Lectures on the Theory of Automorphic Functions, Vol.2 9787040478396, 7040478390
453 76 9MB
English Pages 587 Year 2017
Table of contents :
Why should one open and read Klein-Fricke and Fricke-Klein? -- John Coates
Reference
Preface
Contents
Second Volume: Theory and application of the unique automorphic function of one variable
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 automorphic models
1.10 General single-valuedness theorem. Application to linear differential equations
1.11 ζ as a linearly polymorphic function. The 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 automorphic models 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 φ_d(ζ_1, ζ_2) 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 all multiplicator systems M for a given group Γ
2.10 Example for the determination of the number of the multiplicator systems M, the effect of secondary relations
2.11 Representation of all unbranched automorphic forms
2.12 Existence theorem for single-valued forms φ_d(ζ_1, ζ_2) for given multiplicator system M
2.13 Relations between multiplicator systems inverse to one another
2.14 Integral forms and forms with prescribed poles
2.15 The ζ_1, ζ_2 as linearly-polymorphic forms of the z_1, z_2
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 form of the differential equation for the polymorphic forms ζ_1, ζ_2
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 series with special discussions for the models 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 case. Continuous dependence of the Poincaré series on the group moduli
3.8 Poles of the Poincaré series and the possibility of its vanishing identically. Discussion for the case p = 0
3.9 Construction of one-pole Poincaré series
3.10 One-poled series with poles at elliptic vertices
3.11 Introduction of the elementary forms Ω(ζ_1, ζ_2; ξ_1, ξ_2)
3.12 Behavior of the elementary form Ω(ζ_1, ζ_2; ξ_1, ξ_2) at a parabolic cusp ξ
3.13 Behavior of the elementary forms upon exercise of substitutions of the group Γ on ξ_1, ξ_2. Discussions 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 for models 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 algebraic model
4.3 The polymorphic forms ζ_1, ζ_2 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 p. The p forms Φ_{-2}(ζ_1, ζ_2)
4.9 Concept of conjugate forms. Extended Riemann-Roch theorem and applications of it
4.10 The Poincaré series and the elementary forms for p. Unimultiplicative forms
4.11 Two-poled series of (-2)^{nd} dimension and integrals of the 2^{nd} kind for automorphic models of arbitrary genus p
4.12 The integrals of the first and third kinds. Product 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 from the polygon continua to the group continua
1.9 The discontinuity of the modular group
1.10 The reduced polygons of the character (1, 1)
1.11 The surface Φ_3 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 Φ_3 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 continuum of Riemann surfaces
1.25 Single-valuedness of the mapping between the continuum of groups and the continuum of 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; l_1, l_2)
1.28 Effectuation of the continuity proof for the signature (0, 3; l_1)
1.29 Effectuation of the continuity proof for the signature (1, 1; l_1)
1.30 Effectuation of the continuity proof for the signature (0, 3)
1.31 Representation of the three-dimensional continua B_g and B_f 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 The Green'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 F_ν and their convergence in the principal-circle case
2.9 Mapping of the covering surface onto a circular disc. Proof 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 u_μ
2.12 Mapping of the covering surface by means of the function (u' + iv'). 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 finite models of a circular surface
3.3 The distortion theorem for circular domains
3.4 The distortion theorem for 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 F_n onto a schlicht domain for special reentrant cuts
3.8 Mapping of the surface F_n onto a schlicht domain for arbitrary reentrant cuts
3.9 Introduction of a system of analytic transformations belonging to the domain P_n
3.10 Application of the distortion theorem to the domain P_n
3.11 Application of the consequences of the distortion theorem to the domain P_n
3.12 Effectuation of the convergence proof of the functions η_n(z)
3.13 Proof of the linearity theorem
3.14 Proof of the unicity theorem. Proof 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 G_{360} and the extended ar{G}_{720}
A.7 The two resolvents of sixth degree of the transformation equation
A.8 The discontinuity domains of the Γ_{15} and Γ_{30} belonging to the octahedral and tetrahedral groups
A.9 The two resolvents of the 15^{th} degree of the transformation equation
A.10 Note on the grups Γ_{20} belonging to the ten conjugate G_{18}
A.11 The Riemann surface of the Galois resolvent of the transformation equation
A.12 The curve C_6 in the octahedral coordinate system
A.13 The curve C_6 in the icosahedral coordinate system
A.14 The curve C_6 in the harmonic coordinate system
A.15 The real traces of the C_6 and the character of the points a, b, c
A.16 Further geometrical theorems on the collineation group G_{360}
A.17 The Galois resolvent of the transformation equation
A.18 The solution of the resolvents of 6^{th} and 15^{th} degree
A.19 Solution of the transformation equation of 10^{th} degree
Commentaries
1. Commentary by Richard Borcherds on Elliptic Modular Functions
2. Commentary by Jeremy Gray
3. Commentary by William Harvey on Automorphic Functions
4. Commentary by Barry Mazur
5. Commentary by Series-Mumford-Wright
6. Commentary by Domingo Toledo
7. Commentaries by Other Mathematicians
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