Hilbert Modular Forms With Coefficients In Intersection Homology And Quadratic Base Change

Author : Jayce Getz,Mark Goresky
Publisher : Birkhäuser
Page : 258 pages
File Size : 41,5 Mb
Release : 2012-04-05
Category : Mathematics
ISBN : 3034803524

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Hilbert Modular Forms with Coefficients in Intersection Homology and Quadratic Base Change by Jayce Getz,Mark Goresky Pdf

In the 1970s Hirzebruch and Zagier produced elliptic modular forms with coefficients in the homology of a Hilbert modular surface. They then computed the Fourier coefficients of these forms in terms of period integrals and L-functions. In this book the authors take an alternate approach to these theorems and generalize them to the setting of Hilbert modular varieties of arbitrary dimension. The approach is conceptual and uses tools that were not available to Hirzebruch and Zagier, including intersection homology theory, properties of modular cycles, and base change. Automorphic vector bundles, Hecke operators and Fourier coefficients of modular forms are presented both in the classical and adèlic settings. The book should provide a foundation for approaching similar questions for other locally symmetric spaces.

Author : Jayce Getz
Publisher : Unknown
Page : 196 pages
File Size : 41,5 Mb
Release : 2007
Category : Electronic
ISBN : WISC:89098699309

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Intersection Homology of Hilbert Modular Varieties and Quadratic Base Change by Jayce Getz Pdf

Author : Jayce Getz,Mark Goresky
Publisher : Springer Science & Business Media
Page : 264 pages
File Size : 46,6 Mb
Release : 2012-03-28
Category : Mathematics
ISBN : 9783034803519

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Hilbert Modular Forms with Coefficients in Intersection Homology and Quadratic Base Change by Jayce Getz,Mark Goresky Pdf

In the 1970s Hirzebruch and Zagier produced elliptic modular forms with coefficients in the homology of a Hilbert modular surface. They then computed the Fourier coefficients of these forms in terms of period integrals and L-functions. In this book the authors take an alternate approach to these theorems and generalize them to the setting of Hilbert modular varieties of arbitrary dimension. The approach is conceptual and uses tools that were not available to Hirzebruch and Zagier, including intersection homology theory, properties of modular cycles, and base change. Automorphic vector bundles, Hecke operators and Fourier coefficients of modular forms are presented both in the classical and adèlic settings. The book should provide a foundation for approaching similar questions for other locally symmetric spaces.

Author : Benjamin Howard,Tonghai Yang
Publisher : Springer Science & Business Media
Page : 146 pages
File Size : 54,9 Mb
Release : 2012-01-06
Category : Mathematics
ISBN : 9783642239786

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Intersections of Hirzebruch–Zagier Divisors and CM Cycles by Benjamin Howard,Tonghai Yang Pdf

This monograph treats one case of a series of conjectures by S. Kudla, whose goal is to show that Fourier of Eisenstein series encode information about the Arakelov intersection theory of special cycles on Shimura varieties of orthogonal and unitary type. Here, the Eisenstein series is a Hilbert modular form of weight one over a real quadratic field, the Shimura variety is a classical Hilbert modular surface, and the special cycles are complex multiplication points and the Hirzebruch-Zagier divisors. By developing new techniques in deformation theory, the authors successfully compute the Arakelov intersection multiplicities of these divisors, and show that they agree with the Fourier coefficients of derivatives of Eisenstein series.

Author : Tim Browning
Publisher : Springer Nature
Page : 175 pages
File Size : 52,6 Mb
Release : 2021-11-19
Category : Mathematics
ISBN : 9783030868727

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Cubic Forms and the Circle Method by Tim Browning Pdf

The Hardy–Littlewood circle method was invented over a century ago to study integer solutions to special Diophantine equations, but it has since proven to be one of the most successful all-purpose tools available to number theorists. Not only is it capable of handling remarkably general systems of polynomial equations defined over arbitrary global fields, but it can also shed light on the space of rational curves that lie on algebraic varieties. This book, in which the arithmetic of cubic polynomials takes centre stage, is aimed at bringing beginning graduate students into contact with some of the many facets of the circle method, both classical and modern. This monograph is the winner of the 2021 Ferran Sunyer i Balaguer Prize, a prestigious award for books of expository nature presenting the latest developments in an active area of research in mathematics.

Author : Giovanni Catino,Paolo Mastrolia
Publisher : Springer Nature
Page : 247 pages
File Size : 52,5 Mb
Release : 2020-10-23
Category : Mathematics
ISBN : 9783030571856

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A Perspective on Canonical Riemannian Metrics by Giovanni Catino,Paolo Mastrolia Pdf

This book focuses on a selection of special topics, with emphasis on past and present research of the authors on “canonical” Riemannian metrics on smooth manifolds. On the backdrop of the fundamental contributions given by many experts in the field, the volume offers a self-contained view of the wide class of “Curvature Conditions” and “Critical Metrics” of suitable Riemannian functionals. The authors describe the classical examples and the relevant generalizations. This monograph is the winner of the 2020 Ferran Sunyer i Balaguer Prize, a prestigious award for books of expository nature presenting the latest developments in an active area of research in mathematics.

Author : Xavier Tolsa
Publisher : Springer Science & Business Media
Page : 402 pages
File Size : 49,5 Mb
Release : 2013-12-16
Category : Mathematics
ISBN : 9783319005966

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Analytic Capacity, the Cauchy Transform, and Non-homogeneous Calderón–Zygmund Theory by Xavier Tolsa Pdf

This book studies some of the groundbreaking advances that have been made regarding analytic capacity and its relationship to rectifiability in the decade 1995–2005. The Cauchy transform plays a fundamental role in this area and is accordingly one of the main subjects covered. Another important topic, which may be of independent interest for many analysts, is the so-called non-homogeneous Calderón-Zygmund theory, the development of which has been largely motivated by the problems arising in connection with analytic capacity. The Painlevé problem, which was first posed around 1900, consists in finding a description of the removable singularities for bounded analytic functions in metric and geometric terms. Analytic capacity is a key tool in the study of this problem. In the 1960s Vitushkin conjectured that the removable sets which have finite length coincide with those which are purely unrectifiable. Moreover, because of the applications to the theory of uniform rational approximation, he posed the question as to whether analytic capacity is semiadditive. This work presents full proofs of Vitushkin’s conjecture and of the semiadditivity of analytic capacity, both of which remained open problems until very recently. Other related questions are also discussed, such as the relationship between rectifiability and the existence of principal values for the Cauchy transforms and other singular integrals. The book is largely self-contained and should be accessible for graduate students in analysis, as well as a valuable resource for researchers.

Author : Veronique Fischer,Michael Ruzhansky
Publisher : Birkhäuser
Page : 557 pages
File Size : 45,8 Mb
Release : 2016-03-08
Category : Mathematics
ISBN : 9783319295589

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Quantization on Nilpotent Lie Groups by Veronique Fischer,Michael Ruzhansky Pdf

This book presents a consistent development of the Kohn-Nirenberg type global quantization theory in the setting of graded nilpotent Lie groups in terms of their representations. It contains a detailed exposition of related background topics on homogeneous Lie groups, nilpotent Lie groups, and the analysis of Rockland operators on graded Lie groups together with their associated Sobolev spaces. For the specific example of the Heisenberg group the theory is illustrated in detail. In addition, the book features a brief account of the corresponding quantization theory in the setting of compact Lie groups. The monograph is the winner of the 2014 Ferran Sunyer i Balaguer Prize.

Author : Michael Ruzhansky,Durvudkhan Suragan
Publisher : Springer
Page : 579 pages
File Size : 54,9 Mb
Release : 2019-07-02
Category : Mathematics
ISBN : 9783030028954

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Hardy Inequalities on Homogeneous Groups by Michael Ruzhansky,Durvudkhan Suragan Pdf

This open access book provides an extensive treatment of Hardy inequalities and closely related topics from the point of view of Folland and Stein's homogeneous (Lie) groups. The place where Hardy inequalities and homogeneous groups meet is a beautiful area of mathematics with links to many other subjects. While describing the general theory of Hardy, Rellich, Caffarelli-Kohn-Nirenberg, Sobolev, and other inequalities in the setting of general homogeneous groups, the authors pay particular attention to the special class of stratified groups. In this environment, the theory of Hardy inequalities becomes intricately intertwined with the properties of sub-Laplacians and subelliptic partial differential equations. These topics constitute the core of this book and they are complemented by additional, closely related topics such as uncertainty principles, function spaces on homogeneous groups, the potential theory for stratified groups, and the potential theory for general Hörmander's sums of squares and their fundamental solutions. This monograph is the winner of the 2018 Ferran Sunyer i Balaguer Prize, a prestigious award for books of expository nature presenting the latest developments in an active area of research in mathematics. As can be attested as the winner of such an award, it is a vital contribution to literature of analysis not only because it presents a detailed account of the recent developments in the field, but also because the book is accessible to anyone with a basic level of understanding of analysis. Undergraduate and graduate students as well as researchers from any field of mathematical and physical sciences related to analysis involving functional inequalities or analysis of homogeneous groups will find the text beneficial to deepen their understanding.

Author : Urtzi Buijs,Yves Félix,Aniceto Murillo,Daniel Tanré
Publisher : Springer Nature
Page : 283 pages
File Size : 54,5 Mb
Release : 2020-12-15
Category : Mathematics
ISBN : 9783030544300

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Lie Models in Topology by Urtzi Buijs,Yves Félix,Aniceto Murillo,Daniel Tanré Pdf

Since the birth of rational homotopy theory, the possibility of extending the Quillen approach – in terms of Lie algebras – to a more general category of spaces, including the non-simply connected case, has been a challenge for the algebraic topologist community. Despite the clear Eckmann-Hilton duality between Quillen and Sullivan treatments, the simplicity in the realization of algebraic structures in the latter contrasts with the complexity required by the Lie algebra version. In this book, the authors develop new tools to address these problems. Working with complete Lie algebras, they construct, in a combinatorial way, a cosimplicial Lie model for the standard simplices. This is a key object, which allows the definition of a new model and realization functors that turn out to be homotopically equivalent to the classical Quillen functors in the simply connected case. With this, the authors open new avenues for solving old problems and posing new questions. This monograph is the winner of the 2020 Ferran Sunyer i Balaguer Prize, a prestigious award for books of expository nature presenting the latest developments in an active area of research in mathematics.

Author : Vladimir Turaev,Alexis Virelizier
Publisher : Birkhäuser
Page : 523 pages
File Size : 53,7 Mb
Release : 2017-06-28
Category : Mathematics
ISBN : 9783319498348

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Monoidal Categories and Topological Field Theory by Vladimir Turaev,Alexis Virelizier Pdf

This monograph is devoted to monoidal categories and their connections with 3-dimensional topological field theories. Starting with basic definitions, it proceeds to the forefront of current research. Part 1 introduces monoidal categories and several of their classes, including rigid, pivotal, spherical, fusion, braided, and modular categories. It then presents deep theorems of Müger on the center of a pivotal fusion category. These theorems are proved in Part 2 using the theory of Hopf monads. In Part 3 the authors define the notion of a topological quantum field theory (TQFT) and construct a Turaev-Viro-type 3-dimensional state sum TQFT from a spherical fusion category. Lastly, in Part 4 this construction is extended to 3-manifolds with colored ribbon graphs, yielding a so-called graph TQFT (and, consequently, a 3-2-1 extended TQFT). The authors then prove the main result of the monograph: the state sum graph TQFT derived from any spherical fusion category is isomorphic to the Reshetikhin-Turaev surgery graph TQFT derived from the center of that category. The book is of interest to researchers and students studying topological field theory, monoidal categories, Hopf algebras and Hopf monads.

Author : Angel Cano,Juan Pablo Navarrete,José Seade
Publisher : Springer Science & Business Media
Page : 288 pages
File Size : 48,5 Mb
Release : 2012-11-05
Category : Mathematics
ISBN : 9783034804813

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Complex Kleinian Groups by Angel Cano,Juan Pablo Navarrete,José Seade Pdf

This monograph lays down the foundations of the theory of complex Kleinian groups, a newly born area of mathematics whose origin traces back to the work of Riemann, Poincaré, Picard and many others. Kleinian groups are, classically, discrete groups of conformal automorphisms of the Riemann sphere, and these can be regarded too as being groups of holomorphic automorphisms of the complex projective line CP1. When going into higher dimensions, there is a dichotomy: Should we look at conformal automorphisms of the n-sphere?, or should we look at holomorphic automorphisms of higher dimensional complex projective spaces? These two theories are different in higher dimensions. In the first case we are talking about groups of isometries of real hyperbolic spaces, an area of mathematics with a long-standing tradition. In the second case we are talking about an area of mathematics that still is in its childhood, and this is the focus of study in this monograph. This brings together several important areas of mathematics, as for instance classical Kleinian group actions, complex hyperbolic geometry, chrystallographic groups and the uniformization problem for complex manifolds.​

Author : Antoine Chambert-Loir,Johannes Nicaise,Julien Sebag
Publisher : Springer
Page : 526 pages
File Size : 54,6 Mb
Release : 2018-09-15
Category : Mathematics
ISBN : 9781493978878

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Motivic Integration by Antoine Chambert-Loir,Johannes Nicaise,Julien Sebag Pdf

This monograph focuses on the geometric theory of motivic integration, which takes its values in the Grothendieck ring of varieties. This theory is rooted in a groundbreaking idea of Kontsevich and was further developed by Denef & Loeser and Sebag. It is presented in the context of formal schemes over a discrete valuation ring, without any restriction on the residue characteristic. The text first discusses the main features of the Grothendieck ring of varieties, arc schemes, and Greenberg schemes. It then moves on to motivic integration and its applications to birational geometry and non-Archimedean geometry. Also included in the work is a prologue on p-adic analytic manifolds, which served as a model for motivic integration. With its extensive discussion of preliminaries and applications, this book is an ideal resource for graduate students of algebraic geometry and researchers of motivic integration. It will also serve as a motivation for more recent and sophisticated theories that have been developed since.

Author : Eberhard Freitag
Publisher : Springer Science & Business Media
Page : 255 pages
File Size : 42,5 Mb
Release : 2013-03-09
Category : Mathematics
ISBN : 9783662026380

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Hilbert Modular Forms by Eberhard Freitag Pdf

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.

Author : Friedrich Hirzebruch,Gerard van der Geer
Publisher : Unknown
Page : 200 pages
File Size : 44,7 Mb
Release : 1981
Category : Discontinuous groups
ISBN : UOM:39015015619466

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Lectures on Hilbert Modular Surfaces by Friedrich Hirzebruch,Gerard van der Geer Pdf