书籍详情
θ常数， 黎曼面和模群（影印版）
作者：Hershel M.Farkas,Irwin Kra
出版社：高等教育出版社
出版时间：2019-03-01
ISBN：9787040469042
定价：¥199.00
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内容简介
《θ常数，黎曼面和模群（影印版）》用代数几何的思想和方法来研究θ函数和数论，促进了这些领域的长足进步。但是，作者选择停留在古典观点上。因此，他们的陈述和证明都非常具体。熟悉θ函数和数论的代数几何方法的数学家们，会在书中发现许多有趣想法，以及关于新老结果的详尽解释和推导。该书精彩的部分包括对θ常数恒等式的系统研讨、由模群子群表示的曲面单值化、分拆等式，以及自守函数的傅里叶系数等。该书的预备知识要求对复分析有扎实的理解，熟悉黎曼面、Fuchs群以及椭圆函数，还要对数论感兴趣。该书包含对一些所需材料（尤其是关于θ函数和θ常数）的概述。读者会在书中发现对分析和数论的古典观点的细致论述。该书包含了大量研究级水平的例题和建议，很适合用作研究生教材或者自学。
作者简介
暂缺《θ常数， 黎曼面和模群（影印版）》作者简介
目录
Introduction
Chapter 1．The modular group and elliptic function theory
1．Mobius transformations
2．Riemann surfaces
3．Kleinian groups
3．1．Generalities
3．2．The situation of interest
4．The elliptic paradise
4．1．The family of tori
4．2．The algebraic curve associated to a torus
4．3．Invariants for tori
4．4．Tori with symmetries
4．5．Congruent numbers
4．6．The plumbing construction
4．7．Teichmfiller and moduli spaces for tori
4．8．Fiber spaces-the Teichmuller curve
5．Hyperbolic version of elliptic function theory
5．1．Fuchsian representation
5．2．Symmetries of once punctured tori
5．3．The modular group
5．4．Geometric interpretations
5．5．The period of a punctured torus
5．6．The function of degree two on the once punctured torus
5．7．The quasi-Fuchsian representation
6．Subgroups of the modular group
6．1．Basic properties
6．2．Poincare metric on simply connected domains
6．3．Fundamental domains
6．4．The principal congruence subgroups F（k）
6．5．Adjoining translations： The subgroups G（k）
6．6．The Hecke subgroups Fo（k）
6．7．Structure of F（k，k）
6．8．A two parameter family of groups
7．A geometric test for primality
Chapter 2．Theta functions with characteristics
1．Theta functions and theta constants
1．1．Definitions and basic properties
1．2．The transformation formula
1．3．More transformation formulae
2．Characteristics
2．1．Classes of characteristics
2．2．Integral classes of characteristics
2．3．Rational classes of characteristics
……
Chapter 3．Function theory for the modular group Γ and its subgroups
Chapter 4．Theta constant identities
Chapter 5．Partition theory： Ramanujan congruences
Chapter 6．Identities related to partition functions
Chapter 7．Combinatorial and number theoretic applications
Bibliography
Bibliographical Notes
Index
Chapter 1．The modular group and elliptic function theory
1．Mobius transformations
2．Riemann surfaces
3．Kleinian groups
3．1．Generalities
3．2．The situation of interest
4．The elliptic paradise
4．1．The family of tori
4．2．The algebraic curve associated to a torus
4．3．Invariants for tori
4．4．Tori with symmetries
4．5．Congruent numbers
4．6．The plumbing construction
4．7．Teichmfiller and moduli spaces for tori
4．8．Fiber spaces-the Teichmuller curve
5．Hyperbolic version of elliptic function theory
5．1．Fuchsian representation
5．2．Symmetries of once punctured tori
5．3．The modular group
5．4．Geometric interpretations
5．5．The period of a punctured torus
5．6．The function of degree two on the once punctured torus
5．7．The quasi-Fuchsian representation
6．Subgroups of the modular group
6．1．Basic properties
6．2．Poincare metric on simply connected domains
6．3．Fundamental domains
6．4．The principal congruence subgroups F（k）
6．5．Adjoining translations： The subgroups G（k）
6．6．The Hecke subgroups Fo（k）
6．7．Structure of F（k，k）
6．8．A two parameter family of groups
7．A geometric test for primality
Chapter 2．Theta functions with characteristics
1．Theta functions and theta constants
1．1．Definitions and basic properties
1．2．The transformation formula
1．3．More transformation formulae
2．Characteristics
2．1．Classes of characteristics
2．2．Integral classes of characteristics
2．3．Rational classes of characteristics
……
Chapter 3．Function theory for the modular group Γ and its subgroups
Chapter 4．Theta constant identities
Chapter 5．Partition theory： Ramanujan congruences
Chapter 6．Identities related to partition functions
Chapter 7．Combinatorial and number theoretic applications
Bibliography
Bibliographical Notes
Index
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