Download or read book A First Course in Modular Forms written by Fred Diamond and published by Springer Science & Business Media. This book was released on 2006-03-30 with total page 450 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book Introduction to Modular Forms written by Serge Lang and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 267 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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#

Download or read book Modular Forms A Classical Approach written by Henri Cohen and published by American Mathematical Soc.. This book was released on 2017-08-02 with total page 700 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book Modular Forms a Computational Approach written by William A. Stein and published by American Mathematical Soc.. This book was released on 2007-02-13 with total page 290 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book Modular Forms Basics and Beyond written by Goro Shimura and published by Springer Science & Business Media. This book was released on 2011-11-18 with total page 183 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is an advanced book on modular forms. While there are many books published about modular forms, they are written at an elementary level, and not so interesting from the viewpoint of a reader who already knows the basics. This book offers something new, which may satisfy the desire of such a reader. However, we state every definition and every essential fact concerning classical modular forms of one variable. One of the principal new features of this book is the theory of modular forms of half-integral weight, another being the discussion of theta functions and Eisenstein series of holomorphic and nonholomorphic types. Thus the book is presented so that the reader can learn such theories systematically.

Download or read book Harmonic Maass Forms and Mock Modular Forms Theory and Applications written by Kathrin Bringmann and published by American Mathematical Soc.. This book was released on 2017-12-15 with total page 391 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book Modular Forms written by L J P Kilford and published by World Scientific. This book was released on 2008-08-11 with total page 236 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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

Download or read book Topological Modular Forms written by Christopher L. Douglas and published by American Mathematical Soc.. This book was released on 2014-12-04 with total page 353 pages. Available in PDF, EPUB and Kindle. Book excerpt: The theory of topological modular forms is an intricate blend of classical algebraic modular forms and stable homotopy groups of spheres. The construction of this theory combines an algebro-geometric perspective on elliptic curves over finite fields with techniques from algebraic topology, particularly stable homotopy theory. It has applications to and connections with manifold topology, number theory, and string theory. This book provides a careful, accessible introduction to topological modular forms. After a brief history and an extended overview of the subject, the book proper commences with an exposition of classical aspects of elliptic cohomology, including background material on elliptic curves and modular forms, a description of the moduli stack of elliptic curves, an explanation of the exact functor theorem for constructing cohomology theories, and an exploration of sheaves in stable homotopy theory. There follows a treatment of more specialized topics, including localization of spectra, the deformation theory of formal groups, and Goerss-Hopkins obstruction theory for multiplicative structures on spectra. The book then proceeds to more advanced material, including discussions of the string orientation, the sheaf of spectra on the moduli stack of elliptic curves, the homotopy of topological modular forms, and an extensive account of the construction of the spectrum of topological modular forms. The book concludes with the three original, pioneering and enormously influential manuscripts on the subject, by Hopkins, Miller, and Mahowald.

Download or read book Elliptic Curves Modular Forms and Their L functions written by Alvaro Lozano-Robledo and published by American Mathematical Soc.. This book was released on 2011 with total page 195 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book Hilbert Modular Forms written by Eberhard Freitag and published by Springer Science & Business Media. This book was released on 2013-03-09 with total page 255 pages. Available in PDF, EPUB and Kindle. Book excerpt: Important results on the Hilbert modular group and Hilbert modular forms are introduced and described in this book. In recent times, this branch of number theory has been given more and more attention and thus the need for a comprehensive presentation of these results, previously scattered in research journal papers, has become obvious. The main aim of this book is to give a description of the singular cohomology and its Hodge decomposition including explicit formulae. The author has succeeded in giving proofs which are both elementary and complete. The book contains an introduction to Hilbert modular forms, reduction theory, the trace formula and Shimizu's formulae, the work of Matsushima and Shimura, analytic continuation of Eisenstein series, the cohomology and its Hodge decomposition. Basic facts about algebraic numbers, integration, alternating differential forms and Hodge theory are included in convenient appendices so that the book can be used by students with a knowledge of complex analysis (one variable) and algebra.

Download or read book The 1 2 3 of Modular Forms written by Jan Hendrik Bruinier and published by Springer Science & Business Media. This book was released on 2008-02-10 with total page 273 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book Lectures on Modular Forms written by Robert C. Gunning and published by Princeton University Press. This book was released on 1962-03-21 with total page 116 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book Lectures on Hilbert Modular Varieties and Modular Forms written by Eyal Zvi Goren and published by American Mathematical Soc.. This book was released on 2002 with total page 282 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is devoted to certain aspects of the theory of $p$-adic Hilbert modular forms and moduli spaces of abelian varieties with real multiplication. The theory of $p$-adic modular forms is presented first in the elliptic case, introducing the reader to key ideas of N. M. Katz and J.-P. Serre. It is re-interpreted from a geometric point of view, which is developed to present the rudiments of a similar theory for Hilbert modular forms. The theory of moduli spaces of abelianvarieties with real multiplication is presented first very explicitly over the complex numbers. Aspects of the general theory are then exposed, in particular, local deformation theory of abelian varieties in positive characteristic. The arithmetic of $p$-adic Hilbert modular forms and the geometry ofmoduli spaces of abelian varieties are related. This relation is used to study $q$-expansions of Hilbert modular forms, on the one hand, and stratifications of moduli spaces on the other hand. The book is addressed to graduate students and non-experts. It attempts to provide the necessary background to all concepts exposed in it. It may serve as a textbook for an advanced graduate course.

Download or read book Modular Forms and Fermat s Last Theorem written by Gary Cornell and published by Springer Science & Business Media. This book was released on 2013-12-01 with total page 592 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book Introduction to Elliptic Curves and Modular Forms written by Neal I. Koblitz and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 262 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book Heads in Grammatical Theory written by Greville G. Corbett and published by Cambridge University Press. This book was released on 1993-06-24 with total page 364 pages. Available in PDF, EPUB and Kindle. Book excerpt: A study of the idea of the 'head' or dominating element of a phrase.

Download or read book The Web of Modularity Arithmetic of the Coefficients of Modular Forms and q series written by Ken Ono and published by American Mathematical Soc.. This book was released on 2004 with total page 226 pages. Available in PDF, EPUB and Kindle. Book excerpt: Chapter 1.