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 ?d (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 ?d (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