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Part IV Introduction of quantities of division and transformation and their algebraic relations
1 The division values of doubly periodic functions and the special division equations
1.1 Group-theoretic foundations for the theory of elliptic functions
1.2 Intervening function-theoretic factors. Development of the fundamental problems of the theory of elliptic functions
1.3 Group-theoretic investigation of the nth division values fλ,μ of a doubly-periodic function of the first level
1.4 The division equations belonging to the division values fλ,μ
1.5 The algebraic character of the fλ,μ. Particular properties of the division values ?λ,μ, ? λ,μ
1.6 Methods for the final determination of the coefficients G(g2, g3). The absolute termof the division equation of the ? _λ,μ
1.7 Norming the ?λ,μ, ?_λ,μ. Special considerations for n = 3,4,5
1.8 Introduction of the functions σλ,μ(u | ω1,ω2)
1.9 Detailed group-theoretic investigation of the σ-division values
1.10 Function-theoretic study of the division values σλ,μ
1.11 Special investigation of the σλ,μ for n =2 and 3
1.12 Special study of the σλ,μ for n =5 and 7.Generalization
2 The transformation of the nth order, first level and the transformation equations of the first level
2.1 First introduction of the transformation of the nth order, first level and its group-theoretic investigation. The function-theoretic representatives
2.2 The transformation polygon of the nth order
2.3 Second introduction of the transformation of the nth order and first level. The arithmetic representatives.
2.4 Relation between the function-theoretic and the arithmetic representatives. Extension of the transformation of the nth order
2.5 Branching and genus of the transformation polygon of the nth order
2.6 Function-theoretic study of the transformation. The transformation equation
2.7 Continuation: the transformation equations for J (ω). Concept of the modular equation.
2.8 Replacement of the modular equation for the orders belonging to the genus p =0
2.9 Connection of the principal moduli τ with the transformed discriminant Δ(ω1,ω2)
2.10 Collection of additional formulas for the transformed Δ
2.11 Additional remarks concerning the formulas found for the transformed Δ
2.12 Introduction of the form-theoretic transformation equations adjoint to the first level
2.13 Fundamental theorems concerning the existence of the transformation equations adjoint to the first level in the cases n = q and n = q2
2.14 Formof the transformation equations for the roots of Δ
2.15 Historical notes concerning the transformation equations adjoint to the first level
3 General foundation for the transformation of the nth order of arbitrary modular functions
3.1 General character of a transformedmodular function.
3.2 Group of the transformed modulus z_. Reducibility of the relation f (z_, J ) =0
3.3 Excursus on a general principle of group theory: The case of distinguished subgroups.
3.4 Continuation of the group-theoretic excursus: Case of nondistinguished subgroups.
3.5 Disposal of the question of the conjugacy of the transformed moduli z_(ω)
3.6 Generalities concerning the relation between z_ and z. Exclusion of the non-congruencemoduli
3.7 Investigation of the transformation equation for a congruence modulus of arbitrary level
3.8 Special study of the case ν = 1. The modular equations of higher level
3.9 The representative system of the transformation of the nth order belonging to the mth level
3.10 A special theorem concerning the case of non-relatively prime m, n. Examples
3.11 Calculation of ω for a given modulus by means of a chain of transformations.
4 Setting up the modular equations of higher levels with particular consideration ofm = 5 and 16
4.1 Existence and number of the modular equations for the principal moduli to be studied
4.2 The permutability of the arguments in the left side of the modular equations
4.3 Foundation of the invariant-theoretic method for setting up the modular equations
4.4 The formation of the full formsystems by polarization in the case of cogradience
4.5 The extent of the case of cogradience
4.6 Setting up the full form system for the modular equations of the icosahedron coming up form =5
4.7 Setting up the full form system for the modular equations of ?(ω) coming up form =16
4.8 Additional tools for the final determination of the modular equations. Historical notes
4.9 Collection of a fewmodular equations of the 5th and 16th level. Remark on non-congruencemoduli
4.10 The irrational forms of the Jacobian modular equations, the modular correspondences and the relations between transformed ?-zeros
5 Application of the modular equations of the first level to the theory of integral binary quadratic forms
5.1 Supplementary theorems concerning the binary quadratic forms and their geometrical representation
5.2 Relation of the modular equations of the first level to the quadratic forms of positive determinant. The Smith Curve
5.3 Carrying over the Smith Curve to the nth transformation polygon
5.4 Introduction of the singular moduli of the first level and nth order and the modular functions Hn(ω) of first level
5.5 The relation of the singular moduli to the quadratic forms of negative determinant. Arithmetic determination of the degree of the equation gn(J )= 0
5.6 Function-theoretic determination of the degree of the equation gn(J )= 0. The class-relations of the first level
5.7 The singular moduli coming up for the proper transformation of the nth order. Study of these in the transformation polygon Fψ(n)
5.8 Historical material concerning the equation of the singular moduli properly belonging to the determinant ?Δ. The complex multiplication of elliptic functions
5.9 Specialization of the results up until now for n = 5 and n =7.
6 Application of icosahedral equations to the theory of integral binary quadratic forms
6.1 Relation of the modular equations of the fifth level to the binary quadratic forms of positive determinant
6.2 Transition to the class-number relations of the fifth level and initial steps for their calculations
6.3 Arithmetic enumeration of the zeros of n(ω) lying in the interior of F60
6.4 First part of the function-theoretic investigation of the hi (ω) at the polygon cusps
6.5 Second part of the investigation of the hi (ω) at the polygon cusps
6.6 Collection of the results of the investigations with respect to the polygon cusps
6.7 The system of class-number relations of the fifth level. The class-number relations of the third level
Part V Analytic treatment of the function-theoretic fundamental problem for the congruence groups
1 Introduction to the theory of elliptic normal curves Cn of the nth order
1.1 The definition of the elliptic normal curve of the nth order and its single-valued transformations into itself
1.2 Geometrical deductions concerning the normal curves Cn by means of themethod of projecting
1.3 The canonical coordinate system of the C3 and its generalization for the Cn
1.4 The singular coordinate system for the normal curve C3. Initial introduction of the quantities Xα
1.5 Representation of the Xα coming up for the C3 by means of σ-projects and transformed σ
1.6 Introduction of the singular coordinate polyhedron of the Xα for the elliptic normal curve Cn of arbitrary order
1.7 Preliminary normalization of the Xα. Expression of the collineation group G2n2 by means of the Xα
1.8 Series developments for the functions Xα(u|ω1,ω2) for arbitrary n
1.9 Setting up the n(n?3) 2 linearly independent quadratic relations, subsisting between the general Xα
1.10 Interpretation of problems from the division and transformation theory bymeans of the normal curves
2 The quantities Xα considered as functions of the periodsω1, ω2
2.1 Final choice and representations for the additional factor κ of the Xα(u|ω1,ω2) in the case of an odd n
2.2 Representations of the normalized Xα(u|ω1,ω2) and determination of its level in the case of an odd n
2.3 The modular forms zα, yα, etc. of the nth level and their relation to the division values in the case of an odd n
2.4 Final choice and representation for the additional factor κ of the Xα(u|ω1,ω2) in the case of an odd n
2.5 Representations of the normalized Xα(u|ω1,ω2) and determination of their level in the case of odd n
2.6 Introduction of the modular forms zα, yα, etc. of the 2nth resp. 4nth levels for even values of n
2.7 Transformation of the Xα by modular substitutions, in particular, by S and T , in the case of an odd n
2.8 Effect of the substitutions S,T on the Xα for even n. General remark on the additional factor κ
2.9 Connection to the previous chapter. Final form of the Xα-group for the case of a prime number n = q
2.10 Excursus on the sums ofGauss
2.11 Isomorphic mapping of the Xα-group onto itself by the replacement of ε by εp
2.12 The substitution groups of the modular forms zα, yα and the biquadratic relations of the zα coming up for odd n
3 Formation of new functions of the nth level by means of bilinear combinations of the Xα
3.1 The two- and three-termed bilinear combinations Bα of the Xα
3.2 The three kinds I, II, III of p-termed bilinear combinations B(p,n) α of the Xα
3.3 Analytic developments for the modular forms Aα for odd n prime to 3
3.4 Analytic representation and manifoldness of the B(p,n) α entering in case I. Introduction to the Xα(u, v)
3.5 Analytic representations of the Bα resp. Xα(u, v) coming up in the cases II, III
3.6 Investigation of the quadratic forms of f (ξ,η) entering in the cases II and III and their relative equivalence
3.7 The manifoldness of the function systems Xα(u, v) coming up in case II
3.8 Manifoldness of the function systems Xα(u, v) arising in case III.
3.9 The modular forms Aα, zα, yα,xα, of the nth level, which arise from the general Xα(u, v)
3.10 Series developments for the modular forms Aα, zα, yα etc. in case I
3.11 Series developments of the moduli Aα, etc. in the cases II and III. Rearrangement according to ascending powers of r
4 Form-theoretic-analytic discussions in the domain of the lowest level numbers
4.1 Principles for the study of integral algebraicmodular forms
4.2 Integralmodular forms of the second level and their laws of formation
4.3 The integral modular forms ξ2,ξ4 and_ Δ of the third level. Relation of these to the division values ?λ,μ, ?_λ,μ
4.4 The modular forms_Δ, yα adjoint to the third level. Second representation of the principal modulus ξ(ω).
4.5 The integral modular forms of the fourth level. Representation for_Δ and the Galois principal modulus μ(ω)
4.6 The two digradient binary systems zα of the dimension ?2 for n = 5
4.7 Analytic laws of formation for the Galois principal modulus ζ(ω). Reduction of the ηα to the ζα
4.8 Form-theoretic form of the resolvent of the fifth degree of the icosahedral equation. Relation to the principal modulusτ of the subgroup Γ5
4.9 The ternarymodular system of the fifth level of the Aα
4.10 Analytic representations for the simplest modular functions y(ω) and x(ω) of the sixth level
4.11 Analytic representations for the systems of the zα and Aα of the seventh level considered in Vol. I
4.12 The quaternarymodular system of the Bα for n = 7
5 Themodular systems of the eleventh level and the associated resolvents of eleventh and twelfth degrees
5.1 Introduction of the three modular systems zα of (?2)nd dimension
5.2 Setting-up of the algebraic relations between the zα. Special consideration of the z(1) α
5.3 The invariant combinations of the zα belonging to the G660 and their relation to the modular forms g2, g3,Δ of the first level
5.4 The geometrical models of the 20th, 80th and 50th degrees in the space R4, which correspond to the three zα-systems
5.5 Intersection of the curves C20, C80 and the ruled surface with the spaces Φ(ζ) =0 and Ψ(ζ)= 0
5.6 Choice of a special subgroup G60 of G660, and investigation of the polygon F11 belonging to the G60
5.7 The simplest modular forms and the principal modulus of the Γ chosen
5.8 The two resolvents of eleventh degree in function-theoretic and form-theoretic form.
5.9 The transformation polygon of eleventh order of genus p =1
5.10 The two single-valued forms A,B, adjoint to the polygon F12, and the two-valued function τ(ω)
5.11 The three-valued quantities E(ω1,ω2) and τ_(ω) belonging to the polygon F12. Relation between τ(ω) and τ_(ω)
5.12 Representation of g2, g3,_Δ and their transformed values g _2, g _3,_Δ_ in E, A, B. Expression for the function-theoretic resolvent of 12th degree
5.13 The form-theoretic resolvent of twelfth degree
6 The algebraicmodels for n > 11 defined by themodular equations of the first level
6.1 The transformation polygon belonging to n = 31 and its modular forms of the dimension ?1 and ?2
6.2 The full modular system of the Γ32. The singular moduli and the Smith’s curve of the transformation polygon F32
6.3 The modular forms Z,H of the Γ32. Representation of Δ, g2 and J in the moduli of Γ32
6.4 The surface F48 belonging to n = 35 and its simplest modular system.
6.5 Representation of the principal moduli of the groups Γ8 and Γ belonging to n = 7 and 5 by the moduli of the Γ48 belonging to n = 35
6.6 The transformation polygon F48 belonging to n = 47 and its modular forms of the (?1)st and (?2)nd dimension
6.7 The full modular system of the Γ48 and the associated hyperelliptic relation of the tenth degree
6.8 Foundation for the transformation of the order n =71
Part VI Theory of the modular correspondences and the class-number relations proceeding from them
1 New discussions of Riemann’s theory of algebraic functions
1.1 Introduction of the integrals Qx,y ξ,η of the third kind and, in particular, the normal integrals Πx,y ξ,η
1.2 Periods of the normal integrals Πxyξη Theorem of the permutability of parameter and argument for Π xy ξη
1.3 The Abelian theorem, in particular for the integrals of the first and second kinds
1.4 Foundations of the form-theoreticmethod of consideration. Historical notes.
1.5 The algebraic forms, in particular, the integral algebraic forms G(z1, z2) of the surface Fn
1.6 Representation of all integral forms G(z1, z2) by means of a few of them. Theoremof theminimal basis.
1.7 Form-theoretic representations of the integrals of the surface Fn
1.8 Announcements concerning the form theory to be based on a plane curve.
1.9 The prime formP(x, y) and its fundamental properties
1.10 Continuation: periodicity properties of the prime form. Its relation to the ?-function
1.11 Representation of the algebraic functions and integrals of the Fn by the prime form
1.12 The inversion problemand its solution
2 General theory of algebraic correspondences and the correspondence principle
2.1 The correspondences and the correspondence principle of the geometers
2.2 The general approach. The fundamental relations between the integrals j , as well as between the periods τik
2.3 The totality of the algebraic models of the genus p
2.4 The singular Riemann surfaces. Division of the correspondences into singular and ordinary
2.5 Representation of the valenced correspondences by the prime form. Derivation of the Cayley-Brill correspondence formula.
2.6 The arithmetic foundation of the general correspondence theory
2.7 Formation of aminimal basis for all solution systems (π) of the surface Fn
2.8 Actual existence of all arithmetically possible correspondences. Choice of aminimal basis [Kε] of t correspondences of the Fn
2.9 Representation of all correspondences of the Fn by means of the minimal basis [Kε]. The general correspondence formula
2.10 Riemann surfaces with single-valued transformations into themselves
3 Theory of the integrals of the first kind for the congruence groups of prime level
3.1 The form theory of a surface Fμ on the basis of ω1,ω2
3.2 The modular forms of the first and third kinds in particular. Preliminary choice of the integrals j for a prime level q
3.3 Particularities concerning the integral j (ω|0) of the division polygon. Representation of these by the j (ω|α), j (ω|β)
3.4 A general law of formation for the integrals j (ω|α), j (ω|β) of the qth level
3.5 The principal of the integral development coefficients for the integrals j (ω|α), j (ω|β).
3.6 Introduction of the development functions ψ(m) and χ(m).Minimal bases of integrals j (ω|α), j (ω|β)
3.7 Arithmetic definition of the ψ1(m) for q = 7 and q = 11. The general binary development principle.
3.8 The remaining development functions ψi , χk of the eleventh level. The general quaternary development principle.
3.9 Presentation of a few results concerning the integrals of the lower composite level numbers
3.10 Further questions concerning the integrals of the first kind
4 Special theory of the modular correspondences of nth order of an arbitrary level
4.1 Definition of the modular correspondences of the nth order. Irreducibility, invertibility andmonodromy groups
4.2 The integral equation for the modular correspondences of the sixth level
4.3 The integral relations for the modular correspondences of the seventh level
4.4 Presentation of the integral relations for the correspondences belonging to the Γ96 of the eighth level
4.5 Prime form representation and coincidence number of the modular correspondences in the case of a single development function, explained for q =7
4.6 Approach for the integral relations of the j (ω|α), j (ω|β) for the modular correspondences of an arbitrary prime level q
4.7 Actual setting-up of the integral relations for q = 11
4.8 Preliminary remarks on the choice of a basis of correspondences for an arbitrary prime level q
4.9 Formation of a basis of correspondences for the case of a quadratic residue n
4.10 Formation of a basis of correspondences for the case of a non-residue n of q
4.11 Representation of modular correspondences of the qth level by the prime formin the λ resp. μ correspondences of the basis
4.12 Enumeration of the coincidences for the modular correspondences of the qth level
5 The Class-number relations of an arbitrary prime level, with particular effectuations for q =7 and 11.
5.1 Prepatory theorems for the arithmetic determination of the coincidence numbers νi (n)
5.2 Enumeration of the zeros of h(ω) in the polygon interior. Case distinctions I, II, III to bemade
5.3 Continuation: discussion of the cases I, II, III
5.4 Proof of the lemma on the number of incongruent solutions of a congruence of the second degree.
5.5 Enumeration of the zeros of h(ω) lying at the polygon cusps
5.6 Putting together the results concerning the number ν(n). The special case n =1
5.7 The two systems of class-number relations of the 7th level
5.8 General approach for the class-number relation of the qth level
5.9 Final form of the two systems of the class-number relations of the eleventh level
6 The algebraic representation of themodular correspondences
6.1 Representation of an algebraic function of two surface points x, y by algebraic functions of x or y alone.
6.2 The representation of the algebraic correspondences by algebraic equations
6.3 The arbitrariness of the equation entering in the case of a non-negative valence.
6.4 Choice of the three classes of modular correspondences to be especially treated in the following
6.5 Separation of the zero valenced correspondences for the groups Γ168, Γ96 and Γ384
6.6 Generalities concerning intersection-system correspondences. Separation of those for the curves C4 and C8 of the groups Γ96, Γ384
6.7 Separation of the intersection-system correspondences for the C4 of the group 168
6.8 Invariant-theoretic tools for the construction of the correspondence equations
6.9 Setting up a few biternary invariants for the G168 of the seventh level
6.10 Final calculation of the correspondence equations of the seventh level for n =3,6,19,12
6.11 Disclosure of a few correspondence equations for the groups Γ96 and Γ384
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
