Theta Constants Riemann Surfaces And The Modular Group
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Theta Constants Riemann Surfaces And The Modular Group
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Author : Hershel M. Farkas
language : en
Publisher: American Mathematical Soc.
Release Date : 2001
Theta Constants Riemann Surfaces And The Modular Group written by Hershel M. Farkas and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2001 with Functions, Theta categories.
There are incredibly rich connections between classical analysis and number theory. For instance, analytic number theory contains many examples of asymptotic expressions derived from estimates for analytic functions, such as in the proof of the Prime Number Theorem. In combinatorial number theory, exact formulas for number-theoretic quantities are derived from relations between analytic functions. Elliptic functions, especially theta functions, are an important class of such functions in this context, which had been made clear already in Jacobi's Fundamenta nova. Theta functions are also classically connected with Riemann surfaces and with the modular group $\Gamma = \mathrm{PSL (2,\mathbb{Z )$, which provide another path for insights into number theory. Farkas and Kra, well-known masters of the theory of Riemann surfaces and the analysis of theta functions, uncover here interesting combinatorial identities by means of the function theory on Riemann surfaces related to the principal congruence subgroups $\Gamma(k)$. For instance, the authors use this approach to derive congruences discovered by Ramanujan for the partition function, with the main ingredient being the construction of the same function in more than one way. The authors also obtain a variant on Jacobi's famous result on the number of ways that an integer can be represented as a sum of four squares, replacing the squares by triangular numbers and, in the process, obtaining a cleaner result. The recent trend of applying the ideas and methods of algebraic geometry to the study of theta functions and number theory has resulted in great advances in the area. However, the authors choose to stay with the classical point of view. As a result, their statements and proofs are very concrete. In this book the mathematician familiar with the algebraic geometry approach to theta functions and number theory will find many interesting ideas as well as detailed explanations and derivations of new and old results. Highlights of the book include systematic studies of theta constant identities, uniformizations of surfaces represented by subgroups of the modular group, partition identities, and Fourier coefficients of automorphic functions. Prerequisites are a solid understanding of complex analysis, some familiarity with Riemann surfaces, Fuchsian groups, and elliptic functions, and an interest in number theory. The book contains summaries of some of the required material, particularly for theta functions and theta constants. Readers will find here a careful exposition of a classical point of view of analysis and number theory. Presented are numerous examples plus suggestions for research-level problems. The text is suitable for a graduate course or for independent reading.
Advances In The Theory Of Riemann Surfaces
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Author : Lars Valerian Ahlfors
language : en
Publisher: Princeton University Press
Release Date : 1971-07-21
Advances In The Theory Of Riemann Surfaces written by Lars Valerian Ahlfors and has been published by Princeton University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1971-07-21 with Mathematics categories.
Intended for researchers in Riemann surfaces, this volume summarizes a significant portion of the work done in the field during the years 1966 to 1971.
Advances In The Theory Of Riemann Surfaces Am 66 Volume 66
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Author : Lars Valerian Ahlfors
language : en
Publisher: Princeton University Press
Release Date : 1971-07-01
Advances In The Theory Of Riemann Surfaces Am 66 Volume 66 written by Lars Valerian Ahlfors and has been published by Princeton University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1971-07-01 with Mathematics categories.
Intended for researchers in Riemann surfaces, this volume summarizes a significant portion of the work done in the field during the years 1966 to 1971.
Riemann Surfaces Theta Functions And Abelian Automorphisms Groups
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Author : R.D.M. Accola
language : en
Publisher: Springer
Release Date : 2006-11-14
Riemann Surfaces Theta Functions And Abelian Automorphisms Groups written by R.D.M. Accola and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-11-14 with Mathematics categories.
The Geometry Of Riemann Surfaces And Abelian Varieties
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Author : José María Muñoz Porras
language : en
Publisher: American Mathematical Soc.
Release Date : 2006
The Geometry Of Riemann Surfaces And Abelian Varieties written by José María Muñoz Porras and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006 with Mathematics categories.
Most of the papers in this book deal with the theory of Riemann surfaces (moduli problems, automorphisms, etc.), abelian varieties, theta functions, and modular forms. Some of the papers contain surveys on the recent results in the topics of current interest to mathematicians, whereas others contain new research results.
Theta Functions On Riemann Surfaces
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Author : J. D. Fay
language : en
Publisher: Springer
Release Date : 2006-11-15
Theta Functions On Riemann Surfaces written by J. D. Fay and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-11-15 with Mathematics categories.
These notes present new as well as classical results from the theory of theta functions on Riemann surfaces, a subject of renewed interest in recent years. Topics discussed here include: the relations between theta functions and Abelian differentials, theta functions on degenerate Riemann surfaces, Schottky relations for surfaces of special moduli, and theta functions on finite bordered Riemann surfaces.
Riemann Surfaces And Generalized Theta Functions
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Author : Robert C. Gunning
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Riemann Surfaces And Generalized Theta Functions written by Robert C. Gunning 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.
The investigation of the relationships between compact Riemann surfaces (al gebraic curves) and their associated complex tori (Jacobi varieties) has long been basic to the study both of Riemann surfaces and of complex tori. A Riemann surface is naturally imbedded as an analytic submanifold in its associated torus; and various spaces of linear equivalence elasses of divisors on the surface (or equivalently spaces of analytic equivalence elasses of complex line bundies over the surface), elassified according to the dimensions of the associated linear series (or the dimensions of the spaces of analytic cross-sections), are naturally realized as analytic subvarieties of the associated torus. One of the most fruitful of the elassical approaches to this investigation has been by way of theta functions. The space of linear equivalence elasses of positive divisors of order g -1 on a compact connected Riemann surface M of genus g is realized by an irreducible (g -1)-dimensional analytic subvariety, an irreducible hypersurface, of the associated g-dimensional complex torus J(M); this hyper 1 surface W- r;;;, J(M) is the image of the natural mapping Mg- -+J(M), and is g 1 1 birationally equivalent to the (g -1)-fold symmetric product Mg- jSg-l of the Riemann surface M.
Theta Functions With Applications To Riemann Surfaces
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Author : Harry Ernest Rauch
language : en
Publisher:
Release Date : 1974
Theta Functions With Applications To Riemann Surfaces written by Harry Ernest Rauch and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1974 with Functions, Abelian categories.
Analysis Geometry Number Theory The Mathematics Of Leon Ehrenpreis
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Author : Eric Grinberg
language : en
Publisher: American Mathematical Soc.
Release Date : 2000
Analysis Geometry Number Theory The Mathematics Of Leon Ehrenpreis written by Eric Grinberg and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2000 with Mathematics categories.
This book presents the proceedings from the conference honoring the work of Leon Ehrenpreis. Professor Ehrenpreis worked in many different areas of mathematics and found connections among all of them. For example, one can find his analytic ideas in the context of number theory, geometric thinking within analysis, transcendental number theory applied to partial differential equations, and more. The conference brought together the communities of mathematicians working in the areas of interest to Professor Ehrenpreis and allowed them to share the research inspired by his work. The collection of articles here presents current research on PDEs, several complex variables, analytic number theory, integral geometry, and tomography. The work of Professor Ehrenpreis has contributed to basic definitions in these areas and has motivated a wealth of research results. This volume offers a survey of the fundamental principles that unified the conference and influenced the mathematics of Leon Ehrenpreis.
Extremal Riemann Surfaces
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Author : John R. Quine
language : en
Publisher: American Mathematical Soc.
Release Date : 1997
Extremal Riemann Surfaces written by John R. Quine and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 1997 with Mathematics categories.
Other papers deal with maximizing or minimizing functions defined by the spectrum such as the heat kernel, the zeta function, and the determinant of the Laplacian, some from the point of view of identifying an extremal metric.