend on the Jacquet-Langlands correspondence and on computations in spaces of quaternionic modular forms, both for the case of definite and indefinite quaternion algebras. Several examples are given, and applications to modularity of Galois representations are discussed. The notes by Tim Dokchitser describe the proof, obtained by the author in a joint project with Vladimir Dokchitser, of the parity conjecture for elliptic curves over number fields under the assumption of finiteness of the Tate-Shafarevich group. The statement of the Birch and Swinnerton-Dyer conjecture is included, as well as a detailed study of local and global root numbers of elliptic curves and their classification.

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 : 53,6 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#

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 : 42,8 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.

Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory

Johannes Blümlein,Carsten Schneider,Peter Paule
Elliptic Integrals Elliptic Functions and Modular Forms in Quantum Field Theory Book PDF

Author : Johannes Blümlein,Carsten Schneider,Peter Paule
Publisher : Springer
Page : 509 pages
File Size : 52,7 Mb
Release : 2019-01-30
Category : Computers
ISBN : 9783030044800

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Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory by Johannes Blümlein,Carsten Schneider,Peter Paule Pdf

This book includes review articles in the field of elliptic integrals, elliptic functions and modular forms intending to foster the discussion between theoretical physicists working on higher loop calculations and mathematicians working in the field of modular forms and functions and analytic solutions of higher order differential and difference equations.

Elliptic Curves, Modular Forms, and Their L-functions

Alvaro Lozano-Robledo
Elliptic Curves Modular Forms and Their L functions Book PDF

Author : Alvaro Lozano-Robledo
Publisher : American Mathematical Soc.
Page : 195 pages
File Size : 50,5 Mb
Release : 2011
Category : Mathematics
ISBN : 9780821852422

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Elliptic Curves, Modular Forms, and Their L-functions by Alvaro Lozano-Robledo Pdf

Many problems in number theory have simple statements, but their solutions require a deep understanding of algebra, algebraic geometry, complex analysis, group representations, or a combination of all four. The original simply stated problem can be obscured in the depth of the theory developed to understand it. This book is an introduction to some of these problems, and an overview of the theories used nowadays to attack them, presented so that the number theory is always at the forefront of the discussion. Lozano-Robledo gives an introductory survey of elliptic curves, modular forms, and $L$-functions. His main goal is to provide the reader with the big picture of the surprising connections among these three families of mathematical objects and their meaning for number theory. As a case in point, Lozano-Robledo explains the modularity theorem and its famous consequence, Fermat's Last Theorem. He also discusses the Birch and Swinnerton-Dyer Conjecture and other modern conjectures. The book begins with some motivating problems and includes numerous concrete examples throughout the text, often involving actual numbers, such as 3, 4, 5, $\frac{3344161}{747348}$, and $\frac{2244035177043369699245575130906674863160948472041} {8912332268928859588025535178967163570016480830}$. The theories of elliptic curves, modular forms, and $L$-functions are too vast to be covered in a single volume, and their proofs are outside the scope of the undergraduate curriculum. However, the primary objects of study, the statements of the main theorems, and their corollaries are within the grasp of advanced undergraduates. This book concentrates on motivating the definitions, explaining the statements of the theorems and conjectures, making connections, and providing lots of examples, rather than dwelling on the hard proofs. The book succeeds if, after reading the text, students feel compelled to study elliptic curves and modular forms in all their glory.

Modular Forms and Hecke Operators

A. N. Andrianov,V. G. Zhuravlev
Modular Forms and Hecke Operators Book PDF

Author : A. N. Andrianov,V. G. Zhuravlev
Publisher : American Mathematical Soc.
Page : 334 pages
File Size : 50,5 Mb
Release : 2016-01-29
Category : Electronic
ISBN : 9781470418687

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Modular Forms and Hecke Operators by A. N. Andrianov,V. G. Zhuravlev Pdf

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.

Introduction to Applications of Modular Forms

Zafer Selcuk Aygin
Introduction to Applications of Modular Forms Book PDF

Author : Zafer Selcuk Aygin
Publisher : Unknown
Page : 0 pages
File Size : 48,5 Mb
Release : 2023
Category : Mathematics
ISBN : 3031326318

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Introduction to Applications of Modular Forms by Zafer Selcuk Aygin Pdf

This book is a self-contained treatment for those who study or work on the computational aspects of classical modular forms. The author describes the theory of modular forms and its applications in number theoretic problems such as representations by quadratic forms and the determination of asymptotic formulas for Fourier coefficients of different types of special functions. A detailed account of recent applications of modular forms in number theory with a focus on using computer algorithms is provided. Computer algorithms are included for each presented application to help readers put the theory in context and make new conjectures. In addition, this book: Describes the theory of modular forms and its applications in number theoretic problems Provides a resource for people who study or work on the computational aspects of classical modular forms Includes computer algorithms to help readers conjecture new results and prove them using the presented theory.

Lectures on Modular Forms

Robert C. Gunning
Lectures on Modular Forms Book PDF

Author : Robert C. Gunning
Publisher : Princeton University Press
Page : 116 pages
File Size : 51,6 Mb
Release : 1962-03-21
Category : Education
ISBN : 0691079951

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Lectures on Modular Forms by Robert C. Gunning Pdf

New interest in modular forms of one complex variable has been caused chiefly by the work of Selberg and of Eichler. But there has been no introductory work covering the background of these developments. H. C. Gunning's book surveys techniques and problems; only the simpler cases are treated-modular forms of even weights without multipliers, the principal congruence subgroups, and the Hecke operators for the full modular group alone.