Felix Klein著名的 Erlangen 綱領(lǐng)使得群作用理論成為數(shù)學(xué)的核心部分。在此綱領(lǐng)的精神下，F(xiàn)elix Klein開(kāi)始一個(gè)偉大的計(jì)劃，就是撰寫(xiě)一系列著作將數(shù)學(xué)各領(lǐng)域包括數(shù)論、幾何、復(fù)分析、離散子群等統(tǒng)一起來(lái)。他的一本著作是《二十面體和十五次方程的解》于1884年出版，4年后翻譯成英文版，它將三個(gè)看似不同的領(lǐng)域——二十面體的對(duì)稱性、十五次方程的解和超幾何函數(shù)的微分方程緊密地聯(lián)系起來(lái)。之后Felix Klein和Robert Fricke合作撰寫(xiě)了四卷著作，包括橢圓模函數(shù)兩卷本和自守函數(shù)兩卷本。弗里克、克萊因著季理真主編迪普雷譯的《自守函數(shù)理論講義(第1卷)(英文版)(精)》是對(duì)一本著作的推廣，內(nèi)容包含Poincare 和Klein 在自守形式的高度原創(chuàng)性的工作，它們奠定了Lie群的離散子群、代數(shù)群的算術(shù)子群及自守形式的現(xiàn)代理論的基礎(chǔ)，對(duì)數(shù)學(xué)的發(fā)展起著巨大的推動(dòng)作用。
Preface
0 Introduction． Developments concerning projective determinations of measure
0．1 The projective determinations of measure in the plane and their division into kinds
0．2 The motions belonging to a determination of measure and symmetric transformations of the plane into itself． The variable ζ in the parabolic case
0．3 Setting up all collineations of the conic section zlz3 - z2 =0 into itself． Behavior of the associated ζ
0．4 The group of the "motion and symmetric transformations" for the hyperbolic and elliptic planes
0．5 General definition of the C-values for the points of the projective plane．
0．6 The C-values in the hyperbolic plane． The ζ-halfplane and the ζ-halfplane
0．7 The hyperbolic determination of measure in the ζ-halfplane and on the ζ-halfsphere
0．8 Remarks on surfaces of constant negative curvature
0．9 Illustrations of the motions of the projective plane into itself by figures．
0．10 The elliptic plane and the ζ-plane resp． ζ-sphere
0．11 Transferring the elliptic determination onto the ζ-plane and ζ-sphere
0．12 The hyperbolic determination of measure in space and the associated "motions"
0．13 Connection of the circle-relations with hyperbolic geometry． The rotation subgroups in hyperbolic space
0．14 Mapping of the hyperbolic space onto the ζ-halfplane
0．15 Concluding remarks to the introduction
Part I Foundations for the theory of the discontinuous groups of linear
substitutions of one variable
1 The discontinuity of groups with illustrations by simple examples
1．1 Distinction between continuous and discontinuous substitution groups
1．2 Distinction of properly and improperly discontinuous substitution groups
1．3 Recapitulation and completion regarding the discontinuity domains of cyclic groups
1．4 The groups of the regular solids and the regular divisions of the elliptic plane
1．5 The division of the ζ-halfplane and the hyperbolic plane belonging to the modular group
1．6 Introduction and extension of the Picard group with complex substitution coefficients
1．7 The tetrahedral division of the ζ-halfsphere belonging to the Picard group
1．8 The discontinuity domain and the generation of the Picard group
1．9 Remarks on subgroups of the Picard group． Historical material
The groups without infinitesimal substitutions and their normal discontinuity domains
2．1 The concept of infinitesimal substitutions
2．2 The proper discontinuity of the groups without infinitesimal substitutions
2．3 Introduction of the concept of the polygon-and the polyhedron-groups
2．4 Introduction of the normal discontinuity domains of the projective plane for rotation groups
2．5 The vertices and edges of the normal polygons for principal circle groups． First part： the corners in the interior of the ellipse
2．6 The vertices and edges of the normal polygons for principal circle groups． Second part： the vertices on and outside the ellipse
2．7 The normal polyhedra in the hyperbolic space and their formation in the interior of the sphere
2．8 The normal polyhedra on and outside the sphere
2．9 The behavior of the polygon groups on the surface of the sphere． First part： General
2．10 Continuation： Special consideration of the groups with boundary curves
2．11 The normal discontinuity domains for the groups consisting of substitutions of the first and second kinds
2．12 Carrying over the normal discontinuity domains onto the ζ-plane and into the (-space． Historical material
3 Further approaches to the geometrical theory of the properly discontinuous groups
3．1 The allowed alteration of the discontinuity domains， in particular for principal circle groups
3．2 Continuation： Allowed alteration of the discontinuity domains for polyhedral groups as well as non-principal circle polygon groups
Part II The geometrical theory of the polygon groups of ζ-substitutions
Part III Arithmetic methods of definition of properly discontinuous groups of ζ-substitutions
Commentaries
Index
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