椭圆模函数理论讲义 第1卷
作者：（德）克莱因，（德）弗里克 季理真 （美）迪普雷
出版时间：2017年版
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。《椭圆模函数理论讲义 第一卷 》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。
目录
Part I Introduction to the Study of the Elliptic Modular Functions
1 On the invariants of the binary biquadratic form
1．1 The form f(z1， z2) and its irrational invariants
1．2 The irrational invariants A， B， C of the form f
1．3 Behavior of the A， B， C upon varying the sequence of factors of f
1．4 Equivalence of two forms with the same sequence of factors． First canonical form of f
1．5 The equivalence of a form f with itself
1．6 The equivalence of two forms with arbitrary factor sequence
1．7 The rational invariants of the form f
1．8 The rational invariants in explicit form
1．9 The invariants g2， g3 and the absolute invariant J
1．10 The second canonical form off
1．11 Geometrical observations on the second canonical form
1．12 Third conical form off
1．13 More on the third canonical form． Connection to the theory of the regular solids
1．14 Normal forms of the elliptic integral of the first kind
1．15 Naming the normal forms． History
2 On the periods of the elliptic integral of the first kind
2．1 Pairs of primitive periods of the integral of the first kind
2．2 The periods as invariants． Dependency on the rational invariants．Normalization of the periods
2．3 Setting up the differential equation for the normalized periods
2．4 Fundamental theorems concerning the dependency of the normalized periods on J
2．5 Choice of a special primitive period-pair
2．6 Dissection of the J-plane． Significance of the determinations of the previous paragraph
2．7 Approach to the neighborhood of a singular point
2．8 Preliminary determination of the numbers kl， k2
2．9 Carrying though the investigation for the singular point J = 0
2．10 Carrying the investigation through for the singular point J = 1
2．11 Determinations for the neighborhood of J = oo and associated calculation of Ω2
2．12 Calculation of the limiting value offal for J —— oo
2．13 Disposal of the singular point J = oo． Historical remarks
2．14 Branching of the periods tobto2 over the/-plane
2．15 The periodquotient to as a function ofJ
2．16 Differential equation of the third order for to(J)． The s-functions
Concerning certain conformal mappings and the triangle functions arising from them
3．1 Replacement of the Riemann surface occurring by simpler figures
3．2 Figure for the representation of the connection between A and J
3．3 Carrying the λ-plane onto the surface of the sphere
3．4 Relation to the second chapter
3．5 Figures for illustrating the connection between u and J
3．6 The mapping of a circular arc triangle onto the halfplane of J
3．7 The relation of A to u illustrated through figures
3．8 The circle-relation． Theorems on circular arc triangles
3．9 The symmetry with respect to a circle
3．10 The law of symmetry． Direct and indirect circle-relatedness
3．11 Significance of the law of symmetry for the function u(J)
3．12 General investigation of the function-theoretic significance of the law of symmetry
3．13 Definition and fundamental properties of the triangle- or s-functions
3．14 Series developments for a branch of the s-function
3．15 Differential equation of the third order for the s-function
3．16 Assembly of the triangle functions already appearing
3．17 Division of the triangle functions into kinds
3．18 The s-functions of the first kind
3．19 The s-functions of the second kind
3．20 The s-functions of the third kind
3．21 The triangle figures associated to w(λ) and w(J)
4 Development of the definitions and fundamental problems of a theory of the elliptic modular functions
4．1 The Legendre relation
4．2 The rational invariants g2， g3，△ as functions of the periods to1， to2
4．3 Functional determinants of the forms g2， g3， △
4．4 The periods of integrals of the second kind as functions of w1， w2
4．5 Calculation of the Hessian determinant H(logA)
4．6 Mappings effected by to(J) and ((J)． Icosahedral and modular equation
Part II Treatment of the Group-Theoretic Fundamental Problem
Part III The Function-Theoretic Fundamental Problem
Commentaries
