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　　Felix Klein著名的 Erlangen 綱領使得群作用理論成為數學的核心部分。在此綱領的精神下，Felix Klein開始一個偉大的計劃，就是撰寫一係列著作將數學各領域包括數論、幾何、復分析、離散子群等統一起來。他的一本著作是《二十麵體和十五次方程的解》於1884年齣版，4年後翻譯成英文版，它將三個看似不同的領域——二十麵體的對稱性、十五次方程的解和超幾何函數的微分方程緊密地聯係起來。之後Felix Klein和Robert Fricke閤作撰寫瞭四捲著作，包括橢圓模函數兩捲本和自守函數兩捲本。弗裏剋、剋萊因著季理真主編迪普雷譯的《自守函數理論講義(第1捲)(英文版)(精)》是對一本著作的推廣，內容包含Poincare 和Klein 在自守形式的高度原創性的工作，它們奠定瞭Lie群的離散子群、代數群的算術子群及自守形式的現代理論的基礎，對數學的發展起著巨大的推動作用。

內頁插圖

目錄

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
自守函數理論講義 第一捲 [Lectures on the Theory of Automorphic Functions] 下載 mobi epub pdf txt 電子書

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