The study of the algebraic-solutions problem for a second-order linear ordinary differential equation had brought to light the conceptual importance of considering groups of motions of the sphere, and, in particular, finite groups. Klein connected this study with that of the quintic equation, and so with the theory of transformations of elliptic functions and modular equations as considered by Hermite, Brioschi, and Kronecker around 1858. Klein’s approach to the modular equations was first to obtain a better understanding of the moduli, and this led him to the study of the upper half plane under the action of the group of two by two matrices with integer entries and determinant one; his great achievement was the production of a unified theory of modular functions. Independently of him, Dedekind also investigated these questions from the same standpoint, in response to a paper of Fuchs. So this chapter looks first at Fuchs’s study of elliptic integrals as a function of a parameter, and then at the work of Dedekind. The algebraic study of the modular equation is then discussed, and the chapter concludes with Klein’s unification of these ideas.


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  1. 2.
    Hadamard [1954, 109] said of Hermite: “Methods always seemed to be born in his mind in some mysterious way”.Google Scholar
  2. 3.
    See also Dugac [1976], Mehrtens [1979].Google Scholar
  3. 4.
    Compare Lang [1976], Serre [1973], or Schoeneberg [1974].Google Scholar
  4. 5.
    I cannot find that he ever took an opportunity to do this.Google Scholar
  5. 7.
    The implicit isomorphism between PSL(2, ℤ/5ℤ) and A5 was made explicit by Hermite [1866 = Oeuvres, II, 386-387].Google Scholar
  6. 8.
    See Dugac [1976, 73] for a discussion of priorities, and Dauben [1978, 142] for Kronecker’s tardiness. The matter is fully discussed in H. M. Edwards [1980, 370-371].Google Scholar
  7. 9.
    Klein [1926, I, 366] — strictly, a description of work on automorphic function theory.Google Scholar
  8. 10.
    This highly transitive group inspired Mathieu to look for others, and so led to his discovery of the simple, sporadic groups which bear his name, Mathieu [1860, 1861].Google Scholar
  9. 11.
    The multiplier equation associated to a modular transformation describes M (see Chapter I p.19) as a function of k.Google Scholar
  10. 12.
    Bottazz-ini is editing some unpublished correspondence between Italian and other mathematicians on this topic.Google Scholar
  11. 13.
    Gordan made a study of quintics in his [1878], discussed in Klein [1922, 380-4].Google Scholar
  12. 14.
    Notes of Weierstrass’s lectures were presumably available, see Chapter VI n. 4. Klein based his approach to the modulus of elliptic functions on the treatments in Müller [1867, 1872 a, b] which I have not seen. Hamburger’s report on them in Fortschritte, V, 1873, 256-257, indicates that they are based on Weierstrass’s theory of elliptic functions and the invariants of biquadratic forms.Google Scholar

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© Springer Science+Business Media New York 1986

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  • Jeremy Gray

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