Jeremy J. Gray Linear Differential Equations and Group Theory from Riemann to Poincare Second Edition Birkhäuser Boston Basel Berlin
Introduction to the Second Edition Introduction to the First Edition xiii xv Chapter I: Hypergeometnc Equations and Modular Equations 1.1 Euler and Gauss 1 The hypergeometnc series (hgs) and the hypergeometnc equation (hge) in the work of Euler and Pfaff 1 Gauss's early mathematical isolation 2 Gauss's work on the arithmetico-geometric mean 3 Gauss's published paper on the hgs [1812a] 5-8 and his unpublished one on the hge [1812b] 8-9 in which monodromy considerations are raised 10 1.2 Jacobi and Kummer 11 Legendre's elliptic integrals 11 Jacobi on the transformation of elliptic functions, 12 polynomial and differential equations connecting different moduli 13 Kummer on the hypergeometnc equation, 14 transformations of the variable 15 Monodromy considerations not to be found in Kummer's work 16 His list of 24 Solutions 19 The connection with elliptic functions via Legendre's equation 20 13 Riemann's approach to complex analysis 20 In his papers [1851], [1857c] 21 Riemann regarded a function as a surface and introduced topological considerations of Connectivity; 22 Dirichlet's principle is used to guarantee the existence of prescribed functions 23 1.4 Riemann's P-function 23 P-functions defined 24
i vi Monodromy matrices at the branch-points and products of same, their coefficients expressed in terms of the exponents of the P-functions 25 The P-function shown to satisfy a hge, 26 so the hge is characterised by its exponents at the branch-points 26 1.5 Interiude: Cauchy's theory of differential equations ".28 Cauchy's proofs of the existence of Solutions to differential equations 29 Singular points studied by Briot and Bouquet 30 Weierstrass's unpublished theories 30 Exercises 30-40 Chapter II: Lazarus Fuchs Introduction 41 Fuchs a Student of Kummer and Weierstrass 42 2.1 Fuchs' theory of linear differential equations 42 His approach to differential equations a blend of Riemannian techniques and those of the Berlin school 43 He characterised those equations whose Solutions are everywhere 00 locally of the form (x xo) a Y^ a (x XQ)", as being of the form n=-k d"y F p - X {x)d"- l y F 2(p -i)(x) F n(p - 1} (x) dx" j/(x) dx"' 1 x//(x) 2 " if(x) n y where p is the number of finite Singular points a\,ai,, a p, \jr{x) := (x -a\)(x - a 2 )... (* -a p ), andf s (x) isa polynomial in x of degree at most s. Such equations are 'of the Fuchsian class' 46 Fuchs gave rigorous methods for solving linear differential equations, and a careful analysis of their singular points in terms of the monodromy matrices and their eigenvalues; the case of repeated eigenvalues is introduced 47 The 'indicial' equation for the exponents* 47 Amongst the equations of the Fuchsian class are those all of whose Solutions are algebraic. Fuchs asked how these might be characterised 48 The hypergeometric equation is the only equation of order n > 1 and of the Fuchsian class for which the exponents determine the coefficients exactly. 49 2.2 Generalisations of the hypergeometric equation SO Fuchs showed [1870] that the periods of hyperelliptic integrals, as functions of a parameter, satisfy equations of the Fuchsian class 52 In the case of the elliptic integral he gave a thorough treatment of the monodromy matrices at the singular points 53
vii 23 Conclusion 54 2.4 The new methods of Frobenius and others* 55 Frobenius's [1873] paper simplified the method of Fuchs 56 In subsequent papers Frobenius introduced the idea of irreducibility of a differential equation, and used it to clarify the work of Thome 57 Thome [1873] called the functions which are Solution of equations of the Fuchsian type regulär, and studied equations which have irregulär Solutions... 59 The introduction of Jordan canonical form by Jordan and Hamburger. 62 J. Tannery on Fuchs's work 62 Exercises 63-67 Chapter III: Algebraic Solutions to a Differential Equation 3.1 Schwarz 70 Schwarz [1872] discussed when all the Solutions of a hge are algebraic 70 The Solution is, essentially, that F(a, ß, y, z) is algebraic provided that the quotient of two linearly independent Solutions maps the upper plane onto a triangle whose reflections tessellate the sphere. This condition reduces to the condition on X, ß, and v (rational constants related to a, ß, and y) that A. + [i + v > 1 72 Schwarz depicted a tessellation of the disc by circular-arc triangles corresponding to the case k 5, /x = \, v = j 75 X + (i + v = 1 corresponds to elliptic functions and a tessellation of the plane by triangles 75 3.2 Generalisations 75 Group Theory, invariant theory, and 'hyper-galois' theory 77 Fuchs [1875] obtained necessary but not sufficient conditions for the general second order differential equation to have all its Solutions algebraic, using ad hoc techniques and the methods of invariant theory 77 33 Klein and Gordan 81 Klein and Gordan 81 Klein's calculation of the finite groups of motions of the sphere, [1875/76] 83 The connection with invariant theory. 84 The connection with differential equations: if a second order differential equation has all its Solutions algebraic it can be obtained from the hge by a rational change of variable [1877] 85 Brioschi's calculation of hge's corresponding to Schwarz's allowable X, p, v 87
viii 3.4 The solutions of Gordan and Fuchs 88 Gordan solved Fuchs's problem purely in terms of invariant theory, [1877a, b] 88 Fuchs found his earlier Solution contained invalid items, and removed them [1878] 89 3.5 Jordan's Solution 89 Jordan's main paper, [1878], looks for finite subgroups of SL («; C), for an «th order linear differential equation has all its solutions algebraic if and only if its monodromy group is finite 90 Jordan used Sylow theory, and solved the problem for «= 2 and 3 91-93 He missed the simple group of order 168 93 For general n he established his finiteness theorem, reproved in [1880]* 95 3.6 Equations of higher order 95 Higher order differential equations* 95 Exercises 98-99 Chapter IV: Modular Equations 4.1 Fuchs and Hermite 101 Hermite rekindled Fuchs's interest in the modular function K/K' as a function of k 2 and its inverse 102 Fuchs [1877] studied the inverse of the quotient of two solutions of a differential equation and the idea of functions invariant under sets of monodromy matrices 103 Hermite on a certain matrix group 105 Monodromy matrices were also used by Schläfli [1870] to explain a passage in Hermite's Solution of quintic [1858] concerning modular transformations of order 2 106 4.2 Dedekind 107 Dedekind's career. 107 His paper introducing modular functions 108 The function val(z) and its fundamental domain 110 The role of the Schwarzian and hypergeometric equations 111 Dedekind's t) function 111 Modular equations 112 Dedekind's approach based on the lattice-idea and invariance under SL (2; Z), familiär from the theory of quadratic forms 113 43 Galois theory, groups and fields 115 Galois on solvability by radicals 116 And on the reducibility of the modular equation 117
ix The rediscovery of his work in 1846, culminating in Jordan's theory of permutation groups 117 Kronecker and Dedekind elaborated a Galois theory of groups and equations which led to the concepts of field theory 118-119 Klein's approach to group theory. 120 4.4 The Galois theory of modular equations, c.1858 121 Betti [1853] rederived Galois's result that the modular equation of degree p + 1 reduces to one of degree p when p = 5, 7,11 but for no higher p 121 Hermite [1858] showed that when p = 5 the reduced form of the equation coincides with the general quintic, so solving the quintic equation by modular functions 122 Kronecker [1858a, b] showed that the corresponding reduction when p = 7 produces an equation of degree 7 invariant under 168 permutations of the roots, and conjectured that only these 7th degree equations are solvable by modular equations 124 The Galois groups (for p = 5, 7,11) were calculated explicitly 125 4.5 Klein 126 He interpreted the modular equation in terms of the absolute invariant J of an elliptic integral 127 J agrees with Dedekind's functions val 127 The modular equation interpreted as a Riemann surface over the complex J sphere 128 The particular case when this surface is of genus zero 130 The role of the regulär solids 132 Reduction of the modular equation 134 4.6 A modern treatment of the modular equation 135-140 Exercises 137-140 Chapter V: Some Algebraic Curves 5.1 Algebraic Curves, particularly quartics 141 Plücker [1834] on the 3n(n 2) inflection points of a plane curve of degree n, and, [1839], on its \n(n 2)(n 2 9) bitangents obtained via Plückers formulae 144 Jacobi [1859] derived the estimate for the bitangents independently 144 Plücker showed [1839] that the 28 bitangents to a quartic can all be real 145 Geiser [1869] on the connection between the 28 bitangents and the 27 lines on a cubic surface 146 5.2 Function-theoretic geometry 146
X Weierstrass [1853,1856] on the inversion of Systems of hyper-elliptic integrals... 147 Riemann [1857c] on the theory of Abelian functions 148 Jacobi inversion, and Riemann's 0-function 148 ö-characteristics, 149 the odd characteristics correspond to bitangents 151 53 Klein 152 Klein [1878/79] on the group <7i68 and T(7), a congruence subgroup of PSL (2; Z) 152 The fundamental region for T(7), 155 and the corresponding algebraic curve 155 The curve has genus 3, and may be given the equation x 3 y + y 3 z + z 3 x = 0, which embeds it in CP 2 156 The inflection points, bitangent points, and sextactic points can be identified in the fundamental region 157 Klein's visualisation of the figure 158 Methodological questions, 161 Exercises (and some work of Hesse's) 162-166 Chapter VI: Automorphic Functions 6.1 Lame's Equation 167 Hermite and Fuchs on the Lam6 equation, quasi-periodic functions 167 the work of Picard [1879,1880] 169 and Halphen [1884] 169 Fuchs's generalisation of Jacobi inversion, the introduction of a new class of functions 170 6.2 Poincare 173 Biographical details 173 Correspondence with Fuchs; the birth of Fuchsian functions 174 Poincar6's earliest work on linear differential equations 174 the prize essay of 1880 178 His later account of his discovery of the significance of non-euclidean geometry 180 The three Supplements to the essay 182-184 63 Klein 184 His work in 1880,1881 185 6.4 1881 186
xi Poincare's first papers, Fuchsian groups, theta-fuchsian functions, non-euclidean geometry 186-188 Klein initiates correspondence with Poincar6 188 Klein on fundamental polygons 189 Schwarz's work, and the Dirichlet principle, unknown to Poincare 189 Poincare on the connection with linear differential equations 191 Kleinian groups and functions, three-dimensional non-euclidean geometry. 193 What's in a name? Priorities, Klein v. Fuchs 195 Fuchsian functions 196 Uniformisation of algebraic curves 197 6.5 Klein's response 198 Klein's style 198 The discovery of the 'Grenzkreis' theorem 199 Poincare's approach compared to Klein's 200 Klein's discussion with Schwarz 203 Klein's 'Neue Beiträge' [1882] 203 Klein's health collapsed 205 Poincare's Acte papers of 1882 206 6.6 Poincare's papers of 1883 and 1884 209 Poincare's [1884a] 209 types, Subordination of types 211 class of a differential equation 211 the cases of 4 and 5 Singular points 212 Poincare's [1884b] 213 zeta-fuchsian functions, analogy with Jacobi's work 215 Forsyth's account, [1902] 217 Schlesinger and Plemelj in 1908 219 Conclusion 221 Appendices Appendix 1: Riemann, Schottky, and Schwarz on conformal representation 223 Appendix 2: Riemann's lectures and the Riemann-Hilbert problem 231 Appendix 3: Fuchs's analysis of the nth order equation 247 Appendix 4: On the History of non-euclidean geometry 253 Appendix 5: The uniformisation theorem 261 Appendix 6: Picard-Vessiot Theory 267 Appendix 7: The hypergeometric equation in several variables; Appell and Picard 275
Xll Notes 281 Chapter I 281 Chapter II 284 Chapter III 287 Chapter IV 292 Chapter V 295 Chapter VI 299 Appendices 301 Bibliography 307 Historical List of Names 333 Index 335 Four passages which contain technical details which may not be to everyone's taste have been marked with a star*.