自守函数理论讲义 第二卷 [Lectures on the Theory of Theory of Automorphic Functions,Second Volume] pdf epub mobi txt 电子书 下载 2025

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☆☆☆☆☆

[德] 弗里克，[德] 克莱因 著，[美] 迪普雷 译

图书标签:

• 自守函数
• 自动模形式
• 数论
• 表示论
• 调和分析
• 数学分析
• 复分析
• 朗兰兹纲领
• 模形式
• 代数数论

出版社： 高等教育出版社

ISBN：9787040478396

版次：1

商品编码：12244746

包装：精装

丛书名： 数学经典论题

外文名称：Lectures on the Theory of Theory of Automorphic Functions,Second Volume

开本：16开

出版时间：2017-11-01

用纸：胶

内容简介

　　Felix Klein著名的Erlangen纲领使得群作用理论成为数学的核心部分。在此纲领的精神下，Felix Klein开始一个伟大的计划，就是撰写一系列著作将数学各领域包括数论、几何、复分析、离散子群等统一起来。他的一本著作是《二十面体和十五次方程的解》于1884年出版，4年后翻译成英文版，它将三个看似不同的领域——二十面体的对称性、十五次方程的解和超几何函数的微分方程紧密地联系起来。之后Felix Klein和Robert Fricke合作撰写了四卷著作，包括椭圆模函数两卷本和自守函数两卷本。弗里克、克莱因著季理真主编迪普雷译的《自守函数理论讲义（第2卷）（英文版）（精）》是对一本著作的推广，内容包含Poincare和Klein在自守形式的高度原创性的工作，它们奠定了Lie群的离散子群、代数群的算术子群及自守形式的现代理论的基础，对数学的发展起着巨大的推动作用。

内页插图

目录

Preface
Part I Narrower theory of the single-valued automorphic functions of one variable
Concept， existence and fundamental properties of the automorphic functions
1．1 Definition of the automorphic functions
1．2 Production of an elementary potential of the second kind belonging
to the fundamental domain
1．3 Production of automorphic functions of the group F
1．4 Mapping of the fundamental domain P onto a closed Riemann surface
1．5 The totality of all automorphic functions belonging to a group F and their principal properties
1．6 Classification and closer study of the elementary automorphic functions
1．7 Preparations for the classification of the higher automorphic functions
1．8 Classification and closer study of the higher automorphic functions ．．．
1．9 The integrals of the automorphic models
1．10 General single-valuedness theorem． Application to linear differential equations
1．11 as a linearly polymorphic function． The fundamental problem
1．12 Differential equations of the third order for the polymorphic functions．
1．13 Generalization of the concept of automorphic functions

Form-theoretic discussions for the automorphic models of genus zero
2．1 Shapes of the fundamental domains for the models of genus zero
2．2 Recapitulation of homogeneous variables， substitutions and groups．．．
2．3 General definition of the automorphic forms
2．4 The differentiation process and the principal forms of the models of genus zero
2．5 The family of prime forms and the ground forms for automorphic models with p = 0
2．6 Behavior of the automorphic forms q0d ((1，(2) with respect to the group generators
2．7 The ground forms for the groups of the circular-arc triangles
2．8 The single-valued automorphic forms and their multiplicator systems ．
2．9 The number of all mtflfiplicator systems M for a given group F
2．10 Example for the determination of the number of the multiplicator systems M， the effect of secondary relations
2．11 Representation of all unbranched automorphic forms
2．12 Existence theorem for single-valued forms q0d((1，(2) for given multiplicator system M
2．13 Relations between multiplicator systems inverse to one another．
2．14 Integral forms and forms with prescribed poles
2．15 The (1， (2 as linearly-polymorphic forms of the zl， z2
2．16 Other forms of the polymorphic forms． History
2．17 Differential equations of second order for the polymorphic forms of zero dimension
2．18 Invariant form of the differential equation for the polymorphic forms (1， (2
2．19 Series representation of the polymorphic forms in the case n = 3
2．20 Representation of the polymorphic forms in the case n = 3 by definite integrals

Theory of Poincar6 series with special discussions for the models of genus zero
3．1 The approach to the Poincar6 series
3．2 First convergence study of the Poincar6 series
3．3 Behavior of the Poincar6 series at parabolic cusps
3．4 The Poincar6 series of (-2)nd dimension for groups F with boundary curves
3．5 The Poincar6 series of (-2)nd dimension for principal-circle groups With isolatedly situated boundary points
3．6 Convergence of the Poincar6 series of (-2)nd dimension for certain groups Without boundary curves and Without principal circle
3．7 Second convergence study in the principal-circle case． Continuous dependence of the Poincar6 series on the group moduli
3．8 Poles of the Poincar6 series and the possibility of its vanishing identically． Discussion for the case p = 0
3．9 Construction of one-pole Poincar6 series
3．10 One-poled series with poles at elliptic vertices
3．11 Introduction of the elementary forms ~ ((1， (2; ~ 1， ~2)
3．12 Behavior of the elementary form ~2((1，(2;(1，(2) at a parabolic cusp ( ．．
3．13 Behavior of the elementary forms upon exercise of substitutions of the group F on (1， (2． Discussions for the models of genus p —— 0
3．14 Concerning the representability of arbitrary automorphic forms of genus zero by the elementary forms and the Poincar6 series
The automorphic forms and their analytic representations for models of arbitrary genus
4．1 Recapitulation concerning the groups of arbitrary genus p and their generation
4．2 Recapitulation and extension of the theory of the primeform for an arbitrary algebraic model
4．3 The polymorphic forms (1， (2 for a model of arbitrary genus p
4．4 Differential equations of the polymorphic functions and forms for models with p > 0
4．5 Representation of all unbranched automorphic forms of a group F of arbitrary genus by the prime-and groundforms
4．6 The single-valued automorphic forms and their multiplicator systems for a group of arbitrary genus
4．7 Existence of the single-valued forms for a given multiplicator system in the case of an arbitrary genus
4．8 More on single-valued automorphic forms for arbitrary p． The p forms z~-2((1，~'2)
4．9 Concept of conjugate forms． Extended Riemann-Roch theorem and applications of it
4．10 The Poincard series and the elementary forms for p． Unimultiplicative forms
4．11 Two-poled series of (——2)nd dimension and integrals of the 2nd kind for automorphic models of arbitrary genus p
4．12 The integrals of the first and third kinds． Product representation for the primeform ~
4．13 On the representability of the automorphic forms of arbitrary genus p by the elementary forms and the Poincard series
4．14 Closing remarks

Part II Fundamental theorems concerning the existence of polymorphic functions on Riemann surfaces
1 Continuity studies in the domain of the principal-circle groups
1．1 Recapitulation of the polygon theory of the principal-circle groups
1．2 The polygon continua of the character (0， 3)
1．3 The polygon continua of the character (0， 4)
1．4 The polygon continua of the character (0， n)
1．5 Another representation of the polygon continua of the character (0，4) ．
1．6 The polygon continua of the character (1，1)
1．7 The polygon continua of the character (p， n)
1．8 Transition from the polygon continua to the group continua
1．9 The discontinuity of the modular group
1．10 The reduced polygons of the character (1，1)
1．11 The surface q)3 of third degree coming up for the character (1，1)
1．12 The discontinuity domain of the modular group and the character (1，1)
1．13 Connectivity and boundary of the individual group continuum of the character (1，1)
1．14 The reduced polygons of the character (0， 4)
1．15 The surfaces ~a of the third degree coming up for the character (0，4) ．．
1．16 The discontinuity domain of the modular group and the group continua of the character (0， 4)
1．17 Boundary and connectivity of the individual group continuum of the character (0， 4)
1．18 The normal and the reduced polygons of the character (0， n)
1．19 The continua of the reduced polygons of the character (0， n) for given vertex invariants and fixed vertex arrangement
1．20 The discontinuity domain of the modular group and the group continua of the character (0， n)
1．21 The group continua of the character (p， n)
1．22 Report on the continua of the Riemann surfaces of the genus p
1．23 Report on the continua of the symmetric Riemann surfaces of the genus p
1．24 Continuity of the mapping between the continuum of groups and the continuum of Riemann surfaces
1．25 Single-valuedness of the mapping between the continuum of groups and the continuum of Riemann surfaces
1．26 Generalities on the continuity proof of the fundamental theorem in the domain of the principal-circle groups
1．27 Effectuation of the continuity proof for the signature (0， 3; ll， la)
1．28 Effectuation of the continuity proof for the signature (0， 3; ll)
1．29 Effectuation of the continuity proof for the signature (1，1; 11)
1．30 Effectuation of the continuity proof for the signature (0， 3)
1．31 Representation of the three-dimensional continua Bg and Bf for the signature (1，1)
1．32 Effectuation of the continuity proof for the signature (1，1)

Proof of the principal-circle and the boundary-circle theorem
2．1 Historical information concerning the direct methods of proof of the fundamental theorems
2．2 Theorems on logarithmic potentials and Green's functions
2．3 More on the solution of the boundary-value problem
2．4 The Green's function of a simply connected domain
2．5 Two theorems of Koebe
2．6 Production of the covering surface F~ in the boundary-circle case
2．7 Production of the covering surface in the principal-circle case
2．8 The Green's functions of the domain Fv and their convergence in the principal-circle case
2．9 Mapping of the covering surface onto a circular disc． Proof of the principal-circle theorem
2．10 Introduction of new series of functions in the boundary-circle case
2．11 Connection of the limit functions ur， u" with one another and with Green's functions u~
2．12 Mapping of the covering surface by means of the function

Proof of the boundary-circle theorem
Proof of the reentrant cut theorem
3．1 Theorems on schlicht infinite images of a circular surface
3．2 Theorems on schlicht finite models of a circular surface
3．3 The distortion theorem for circular domains
3．4 The distortion theorem for arbitrary domains
3．5 Consequences of the distortion theorem
3．6 Production of the covering surface Foo for a Riemann surface provided with p reentrant cuts
3．7 Mapping of the surface Fn onto a schlicht domain for special reentrant cuts
3．8 Mapping of the surface Fn onto a schlicht domain for arbitrary reentrant cuts
3．9 Introduction of a system of analytic transformations belonging to the domain Pn
3．10 Application of the distortion theorem to the domain Pn
3．11 Application of the consequences of the distortion theorem to the domain Pn
3．12 Effectuation of the convergence proof of the functions r/n (z)
3．13 Proof of the linearity theorem
3．14 Proof of the unicity theorem． Proof of the reentrant cut theorem
3．15 Koebe's proof of the general Kleinian fundamental theorem

An addition to the transformation theory of automorphic functions
A． 1 General approach to the transformation of single-valued automorphic functions
A．2 The arithmetic character of the group of the signature (0， 3; 2， 4， 5)
A．3 Introduction of the transformation of third degree
A．4 Setting up the transformation equation of tenth degree
A．5 The Galois group of the transformation equation and its cyclic subgroups
A．6 The non-cyclic subgroups of the Ga60 and the extended G720
A．7 The two resolvents of sixth degree of the transformation equation
A．8 The discontinuity domains of the F15 and F30 belonging to the octahedral and tetrahedral groups
A．9 The two resolvents of the 15th degree of the transformation equation ．．
A．10 Note on the grups F20 belonging to the ten conjugate G18
A．11 The Riemann surface of the Galois resolvent of the transformation equation
A．12 The curve C6 in the octahedral coordinate system
A．13 The curve C6 in the icosahedral coordinate system
A．14 The curve C6 in the harmonic coordinate system
A．15 The real traces of the C6 and the character of the points a， b， c
A．16 Further geometrical theorems on the collineation group G360
A．17 The Galois resolvent of the transformation equation
A．18 The solution of the resolvents of 6th and 15th degree
A．19 Solution of the transformation equation of 10th degree

Commentaries
1 Commentary by Richard Borcherds on Elliptic Modular Functions
2 Commentary by leremy Gray
3 Commentary by William Harvey on Automorphic Functions
4 Commentary by Barry Mazur
5 Commentary by Series-Mumford-Wright
6 Commentary by Domingo Toledo
7 Commentaries by Other Mathematicians

好的，以下是根据您的要求，针对一本假设存在的、名为《自守函数理论讲义 第二卷》的书籍，撰写的详细且不包含任何原主题内容的图书简介。 --- 《黎曼曲面与模空间导论》 第一版 | 2024年 作者：[在此处填写一位虚构的数学家姓名] 出版社：[在此处填写一家虚构的学术出版社名称] --- 内容简介：跨越几何与代数的桥梁 本书《黎曼曲面与模空间导论》旨在为数学、理论物理及相关领域的研究生和高级本科生提供一个全面且深入的框架，用以理解现代几何分析中的两大核心构件：黎曼曲面（Riemann Surfaces）及其相关的模空间（Moduli Spaces）。全书共分为五个主要部分，结构严谨，逻辑清晰，旨在将抽象的拓扑概念与具体的分析工具相结合，为读者构建起一座从基础到前沿的坚实桥梁。 本书的特色在于其独特的叙事方式，它不满足于仅介绍既有理论，更致力于探究这些理论背后的几何直觉和代数结构。我们从最基础的拓扑概念出发，逐步引入复分析的强大工具，最终导向对高维几何对象的深刻洞察。 --- 第一部分：黎曼曲面的拓扑基础与复结构（The Topological and Complex Foundations of Riemann Surfaces） 本部分聚焦于黎曼曲面的底层结构。我们首先回顾紧致流形的基本概念，特别是二维实流形上的拓扑不变量，如欧拉示性数和基本群。重点将放在如何通过拓扑结构来分类所有可能的黎曼曲面类型上。 随后，我们将引入局部坐标和复结构的概念。详细讨论了哪些拓扑结构可以“承载”一个良定义的复解析结构。我们深入研究了完备性的概念，并阐述了椭圆曲线（Genus 1的黎曼曲面）作为最简单非平凡案例的全面分析。读者将学习如何利用双曲几何的视角来理解单连通黎曼曲面（如单位圆盘），并掌握狄利克雷有限区域（Dirichlet Fundamental Domains）的概念，这是理解模空间结构的关键先驱。 第二部分：调和分析与微分形式（Harmonic Analysis and Differential Forms on Surfaces） 在建立了黎曼曲面的框架之后，本部分将引入分析学的工具。我们详尽阐述了德拉姆上同调（de Rham Cohomology）在曲面上的应用，并展示了拓扑（奇异上同调群）与微分形式（闭微分形式）之间的深刻联系。 核心内容聚焦于调和微分形式（Harmonic Differentials）。读者将学习霍奇分解（Hodge Decomposition）在曲面上的具体表现形式，理解其对曲面结构信息编码的重要性。我们详细讨论了韦纳-科斯塔斯定理（Weitzenböck’s formula）及其在计算曲面几何量（如面积和黎曼度量）中的应用。通过对Green’s Functions在曲面上的性质的分析，本部分为理解调和函数理论奠定了坚实的基础。 第三部分：自伴随算子与谱理论（Self-Adjoint Operators and Spectral Theory） 本部分将目光投向了曲面上的微分算子，特别是拉普拉斯-贝特拉米算子（Laplace-Beltrami Operator）。我们将分析该算子在不同边界条件下的谱性质。 专题讨论了黎曼曲面的谱（The Spectrum of a Riemann Surface）及其与曲面几何的关联。读者将学习如何利用谱几何的视角来理解曲面的拓扑和度量信息，特别是Weyl律和克罗夫-拉特勒不等式。本部分详尽分析了热核展开（Heat Kernel Expansion）在曲面上的性质，并探讨了谱不变量如何决定曲面的基本特征，例如其体积和面积。 第四部分：模空间的构造与几何（Construction and Geometry of Moduli Spaces） 这是本书的理论高峰，将前三部分的内容融会贯通，引入模空间的概念。我们首先明确模空间的定义：它是对一族具有特定几何结构的对象的集合，并赋予其一个拓扑结构（通常是拓扑或复拓扑）。 本书主要关注亏格 $g$ 的黎曼曲面模空间 $mathcal{M}_g$ 的构造。我们将采用Teichmüller空间作为起点，详细介绍如何通过Fenchel-Nielsen坐标来参数化具有特定边界长度和扭转角的双曲结构。随后，通过对Weil–Petersson度量的介绍，我们探讨了模空间的内蕴几何，并分析了其奇异性结构，特别是自交曲线（non-separating geodesics）所导致的边界。 第五部分：模空间的紧化与应用（Compactification and Applications of Moduli Spaces） 为了在 $mathcal{M}_g$ 上进行分析，通常需要引入紧化（Compactification）的概念。本部分详细讨论了Deligne–Mumford 紧化 $overline{mathcal{M}}_g$ 的构造，它包含了带尖点（nodal curves）的黎曼曲面。读者将深入理解尖点如何通过“收缩”一个短的测地线来生成，并学习如何构造这些尖点处的局部坐标系。 最后，本书将简要回顾模空间理论在现代数学物理中的应用，包括其与弦理论中界面几何的联系，以及在代数几何中对曲线模的刻画。通过对黎曼曲面及其模空间的全面、细致的解析，本书旨在使读者不仅掌握理论工具，更能培养出对高维复几何结构深刻而直观的理解。 --- 目标读者： 具有扎实的复分析、微分几何和基础拓扑学背景的研究生、博士后及专业研究人员。 先决条件： 读者应熟悉复变函数论、基础拓扑学和经典微分几何的基本概念。 ---

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《自守函数理论讲义 第二卷》是一本让我充满期待的书。我对手写的笔记风格总是情有独钟，因为它常常蕴含着作者在教学过程中最真挚的想法和最精辟的讲解。这本书如果能保留一部分这样的风格，无疑会大大增强其亲切感和引导性。我曾听说自守函数理论在近现代数学中扮演着至关重要的角色，尤其是在数论和代数几何的深刻联系上。因此，我非常期待在这本书中看到作者如何巧妙地揭示这些联系，并以一种清晰易懂的方式呈现给读者。无论是复杂的积分表示，还是奇妙的模形式性质，亦或是与某些特殊函数（例如theta函数）的联系，都是我急切想了解的内容。我相信，这本讲义的深度和广度，定能满足我作为一名对数学理论充满好奇的读者的需求。

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这本《自守函数理论讲义 第二卷》真是让人爱不释手！从封面设计到纸张触感，都透露着一股严谨而又不失艺术的气息，这无疑为阅读体验打下了坚实的基础。我尤其欣赏它在排版上的用心，每一页的布局都经过深思熟虑，文字清晰，公式规范，让我能够毫无阻碍地沉浸在自守函数的世界里。虽然我尚未深入到每一个定理的细节，但仅仅是翻阅目录和初步浏览，就能感受到作者驾驭这宏大理论的深厚功力。那些抽象的概念，在作者的笔下，似乎变得触手可及。我特别期待书中对某些关键概念的深入阐释，比如黎曼曲面与自守群之间的深刻联系，以及在数论、几何学等领域中自守函数的应用。这种理论与实践的结合，往往是打开更深层理解之门的钥匙。这本书不仅是一本教材，更像是一份精心准备的学术盛宴，让人跃跃欲试，渴望在其中汲取知识的养分。