The Theory Of Jacobi Forms

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The Theory Of Jacobi Forms

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Author : Martin Eichler
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-12-14

The Theory Of Jacobi Forms written by Martin Eichler and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-12-14 with Mathematics categories.

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.

The Theory Of Jacobi Forms

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Author : Martin Eichler
language : en
Publisher:
Release Date : 2014-09-01

The Theory Of Jacobi Forms written by Martin Eichler and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-09-01 with categories.

Jacobi Forms Finite Quadratic Modules And Weil Representations Over Number Fields

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Author : Hatice Boylan
language : en
Publisher: Springer
Release Date : 2014-12-05

Jacobi Forms Finite Quadratic Modules And Weil Representations Over Number Fields written by Hatice Boylan and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-12-05 with Mathematics categories.

The new theory of Jacobi forms over totally real number fields introduced in this monograph is expected to give further insight into the arithmetic theory of Hilbert modular forms, its L-series, and into elliptic curves over number fields. This work is inspired by the classical theory of Jacobi forms over the rational numbers, which is an indispensable tool in the arithmetic theory of elliptic modular forms, elliptic curves, and in many other disciplines in mathematics and physics. Jacobi forms can be viewed as vector valued modular forms which take values in so-called Weil representations. Accordingly, the first two chapters develop the theory of finite quadratic modules and associated Weil representations over number fields. This part might also be interesting for those who are merely interested in the representation theory of Hilbert modular groups. One of the main applications is the complete classification of Jacobi forms of singular weight over an arbitrary totally real number field.

Elements Of The Representation Theory Of The Jacobi Group

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Author : Rolf Berndt
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-01-05

Elements Of The Representation Theory Of The Jacobi Group written by Rolf Berndt and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-01-05 with Mathematics categories.

Combining algebraic groups and number theory, this volume gathers material from the representation theory of this group for the first time, doing so for both local (Archimedean and non-Archimedean) cases as well as for the global number field case.

The Moduli Space Of Curves

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Author : Robert H. Dijkgraaf
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06

The Moduli Space Of Curves written by Robert H. Dijkgraaf and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.

This generalization of geometry is bound to have wide spread repercussions for mathematics as well as physics. The unearthing of it will entail a new golden age in the interaction of mathematics and physics. E. Witten (1986) The idea that the moduli space Mg of curves of fixed genus 9 - that is, the algebraic variety that parametrizes all curves of genus 9 - is an intriguing object in its own right seems to have come slowly. Although the para meters or moduli of curves surface in Riemann's famous memoir on abelian functions (from 1857) and in work of Hurwitz and later were considered by the geometers of the Italian school, for a long time they attracted attention only in the special case 9 = 1, where they were studied in the framework of the theory of modular functions. The work of Grothendieck, who in the early sixties pointed the way towards the right approach, and the subsequent construction (in 1965) of the moduli space Mg by Mumford were the first foundational work, to be followed by the construction of a compactification Mg by Deligne and Mumford in 1969. The theorem of Harris and Mumford saying that for 9 sufficiently large the space Mg is of general type was the first big insight in its structure.

Problems In The Theory Of Modular Forms

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Author : M. Ram Murty
language : en
Publisher: Springer
Release Date : 2016-11-25

Problems In The Theory Of Modular Forms written by M. Ram Murty and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-11-25 with Mathematics categories.

This book introduces the reader to the fascinating world of modular forms through a problem-solving approach. As such, besides researchers, the book can be used by the undergraduate and graduate students for self-instruction. The topics covered include q-series, the modular group, the upper half-plane, modular forms of level one and higher level, the Ramanujan τ-function, the Petersson inner product, Hecke operators, Dirichlet series attached to modular forms and further special topics. It can be viewed as a gentle introduction for a deeper study of the subject. Thus, it is ideal for non-experts seeking an entry into the field.

The 1 2 3 Of Modular Forms

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Author : Jan Hendrik Bruinier
language : en
Publisher: Springer Science & Business Media
Release Date : 2008-02-10

The 1 2 3 Of Modular Forms written by Jan Hendrik Bruinier and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008-02-10 with Mathematics categories.

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 And Related Topics In Number Theory

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Author : B. Ramakrishnan
language : en
Publisher: Springer Nature
Release Date : 2020-11-24

Modular Forms And Related Topics In Number Theory written by B. Ramakrishnan and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-11-24 with Mathematics categories.

This book collects the papers presented at the Conference on Number Theory, held at the Kerala School of Mathematics, Kozhikode, Kerala, India, from December 10–14, 2018. The conference aimed at bringing the active number theorists and researchers in automorphic forms and allied areas to demonstrate their current research works. This book benefits young research scholars, postdoctoral fellows, and young faculty members working in these areas of research.

Developments In The Theory Of Jacobi Forms

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Author : N.-P. Skoruppa
language : en
Publisher:
Release Date : 1989

Developments In The Theory Of Jacobi Forms written by N.-P. Skoruppa and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1989 with categories.