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自守函数理论讲义 第二卷 (Lectures on the Theory of Automorphic Functions,Second Volume) 英文

样章
作者：
Art Dupre
定价：
168.00 元
版面字数：
910千字
开本：
16开
装帧形式：
精装
版次：
1
最新版次
印刷时间：
2017年
ISBN：
978-7-04-047839-6
物料号：
47839-00
出版时间：
2017-10-19
读者对象：
学术著作
一级分类：
自然科学
二级分类：
数学与统计
三级分类：
函数论
暂无
  • 目录
    • Front Matter
      • Part I Narrower theory of the single-valued automorphic functions of one variable
        • 1 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 Γ
                • 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 Γ 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 automorphicmodels
                            • 1.10 General single-valuedness theoremApplication to linear differential equations
                              • 1.11 ζ as a linearly polymorphic functionThe fundamental problem
                                • 1.12 Differential equations of the third order for the polymorphic functions
                                  • 1.13 Generalization of the concept of automorphic functions
                                  • 2 Form-theoretic discussions for the automorphicmodels 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 ϕd (ζ1,ζ2) 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 allmultiplicator systems Mfor a given group Γ
                                                      • 2.10 Example for the determination of the number of the multiplicator systemsM, the effect of secondary relations
                                                        • 2.11 Representation of all unbranched automorphic forms
                                                          • 2.12 Existence theorem for single-valued forms ϕd (ζ1,ζ2) for given multiplicator systemM
                                                            • 2.13 Relations between multiplicator systems inverse to one another
                                                              • 2.14 Integral forms and formswith prescribed poles
                                                                • 2.15 The ζ1, ζ2 as linearly-polymorphic forms of the z1, 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 formof the differential equation for the polymorphic forms ζ1, ζ2
                                                                        • 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
                                                                          • 3 Theory of Poincaré serieswith special discussions for themodels of genus zero
                                                                            • 3.1 The approach to the Poincaré series
                                                                              • 3.2 First convergence study of the Poincaré series
                                                                                • 3.3 Behavior of the Poincaré series at parabolic cusps
                                                                                  • 3.4 The Poincaré series of (−2)nd dimension for groups Γ with boundary curves
                                                                                    • 3.5 The Poincaré series of (−2)nd dimension for principal-circle groups with isolatedly situated boundary points
                                                                                      • 3.6 Convergence of the Poincaré series of (−2)nd dimension for certain groups without boundary curves and without principal circle
                                                                                        • 3.7 Second convergence study in the principal-circle caseContinuous dependence of the Poincaré series on the groupmoduli
                                                                                          • 3.8 Poles of the Poincaré series and the possibility of its vanishing identicallyDiscussion for the case p =0
                                                                                            • 3.9 Construction of one-pole Poincaré series
                                                                                              • 3.10 One-poled serieswith poles at elliptic vertices
                                                                                                • 3.11 Introduction of the elementary forms Ω(ζ1,ζ2;ξ1,ξ2)
                                                                                                  • 3.12 Behavior of the elementary formΩ(ζ1,ζ2;ξ1,ξ2) at a parabolic cusp ξ
                                                                                                    • 3.13 Behavior of the elementary forms upon exercise of substitutions of the group Γ on ξ1,ξ2Discussions 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 Poincaré series
                                                                                                      • 4 The automorphic forms and their analytic representations formodels 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 algebraicmodel
                                                                                                            • 4.3 The polymorphic forms ζ1,ζ2 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 Γ 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 pThe p forms Φ−2(ζ1,ζ2)
                                                                                                                        • 4.9 Concept of conjugate formsExtended Riemann-Roch theorem and applications of it
                                                                                                                          • 4.10 The Poincaré series and the elementary forms for pUnimultiplicative 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 kindsProduct representation for the primeform
                                                                                                                                • 4.13 On the representability of the automorphic forms of arbitrary genus p by the elementary forms and the Poincaré 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 fromthe polygon continua to the group continua
                                                                                                                                                    • 1.9 The discontinuity of themodular group
                                                                                                                                                      • 1.10 The reduced polygons of the character (1,1)
                                                                                                                                                        • 1.11 The surface Φ3 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 Φ3 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 continuumof Riemann surfaces
                                                                                                                                                                                    • 1.25 Single-valuedness of the mapping between the continuum of groups and the continuumof 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; l1, l2)
                                                                                                                                                                                          • 1.28 Effectuation of the continuity proof for the signature (0, 3; l1)
                                                                                                                                                                                            • 1.29 Effectuation of the continuity proof for the signature (1, 1; l1)
                                                                                                                                                                                              • 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)
                                                                                                                                                                                                  • 2 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 TheGreen’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 Fν and their convergence in the principal-circle case
                                                                                                                                                                                                                    • 2.9 Mapping of the covering surface onto a circular discProof 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 u_,u__ with one another and with Green’s functions uμ
                                                                                                                                                                                                                          • 2.12 Mapping of the covering surface by means of the function (u_ +i v_). Proof of the boundary-circle theorem
                                                                                                                                                                                                                          • 3 Proof of the reentrant cut theorem
                                                                                                                                                                                                                            • 3.1 Theorems on schlicht infinite images of a circular surface
                                                                                                                                                                                                                              • 3.2 Theorems on schlicht finitemodels of a circular surface
                                                                                                                                                                                                                                • 3.3 The distortion theoremfor circular domains
                                                                                                                                                                                                                                  • 3.4 The distortion theoremfor arbitrary domains
                                                                                                                                                                                                                                    • 3.5 Consequences of the distortion theorem
                                                                                                                                                                                                                                      • 3.6 Production of the covering surface F∞ 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 theoremto 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 ηn(z)
                                                                                                                                                                                                                                                    • 3.13 Proof of the linearity theorem
                                                                                                                                                                                                                                                      • 3.14 Proof of the unicity theoremProof of the reentrant cut theorem
                                                                                                                                                                                                                                                        • 3.15 Koebe’s proof of the general Kleinian fundamental theorem
                                                                                                                                                                                                                                                        • A 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 G360 and the extended G720
                                                                                                                                                                                                                                                                      • A.7 The two resolvents of sixth degree of the transformation equation
                                                                                                                                                                                                                                                                        • A.8 The discontinuity domains of the Γ15 and Γ30 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 Γ20 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 TheGalois 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 EllipticModular Functions
                                                                                                                                                                                                                                                                                                • 2 Commentary by Jeremy Gray
                                                                                                                                                                                                                                                                                                  • 3 Commentary byWilliam Harvey on Automorphic Functions
                                                                                                                                                                                                                                                                                                    • 4 Commentary by BarryMazur
                                                                                                                                                                                                                                                                                                      • 5 Commentary by Series-Mumford-Wright
                                                                                                                                                                                                                                                                                                        • 6 Commentary by Domingo Toledo
                                                                                                                                                                                                                                                                                                          • 7 Commentaries by OtherMathematicians

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