Front Matter
Part I Introduction to the Study of the EllipticModular Functions
1 On the invariants of the binary biquadratic form
1.1 The form f (z1, z2) and its irrational invariants
1.2 The irrational invariants A,B,C of the form f
1.3 Behavior of the A,B,C upon varying the sequence of factors of f
1.4 Equivalence of two forms with the same sequence of factors. First canonical formof f
1.5 The equivalence of a form f with itself
1.6 The equivalence of two formswith arbitrary factor sequence
1.7 The rational invariants of the form f
1.8 The rational invariants in explicit form
1.9 The invariants g2, g3 and the absolute invariant J
1.10 The second canonical formof f
1.11 Geometrical observations on the second canonical form
1.12 Third conical formof f
1.13 More on the third canonical form. Connection to the theory of the regular solids
1.14 Normal forms of the elliptic integral of the first kind
1.15 Naming the normal forms.History
2 On the periods of the elliptic integral of the first kind
2.1 Pairs of primitive periods of the integral of the first kind
2.2 The periods as invariants. Dependency on the rational invariants. Normalization of the periods
2.3 Setting up the differential equation for the normalized periods
2.4 Fundamental theorems concerning the dependency of the normalized periods on J
2.5 Choice of a special primitive period-pair
2.6 Dissection of the J-plane. Significance of the determinations of the previous paragraph
2.7 Approach to the neighborhood of a singular point
2.8 Preliminary determination of the numbers k1,k2
2.9 Carrying though the investigation for the singular point J =0
2.10 Carrying the investigation through for the singular point J = 1
2.11 Determinations for the neighborhood of J =∞ and associated calculation of Ω2
2.12 Calculation of the limiting value of Ω1 for J =∞
2.13 Disposal of the singular point J =∞.Historical remarks
2.14 Branching of the periods ω1,ω2 over the J-plane
2.15 The periodquotient ω as a function of J
2.16 Differential equation of the third order for ω(J ). The s-functions
3 Concerning certain conformalmappings and the triangle functions arising fromthem
3.1 Replacement of the Riemann surface occurring by simpler figures
3.2 Figure for the representation of the connection between λ and J
3.3 Carrying the λ-plane onto the surface of the sphere
3.4 Relation to the second chapter
3.5 Figures for illustrating the connection between μ and J
3.6 The mapping of a circular arc triangle onto the halfplane of J
3.7 The relation of λ to μ illustrated through figures
3.8 The circle-relation. Theorems on circular arc triangles
3.9 The symmetrywith respect to a circle
3.10 The lawof symmetry.Direct and indirect circle-relatedness
3.11 Significance of the law of symmetry for the function μ(J)
3.12 General investigation of the function-theoretic significance of the law of symmetry
3.13 Definition and fundamental properties of the triangle- or s-functions
3.14 Series developments for a branch of the s-function
3.15 Differential equation of the third order for the s-function
3.16 Assembly of the triangle functions already appearing
3.17 Division of the triangle functions into kinds
3.18 The s-functions of the first kind
3.19 The s-functions of the second kind
3.20 The s-functions of the third kind
3.21 The triangle figures associated to ω(λ) and ω(J)
4 Development of the definitions and fundamental problems of a theory of the ellipticmodular functions
4.1 The Legendre relation
4.2 The rational invariants g2, g3, Δ as functions of the periods ω1, ω2
4.3 Functional determinants of the forms g2, g3, Δ
4.4 The periods of integrals of the second kind as functions of ω1, ω2
4.5 Calculation of the Hessian determinant H(logΔ)
4.6 Mappings effected by ω(J) and ζ(J ). Icosahedral andmodular equation
4.7 Form-theoretic comparison of the icosahedral andmodular equations
4.8 Further comparison of the icosahedral andmodular forms
4.9 Analogy in the function-theoretic treatment of the respective form problems
4.10 Algebraic equations with a variable parameter
4.11 Transfer of algebraic concepts to transcendental equations
4.12 Group-theoretic comparison of the icosahedral andmodular equations
4.13 Formulation of the general fundamental task
4.14 The group-theoretic fundamental problem
4.15 The function-theoretic fundamental problem
4.16 The ellipticmodular forms
5 Analytic representations for doubly periodic functions andmodular forms
5.1 Deviations of Terminology
5.2 The functions ?(u) and ? (u).Doubly periodic functions
5.3 Analytic representations for the functions ?(u), ? (u)
5.4 Doubly infinite series for g2 and g3
5.5 Simply infinite series for g2, g3 and the periods η1, η2
5.6 Product representation of the discriminant Δ. Themodular equation in explicit form
5.7 The function σ(u |ω1,ω2)
5.8 Product representation of the σ-function
5.9 Representation of doubly periodic functions by σ(u)
5.10 The functions σλ,μ(u |ω1,ω2)
5.11 Transition to the ?-functions
Part II Treatment of the Group-Theoretic Fundamental Problem
1 Of the linear substitutions of one variable and their geometric interpretation
1.1 Division into kinds of the linear substitutions of one variable
1.2 Geometrical interpretation of the substitutions with separately situated fixed points for special positions of the latter
1.3 Projection of the figures obtained onto the sphere
1.4 Orbit curves and level lines in the case of a general position of the fixed points
1.5 Disposal of the parabolic substitution
1.6 Concerning the substitutions arising from the s-functions of the first and second kind
1.7 The substitutions of the s-function s 12 , 13, 17; J1521.8 The kinds ofmodular substitutions taken from the modular division
1.9 A preliminary arithmetical consideration of the modular substitutions
1.10 The concept of equivalence and of the fundamental domain in for a group of linear substitutions
1.11 Form of the fundamental domain of a cyclic group in the hyperbolic and parabolic cases
1.12 Continuation: Case of an elliptic substitution VArbitrariness of the shape of the fundamental domain of a cyclic group
1.13 Special explanations for the ellipticmodular substitutions
1.14 Continuation: Determinations for the parabolic and hyperbolic substitutions
1.15 The substitutions of the variable z, which signify indirect circle-relations
1.16 Extension of a cyclic group of non-loxodromic substitutions of the first kind by associated reflections
1.17 Fundamental domains for the extended groups just considered
2 The modular group and its corresponding division of the ω-halfplane
2.1 Preliminary results on the fundamental domain of the modular group
2.2 Closer consideration of the fundamental domain. Negative part of the proof
2.3 Continuation: positive part of the proof
2.4 Simply andmultiply equivalent points
2.5 The substitutions S and T as generators of themodular group
2.6 Covering of the ω-halfplane with equivalent circular arc triangles with angles π/3, π/3, 0
2.7 Extension of themodular group by reflections
2.8 The fundamental domain of the extendedmodular group
2.9 Comparison of the fundamental domains of the original and extended modular groups
2.10 The generating operations of the extended group
2.11 Transformations of the modular division into itself. Domains, which streamout froma rational point
2.12 Projection of the modular division into a rectilinear figure of triangles
3 The integral binary quadratic forms and the conjugation of modular substitutions
3.1 Naming the quadratic forms
3.2 The equivalence of quadratic forms
3.3 Representation of forms of negative determinant and their reduction
3.4 The number of the substitutions, which effect the equivalence of two forms with D <0
3.5 External characterization of reduced forms. Finiteness of the class number
3.6 Representation of forms of positive determinant
3.7 First approach to the transformation of a formof positive determinant into itself
3.8 Report on the Pell equation
3.9 Production of all substitutions, which transform a form of positive determinant into itself
3.10 Position in the ω-halfplane of a semicircle representing a form of positive determinant
3.11 The reduced forms and their periods. Disposal of the problem of equivalence
3.12 External characterization of reduced forms. Finiteness of the class number
3.13 Existence proof for the smallest positive solution T,U of the Pell equation
3.14 Transformation of themodular substitutions
3.15 Conjugacy in the case of elliptic and parabolic substitutions
3.16 Conjugacy of hyperbolic substitutions
3.17 Conjugacy of the cyclic subgroups contained in the modular group
4 Discussion of a special subgroup contained in themodular group
4.1 Definition of the subgroups Γ and Γ
4.2 The fundamental domain for Γ
4.3 The generators of Γ and Γ
4.4 The simplest fundamental domain for Γ
4.5 Approach to relating the subgroup Γ to the total group Γ
4.6 System of representatives and index for the subgroup ΓNotation Γ6for Γ
4.7 The Γ6 as distinguished subgroup
4.8 The finite group G6 which corresponds to Γ6. Γ2 and the three conjugates Γ3
4.9 The fundamental domains F3 of the conjugate Γ3
4.10 Renewed consideration of the fundamental domain F6 of subgroup Γ6
4.11 Folding of the fundamental domain of F6 into a dihedrally divided sphere. Relation of the dihedral division to the modular division
4.12 Explaining the group G6 by means of the dihedally divided sphere. Regularity of the fundamental domain F6
4.13 Folding of the fundamental domains F3 into simply covered planes. Irregularity of the F3
4.14 Symmetry of the domains F3. The regular-symmetric domain F6
4.15 Preliminary remarks on the function-theoretic significance of the subgroup Γ6
5 General approach for the treatment of the subgroups of themodular group
5.1 Index and systemof representatives for a given subgroup
5.2 Production of a fundamental domain Fμ belonging to a given subgroup Γμ
5.3 Covering the ω-halfplane with polygons, which correspond to the subgroup Γμ. Generation of Γμ
5.4 Transformation of subgroups. Conjugate and distinguished subgroups
5.5 The fundamental polygons of conjugate and distinguished subgroups
5.6 The finite groups Gμ and G2μ, which correspond to a distinguished subgroup Γμ
5.7 General viewpoint for the decomposition of the groups Gμ into their subgroups
5.8 Significance of the previous paragraph for our group-theoretic fundamental problem. Disposal of the latter according to plan
5.9 Bending together the fundamental domain Fμ to a closed surface. Genus p of a subgroup Γμ. Relation to the ω-halfplane
5.10 Special investigation for the distinguished subgroups Γμ. Regularity and symmetry of the associated surfaces Fμ
5.11 Partial regularity or symmetry of the surfaces Fμ for relatively distinguished subgroups Γμ
5.12 Rules for the calculation of the genus p of a subgroup Γμ. Diophantine equation for distinguished subgroups
6 Definition of all subgroups of themodular group bymeans of the surfaces Fμ
6.1 Methods of defining subgroups bymeans of fundamental polygons or surfaces Fμ: the branching theorem
6.2 Production of a mapping between the divided surface Fμ and the ω-halfplane
6.3 Spreading out the surface Fμ in the ω-halfplane. Proof of the branching theorem. Immediate applications
6.4 The spherical nets of the regular solids and the distinguished subgroups of the modular group of genus p =0
6.5 The subgroups Γ{n} corresponding to the functions s(12, 13, 1n; J)
6.6 Significance of Γ{n} for the solution of the group-theoretic fundamental problem.Division of the subgroups into classes
6.7 Discussion of a special distinguished subgroup of the sixth class of index 72
6.8 Definition of a special distinguished subgroup of the seventh class of index 168
6.9 The finite group G168. The subgroups G7 and G21 of G168
6.10 The 28 symmetry lines of F168 and the subgroups G3 and G6 of G168
6.11 The 21 shortest lines of F168 and the subgroups G8,G4,G2 in G168
6.12 The four-groups G4 and octahedral groups G24 in G168
7 The congruence groups of the nth level contained in themodular group
7.1 The principal congruence group of the nth level
7.2 The modular substitutions considered modulo n. The group Gμ(n)
belonging to Γμ(n)
7.3 The homogenous modular substitutions and their groups. The
homogenous principal congruence groups of nth level
7.4 Calculation of the index μ(n) of the principal congruence group of nth level
7.5 Comparison of Γ{n} and the principal congruence groups Γμ(n)Principal congruence groups of the previous chapter
7.6 Generalities on congruence groups of nth level
7.7 An important principle of group theory
7.8 Illustration of the developments of the previous paragraph by the G72 belonging to the sixth level
7.9 Reduction of the problem of the decomposition of the groups Gμ(n), resp. G2μ(n).History
7.10 The branching schema of distinguished subgroups
7.11 The congruence character of subgroups of the nth class. Range of thecongruence groups
8 The cyclic subgroups in the groups G q(q2?1)2, Gq(q2?1) and Gq(q2?1)belonging to the prime level q
8.1 TheGalois imaginary numbers
8.2 Imaginary form of the group G q(q2?1)2
8.3 The cyclic subgroups Gq of order q
8.4 The cyclic subgroups G q?12of order q?12
8.5 The cyclic subgroups G q+12of order q+12
8.6 The totality of cyclic subgroups. Preliminary reference to the surface F q(q2?1)2
8.7 The cyclic subgroups of the homogeneous Gq(q2?1)
8.8 Cyclic subgroups of Gq(q2?1) for the case q = 4h+1
8.9 Cyclic subgroups of Gq(q2?1) for the case q = 4h?1
8.10 The symmetric transformations of the surface F q(q2?1)2into itself
9 Enumeration of all non-cyclic subgroups of the group G q(q2?1)2belongingto prime level q
9.1 The generating substitutions of a group G{n} belonging to an arbitrarynumber n
9.2 Dyck’s theorem on the generators of the groups holohedricallyisomorphic with G{n}
9.3 Non-cyclic subgroups of G q(q2?1)2and Gq(q2?1), in which a cyclic Gq takes part
9.4 The semimetacylic subgroups for arbitrary level number n
9.5 Decomposition of the surface F q(q2?1)2into q?2 polygonwreaths
9.6 The subgroups of dihedral type contained in G q(q2?1)2
9.7 Putting the polygon wreaths together into the surface F q(q2?1)2
9.8 The four-groups contained in G q(q2?1)2
9.9 The subgroups of G q(q2?1) 2 of tetrahedral and octahedral type
9.10 Setting up and enumeration of the subgroups of icosahedral type contained in G q(q2?1)2
9.11 General equations of condition for the order of a subgroup of G q(q2?1)2
9.12 Proof of the completeness of the given decomposition of G q(q2?1)2
9.13 Simplicity of G q(q2?1)2. TheGalois theorem. Concluding remarks
Part III The Function-Theoretic Fundamental Problem
1 Foundation of Riemann’s theory of algebraic functions and their integrals
1.1 The many-sheeted Riemann surface Fn over a plane
1.2 The algebraic functions belonging to the surfaces Fn
1.3 The integrals belonging to the surface Fn under consideration
1.4 The potentials belonging to the surface Fn under consideration
1.5 Formulation of the existence theoremfor an arbitrarily given Riemann surface Fn. Plan of the proof
1.6 Solution of the boundary value problemfor circular domains
1.7 Description of the combinationmethod in a special case
1.8 Production of the potentials of the third and first kinds
1.9 Two lemmas concerning potentials and integrals of the first kind
1.10 The 2p potentials and the p integrals of the first kind of Fn
1.11 The p normal integrals of the first kind of Fn
1.12 The integrals, in particular the normal ones, of the second kind of Fn
2 Continuation of Riemann’s theory of algebraic functions
2.1 The algebraic functions on a Riemann surface Fn of genus p =0
2.2 The algebraic functions on a Riemann surface Fn of genus p =1
2.3 The algebraic functions on a Riemann surface Fn of an arbitrary genus
2.4 The functions ? of an Fn of higher genus
2.5 The Riemann-Roch theorem
2.6 Extension of the Riemann-Roch theorem to p =0 and p = 1. A special application for p >1
2.7 The Brill-N?ther reciprocity theorem. The special functions
2.8 Introduction of the language of analytic geometry
2.9 The curve in the space Rv of v dimensions
2.10 The equivalent point systems. Homogeneous coordinates. The projective conception
2.11 The normal curves Cm
2.12 The rational and the elliptic normal curves in particular
2.13 The cases p > 1: the normal curve of the ?
2.14 The cases p >1: the hyperelliptic case
2.15 Concluding remarks
3 General solution of the function-theoretic fundamental problem
3.1 Transformation of the polygons, resp., the closed surfaces Fμ, into Riemann surfaces
3.2 The functions ω(J) and s(J) on the Riemann surface Fμ
3.3 The functions of F(J )μ considered in their dependency on ω
3.4 Character of the z(ω), j (ω) as the modular functions sought for the Γμ
3.5 The full modular system of the subgroups Γμ and the associated algebraic resolvents of themodular equation
3.6 General investigations concerning the symmetric subgroups
3.7 Principal moduli and systems of moduli for symmetric subgroups
3.8 Generalities concerning the moduli of conjugate and distinguished subgroups
3.9 Special considerations for the principal moduli and systems of moduli of distinguished subgroups
3.10 The Galois problems and their resolvents. Plan of the further developments
4 The modular functions belonging to the distinguished subgroups of genus p = 0
4.1 Fixing the Galois principal moduli and collection of associated formulas
4.2 Introduction of the modular forms belonging to the Galois principal moduli
4.3 Final determination of the modular forms λ1,λ2, etc
4.4 Relations between the modular forms λ1,λ2, etc. and g2, g3. The form problem
4.5 The singlevalued modular forms 3 Δ, 4 Δ and 12 Δ
4.6 Determination of the homogeneous subgroups belonging to the modular forms λ1,λ2, etc
4.7 The six conjugate Γ6 of the fifth level and the associated F6
4.8 Setting up the resolvent of the sixth degree
4.9 Details concerning the resolvent of the sixth degree. The question of its affect
4.10 The system of the moduli of the A. Affect of the resolvent of the twelfth degree
4.11 The five conjugate Γ5 of the fifth level and the associated surface F5
4.12 Setting up the resolvent of the fifth degree
5 Modular functions, which let themselves be produced from the Galois principalmoduli
5.1 The distinguished Γ48 and Γ120 leading to hyperellipticmodels
5.2 The singlevalued modular functions n λ, n 1?λ, etc
5.3 Enumeration of all congruence moduli contained among thequantities n λ, etc
5.4 Special investigation of the congruence moduli 8 λ, 8 1?λ
5.5 Behavior of ?,ψ,χ with regard to arbitrarymodular substitutions
5.6 Putting together additional congruence moduli
5.7 TheGalois systems to be built fromthemoduli considered
5.8 Setting up a fewcongruence groups of the sixth level
5.9 The congruencemoduli y(ω) and x(ω) of the sixth level
5.10 The 72 transformations of the C3 into itself. Geometrical theorems
5.11 The moduli 3 λ(λ?1), ξ3 ?1. The distinguished Γ6
6 The systems ofmoduli zα and Aγ of the seventh level
6.1 Introduction of the modular forms zα and the curve C4
6.2 Geometrical significance of the points a, b, c on the C4
6.3 The eight inflection triangles and the eight G21. Choice of special zα
6.4 Setting up the equation of the C4. The real trace
6.5 Setting up the 168 ternary substitutions. Additional remarks
6.6 The collineations of periods 2 and 3 contained in the G168
6.7 The collineation groups G6, G4 and G24 in the G168
6.8 The three kinds of conic sections through eight points b
6.9 The tangent-C3 of the C4 and its distinguished system
6.10 Introduction of the system ofmoduli of the Aγ
6.11 Relations between δν and Aγ. The system of substitutions of the Aγ
6.12 The space curve of sixth order of the Aγ
6.13 The C6 of the Aγ as the conic vertices of a bundle of surfaces of the second order
7 The Galois problem of 168th degree and its resolvents of the 8th and 7th degrees.—Concluding remarks
7.1 The three covariants of the ternary biquadratic form f (zα)
7.2 The φ, ψ, X as modular forms of the first level. The problem of the 168th degree
7.3 The problem of the Aγ. Mention of the extended problems
7.4 Setting up the function-theoretic resolvents of the eighth degree
7.5 Expressions of the moduli τ in the zα. The form-theoretic resolvents of the eighth degree
7.6 The two form-theoretic resolvents of seventh degree
7.7 Representation of the moduli τ belonging to the Γ7 in terms of the zα
7.8 The setting up and investigation of the form-theoretic resolvent of the seventh degree
7.9 Comparison of the levels q = 5 and q = 7. Plan of the further development
7.10 The significance of the modular functions for the theory of the general linearly-automorphic functions
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
