Elliptic Modular Functions

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Elliptic Modular Functions

B. Schoeneberg
Elliptic Modular Functions Book PDF

Author : B. Schoeneberg
Publisher : Springer Science & Business Media
Page : 244 pages
File Size : 41,8 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783642656637

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Elliptic Modular Functions by B. Schoeneberg Pdf

This book is a fully detailed introduction to the theory of modular functions of a single variable. I hope that it will fill gaps which in view ofthe lively development ofthis theory have often been an obstacle to the students' progress. The study of the book requires an elementary knowledge of algebra, number theory and topology and a deeper knowledge of the theory of functions. An extensive discussion of the modular group SL(2, Z) is followed by the introduction to the theory of automorphic functions and auto morphic forms of integral dimensions belonging to SL(2,Z). The theory is developed first via the Riemann mapping theorem and then again with the help of Eisenstein series. An investigation of the subgroups of SL(2, Z) and the introduction of automorphic functions and forms belonging to these groups folIows. Special attention is given to the subgroups of finite index in SL (2, Z) and, among these, to the so-called congruence groups. The decisive role in this setting is assumed by the Riemann-Roch theorem. Since its proof may be found in the literature, only the pertinent basic concepts are outlined. For the extension of the theory, special fields of modular functions in particular the transformation fields of order n-are studied. Eisen stein series of higher level are introduced which, in case of the dimension - 2, allow the construction of integrals of the 3 rd kind. The properties of these integrals are discussed at length.

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 : 49,5 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.

Vorlesungen Uber Die Theorie For Elliptischen Modulfunctionen

Dr Robert Fricke
Vorlesungen Uber Die Theorie For Elliptischen Modulfunctionen Book PDF

Author : Dr Robert Fricke
Publisher : Legare Street Press
Page : 0 pages
File Size : 44,5 Mb
Release : 2023-07-18
Category : Electronic
ISBN : 101961658X

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Vorlesungen Uber Die Theorie For Elliptischen Modulfunctionen by Dr Robert Fricke Pdf

Fricke's groundbreaking study of the theory of elliptic modular functions is a must-read for anyone interested in the foundations of modern mathematics. With clear explanations and insightful examples, Fricke offers a comprehensive overview of this complex and fascinating subject. This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work is in the "public domain in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant.

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 : 46,5 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.

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 : 48,6 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.

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 : 54,8 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.

Modular Functions and Dirichlet Series in Number Theory

Tom M. Apostol
Modular Functions and Dirichlet Series in Number Theory Book PDF

Author : Tom M. Apostol
Publisher : Springer Science & Business Media
Page : 218 pages
File Size : 49,5 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461209997

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Modular Functions and Dirichlet Series in Number Theory by Tom M. Apostol Pdf

A new edition of a classical treatment of elliptic and modular functions with some of their number-theoretic applications, this text offers an updated bibliography and an alternative treatment of the transformation formula for the Dedekind eta function. It covers many topics, such as Hecke’s theory of entire forms with multiplicative Fourier coefficients, and the last chapter recounts Bohr’s theory of equivalence of general Dirichlet series.

Elliptic Functions

Komaravolu Chandrasekharan
Elliptic Functions Book PDF

Author : Komaravolu Chandrasekharan
Publisher : Springer Science & Business Media
Page : 199 pages
File Size : 46,8 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783642522444

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Elliptic Functions by Komaravolu Chandrasekharan Pdf

This book has grown out of a course of lectures on elliptic functions, given in German, at the Swiss Federal Institute of Technology, Zurich, during the summer semester of 1982. Its aim is to give some idea of the theory of elliptic functions, and of its close connexion with theta-functions and modular functions, and to show how it provides an analytic approach to the solution of some classical problems in the theory of numbers. It comprises eleven chapters. The first seven are function-theoretic, and the next four concern arithmetical applications. There are Notes at the end of every chapter, which contain references to the literature, comments on the text, and on the ramifications, old and new, of the problems dealt with, some of them extending into cognate fields. The treatment is self-contained, and makes no special demand on the reader's knowledge beyond the elements of complex analysis in one variable, and of group theory.

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 : 54,6 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.

Elliptic Functions

Serge Lang
Elliptic Functions Book PDF

Author : Serge Lang
Publisher : Springer Science & Business Media
Page : 319 pages
File Size : 54,8 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461247524

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Elliptic Functions by Serge Lang Pdf

Elliptic functions parametrize elliptic curves, and the intermingling of the analytic and algebraic-arithmetic theory has been at the center of mathematics since the early part of the nineteenth century. The book is divided into four parts. In the first, Lang presents the general analytic theory starting from scratch. Most of this can be read by a student with a basic knowledge of complex analysis. The next part treats complex multiplication, including a discussion of Deuring's theory of l-adic and p-adic representations, and elliptic curves with singular invariants. Part three covers curves with non-integral invariants, and applies the Tate parametrization to give Serre's results on division points. The last part covers theta functions and the Kronecker Limit Formula. Also included is an appendix by Tate on algebraic formulas in arbitrary charactistic.

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 : 40,6 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.

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 : 55,6 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.

Modular Forms and Functions

Robert A. Rankin
Modular Forms and Functions Book PDF

Author : Robert A. Rankin
Publisher : Cambridge University Press
Page : 0 pages
File Size : 54,6 Mb
Release : 2008-12-04
Category : Mathematics
ISBN : 0521091683

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Modular Forms and Functions by Robert A. Rankin Pdf

This book provides an introduction to the theory of elliptic modular functions and forms, a subject of increasing interest because of its connexions with the theory of elliptic curves. Modular forms are generalisations of functions like theta functions. They can be expressed as Fourier series, and the Fourier coefficients frequently possess multiplicative properties which lead to a correspondence between modular forms and Dirichlet series having Euler products. The Fourier coefficients also arise in certain representational problems in the theory of numbers, for example in the study of the number of ways in which a positive integer may be expressed as a sum of a given number of squares. The treatment of the theory presented here is fuller than is customary in a textbook on automorphic or modular forms, since it is not confined solely to modular forms of integral weight (dimension). It will be of interest to professional mathematicians as well as senior undergraduate and graduate students in pure mathematics.

Geometric Modular Forms and Elliptic Curves

Haruzo Hida
Geometric Modular Forms and Elliptic Curves Book PDF

Author : Haruzo Hida
Publisher : World Scientific
Page : 468 pages
File Size : 54,7 Mb
Release : 2011
Category : Mathematics
ISBN : 9789814368650

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Geometric Modular Forms and Elliptic Curves by Haruzo Hida Pdf

1. An algebro-geometric tool box. 1.1. Sheaves. 1.2. Schemes. 1.3. Projective schemes. 1.4. Categories and functors. 1.5. Applications of the key-lemma. 1.6. Group schemes. 1.7. Cartier duality. 1.8. Quotients by a group scheme. 1.9. Morphisms. 1.10. Cohomology of coherent sheaves. 1.11. Descent. 1.12. Barsotti-Tate groups. 1.13. Formal scheme -- 2. Elliptic curves. 2.1. Curves and divisors. 2.2. Elliptic curves. 2.3. Geometric modular forms of level 1. 2.4. Elliptic curves over C. 2.5. Elliptic curves over p-adic fields. 2.6. Level structures. 2.7. L-functions of elliptic curves. 2.8. Regularity. 2.9. p-ordinary moduli problems. 2.10. Deformation of elliptic curves -- 3. Geometric modular forms. 3.1. Integrality. 3.2. Vertical control theorem. 3.3. Action of GL(2) on modular forms -- 4. Jacobians and Galois representations. 4.1. Jacobians of stable curves. 4.2. Modular Galois representations. 4.3. Fullness of big Galois representations -- 5. Modularity problems. 5.1. Induced and extended Galois representations. 5.2. Some other solutions. 5.3. Modularity of Abelian Q-varieties

Modular Forms and Fermat’s Last Theorem

Gary Cornell,Joseph H. Silverman,Glenn Stevens
Modular Forms and Fermat s Last Theorem Book PDF

Author : Gary Cornell,Joseph H. Silverman,Glenn Stevens
Publisher : Springer Science & Business Media
Page : 592 pages
File Size : 49,6 Mb
Release : 2013-12-01
Category : Mathematics
ISBN : 9781461219743

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Modular Forms and Fermat’s Last Theorem by Gary Cornell,Joseph H. Silverman,Glenn Stevens Pdf

This volume contains the expanded lectures given at a conference on number theory and arithmetic geometry held at Boston University. It introduces and explains the many ideas and techniques used by Wiles, and to explain how his result can be combined with Ribets theorem and ideas of Frey and Serre to prove Fermats Last Theorem. The book begins with an overview of the complete proof, followed by several introductory chapters surveying the basic theory of elliptic curves, modular functions and curves, Galois cohomology, and finite group schemes. Representation theory, which lies at the core of the proof, is dealt with in a chapter on automorphic representations and the Langlands-Tunnell theorem, and this is followed by in-depth discussions of Serres conjectures, Galois deformations, universal deformation rings, Hecke algebras, and complete intersections. The book concludes by looking both forward and backward, reflecting on the history of the problem, while placing Wiles'theorem into a more general Diophantine context suggesting future applications. Students and professional mathematicians alike will find this an indispensable resource.