Quantization And Non Holomorphic Modular Forms

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Quantization And Non Holomorphic Modular Forms

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Author : André Unterberger
language : en
Publisher: Springer
Release Date : 2007-05-06

Quantization And Non Holomorphic Modular Forms written by André Unterberger and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2007-05-06 with Mathematics categories.

This is a new approach to the theory of non-holomorphic modular forms, based on ideas from quantization theory or pseudodifferential analysis. Extending the Rankin-Selberg method so as to apply it to the calculation of the Roelcke-Selberg decomposition of the product of two Eisenstein series, one lets Maass cusp-forms appear as residues of simple, Eisenstein-like, series. Other results, based on quantization theory, include a reinterpretation of the Lax-Phillips scattering theory for the automorphic wave equation, in terms of distributions on R2 automorphic with respect to the linear action of SL(2,Z).

Quantization And Non Holomorphic Modular Forms

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Author : André Unterberger
language : en
Publisher:
Release Date : 2014-01-15

Quantization And Non Holomorphic Modular Forms written by André Unterberger and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-01-15 with categories.

Pseudodifferential Analysis Automorphic Distributions In The Plane And Modular Forms

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Author : André Unterberger
language : en
Publisher: Springer Science & Business Media
Release Date : 2011-08-06

Pseudodifferential Analysis Automorphic Distributions In The Plane And Modular Forms written by André Unterberger 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 2011-08-06 with Mathematics categories.

Pseudodifferential analysis, introduced in this book in a way adapted to the needs of number theorists, relates automorphic function theory in the hyperbolic half-plane Π to automorphic distribution theory in the plane. Spectral-theoretic questions are discussed in one or the other environment: in the latter one, the problem of decomposing automorphic functions in Π according to the spectral decomposition of the modular Laplacian gives way to the simpler one of decomposing automorphic distributions in R2 into homogeneous components. The Poincaré summation process, which consists in building automorphic distributions as series of g-transforms, for g E SL(2;Z), of some initial function, say in S(R2), is analyzed in detail. On Π, a large class of new automorphic functions or measures is built in the same way: one of its features lies in an interpretation, as a spectral density, of the restriction of the zeta function to any line within the critical strip. The book is addressed to a wide audience of advanced graduate students and researchers working in analytic number theory or pseudo-differential analysis.

Pseudodifferential Operators With Automorphic Symbols

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Author : André Unterberger
language : en
Publisher: Birkhäuser
Release Date : 2015-06-22

Pseudodifferential Operators With Automorphic Symbols written by André Unterberger and has been published by Birkhäuser this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-06-22 with Mathematics categories.

The main results of this book combine pseudo differential analysis with modular form theory. The methods rely for the most part on explicit spectral theory and the extended use of special functions. The starting point is a notion of modular distribution in the plane, which will be new to most readers and relates under the Radon transformation to the classical one of modular form of the non-holomorphic type. Modular forms of the holomorphic type are addressed too in a more concise way, within a general scheme dealing with quantization theory and elementary, but novel, representation-theoretic concepts.

Pseudodifferential Methods In Number Theory

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Author : André Unterberger
language : en
Publisher: Birkhäuser
Release Date : 2018-07-16

Pseudodifferential Methods In Number Theory written by André Unterberger and has been published by Birkhäuser this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-07-16 with Mathematics categories.

Classically developed as a tool for partial differential equations, the analysis of operators known as pseudodifferential analysis is here regarded as a possible help in questions of arithmetic. The operators which make up the main subject of the book can be characterized in terms of congruence arithmetic. They enjoy a Eulerian structure, and are applied to the search for new conditions equivalent to the Riemann hypothesis. These consist in the validity of certain parameter-dependent estimates for a class of Hermitian forms of finite rank. The Littlewood criterion, involving sums of Möbius coefficients, and the Weil so-called explicit formula, which leads to his positivity criterion, fit within this scheme, using in the first case Weyl's pseudodifferential calculus, in the second case Fuchs'. The book should be of interest to people looking for new possible approaches to the Riemann hypothesis, also to new perspectives on pseudodifferential analysis and on the way it combines with modular form theory. Analysts will have no difficulty with the arithmetic aspects, with which, save for very few exceptions, no previous acquaintance is necessary.

Automorphic Pseudodifferential Analysis And Higher Level Weyl Calculi

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Author : André Unterberger
language : en
Publisher: Birkhäuser
Release Date : 2012-12-06

Automorphic Pseudodifferential Analysis And Higher Level Weyl Calculi written by André Unterberger and has been published by Birkhäuser this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.

Award-winning monograph of the Ferran Sunyer i Balaguer Prize 2002. The subject of this book is the study of automorphic distributions, by which is meant distributions on R2 invariant under the linear action of SL(2,Z), and of the operators associated with such distributions under the Weyl rule of symbolic calculus. Researchers and postgraduates interested in pseudodifferential analyis, the theory of non-holomorphic modular forms, and symbolic calculi will benefit from the clear exposition and new results and insights.

Orthogonal Polynomials And Special Functions

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Author : Francisco Marcellàn
language : en
Publisher: Springer
Release Date : 2006-10-18

Orthogonal Polynomials And Special Functions written by Francisco Marcellàn and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-10-18 with Mathematics categories.

Special functions and orthogonal polynomials in particular have been around for centuries. Can you imagine mathematics without trigonometric functions, the exponential function or polynomials? The present set of lecture notes contains seven chapters about the current state of orthogonal polynomials and special functions and gives a view on open problems and future directions.

Splitting Deformations Of Degenerations Of Complex Curves

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Author : Shigeru Takamura
language : en
Publisher: Springer
Release Date : 2006-10-11

Splitting Deformations Of Degenerations Of Complex Curves written by Shigeru Takamura and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-10-11 with Mathematics categories.

Here is a deformation theory for degenerations of complex curves; specifically, discussing deformations which induce splitting of the singular fiber of a degeneration. The author constructs a deformation of the degeneration in such a way that a subdivisor is "barked," or peeled off from the singular fiber. "Barking deformations" are related to deformations of surface singularities, in particular, cyclic quotient singularities, as well as the mapping class groups of Riemann surfaces via monodromies.

The Lace Expansion And Its Applications

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Author : Gordon Slade
language : en
Publisher: Springer Science & Business Media
Release Date : 2006-05-17

The Lace Expansion And Its Applications written by Gordon Slade 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 2006-05-17 with Mathematics categories.

The lace expansion is a powerful and flexible method for understanding the critical scaling of several models of interest in probability, statistical mechanics, and combinatorics, above their upper critical dimensions. These models include the self-avoiding walk, lattice trees and lattice animals, percolation, oriented percolation, and the contact process. This volume provides a unified and extensive overview of the lace expansion and its applications to these models.

The Wulff Crystal In Ising And Percolation Models

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Author : Raphaël Cerf
language : en
Publisher: Springer
Release Date : 2006-08-29

The Wulff Crystal In Ising And Percolation Models written by Raphaël Cerf and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-08-29 with Mathematics categories.

This volume is a synopsis of recent works aiming at a mathematically rigorous justification of the phase coexistence phenomenon, starting from a microscopic model. It is intended to be self-contained. Those proofs that can be found only in research papers have been included, whereas results for which the proofs can be found in classical textbooks are only quoted.