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椭圆模函数理论讲义 第一卷 (Lectures on the Theory of Elliptic Modular Functions, First Volu

样章
作者：
Art Dupre
定价：
168.00元
ISBN：
978-7-04-047872-3
版面字数：
1060.000千字
开本：
16开
全书页数：
暂无
装帧形式：
精装
重点项目：
暂无
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
  • 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 ϑ-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
      • 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
      • 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 ϕ 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-Nöther 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 ϕ
      • 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 ϕ,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

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