Modular Forms With Integral And Half Integral Weights
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| Author | : Xueli Wang |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 436 |
| Release | : 2013-02-20 |
| Genre | : Mathematics |
| ISBN | : 3642293026 |
Download Modular Forms with Integral and Half-Integral Weights Book in PDF, Epub and Kindle
"Modular Forms with Integral and Half-Integral Weights" focuses on the fundamental theory of modular forms of one variable with integral and half-integral weights. The main theme of the book is the theory of Eisenstein series. It is a fundamental problem to construct a basis of the orthogonal complement of the space of cusp forms; as is well known, this space is spanned by Eisenstein series for any weight greater than or equal to 2. The book proves that the conclusion holds true for weight 3/2 by explicitly constructing a basis of the orthogonal complement of the space of cusp forms. The problem for weight 1/2, which was solved by Serre and Stark, will also be discussed in this book. The book provides readers not only basic knowledge on this topic but also a general survey of modern investigation methods of modular forms with integral and half-integral weights. It will be of significant interest to researchers and practitioners in modular forms of mathematics. Dr. Xueli Wang is a Professor at South China Normal University, China. Dingyi Pei is a Professor at Guangzhou University, China.
| Author | : Willem Kuyk |
| Publisher | : |
| Total Pages | : |
| Release | : 1973 |
| Genre | : Modular functions |
| ISBN | : |
Download Modular Functions of One Variable, I-IV Book in PDF, Epub and Kindle
| Author | : Martin Eichler |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 156 |
| Release | : 2013-12-14 |
| Genre | : Mathematics |
| ISBN | : 1468491628 |
Download The Theory of Jacobi Forms Book in PDF, Epub and Kindle
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.
| Author | : Peter Sarnak |
| Publisher | : Cambridge University Press |
| Total Pages | : 124 |
| Release | : 1990-11-15 |
| Genre | : Mathematics |
| ISBN | : 1316582442 |
Download Some Applications of Modular Forms Book in PDF, Epub and Kindle
The theory of modular forms and especially the so-called 'Ramanujan Conjectures' have been applied to resolve problems in combinatorics, computer science, analysis and number theory. This tract, based on the Wittemore Lectures given at Yale University, is concerned with describing some of these applications. In order to keep the presentation reasonably self-contained, Professor Sarnak begins by developing the necessary background material in modular forms. He then considers the solution of three problems: the Ruziewicz problem concerning finitely additive rotationally invariant measures on the sphere; the explicit construction of highly connected but sparse graphs: 'expander graphs' and 'Ramanujan graphs'; and the Linnik problem concerning the distribution of integers that represent a given large integer as a sum of three squares. These applications are carried out in detail. The book therefore should be accessible to a wide audience of graduate students and researchers in mathematics and computer science.
| Author | : Goro Shimura |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 183 |
| Release | : 2011-11-18 |
| Genre | : Mathematics |
| ISBN | : 146142125X |
Download Modular Forms: Basics and Beyond Book in PDF, Epub and Kindle
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.
| Author | : W. Kohnen |
| Publisher | : |
| Total Pages | : 39 |
| Release | : 1984 |
| Genre | : |
| ISBN | : |
Download Fourier Coefficients of Modular Forms of Half-integral Weight Book in PDF, Epub and Kindle
| Author | : YoungJu Choie |
| Publisher | : Springer Nature |
| Total Pages | : 296 |
| Release | : 2019-11-20 |
| Genre | : Mathematics |
| ISBN | : 3030291235 |
Download Jacobi-Like Forms, Pseudodifferential Operators, and Quasimodular Forms Book in PDF, Epub and Kindle
This book explores various properties of quasimodular forms, especially their connections with Jacobi-like forms and automorphic pseudodifferential operators. The material that is essential to the subject is presented in sufficient detail, including necessary background on pseudodifferential operators, Lie algebras, etc., to make it accessible also to non-specialists. The book also covers a sufficiently broad range of illustrations of how the main themes of the book have occurred in various parts of mathematics to make it attractive to a wider audience. The book is intended for researchers and graduate students in number theory.
| Author | : Ken Ono |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 226 |
| Release | : 2004 |
| Genre | : Mathematics |
| ISBN | : 0821833685 |
Download The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and $q$-series Book in PDF, Epub and Kindle
Chapter 1.
| Author | : |
| Publisher | : |
| Total Pages | : |
| Release | : 2007 |
| Genre | : |
| ISBN | : |
Download Distribution of Integral Fourier Coefficients of a Modular Form of Half Integral Weight Modulo Primes Book in PDF, Epub and Kindle
Recently, Bruinier and Ono classified cusp forms $f(z) := \sum_{n=0}^{\infty} a_f(n)q ^n \in S_{\lambda+1/2}(\Gamma_0(N), \chi)\cap \mathbb{Z}[[q]]$ that does not satisfy a certain distribution property for modulo odd primes $p$. In this paper, using Rankin-Cohen Bracket, we extend this result to modular forms of half integral weight for primes $p \geq 5$. As applications of our main theorem we derive distribution properties, for modulo primes $p\geq5$, of traces of singular moduli and Hurwitz class number. We also study an analogue of Newman's conjecture for overpartitions.
| Author | : Kathrin Bringmann |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 409 |
| Release | : 2017-12-15 |
| Genre | : Mathematics |
| ISBN | : 1470419440 |
Download Harmonic Maass Forms and Mock Modular Forms: Theory and Applications Book in PDF, Epub and Kindle
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.
