# Automorphic Functions

Chapter

## Abstract

This final chapter is, naturally, concerned with the triumphant accomplishments of Poincaré: the creation of the theory of Fuchsian groups and automorphic functions. These developments brought together the theory of linear differential equations and the group-theoretic approach to the study of Riemann surfaces, so this account draws on all of the preceding material. It begins with a significant stage intermediate between the embryonic general theory and the developed Fuchsian theory: Lamé’s equation.

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## Notes

- 1.The Lame equation is discussed in Whittaker and Watson, Ch. XXIII.Google Scholar
- 2.The applications of elliptic functions are discussed in Houzel [1978, §15].Google Scholar
- 4.I have not seen [Mittag-Leff 1er, 1876] which is in Swedish. A propos of the Weierstrassian theory Halphen commented [69n]: “Il existe des tableaux de formules lithographiées qui ont été rédigées d’après les leçons de M. Weierstrass, et qui sont entre les mains de presque tous les géomètres allemands.”.Google Scholar
- 5.This information is taken from Darboux, Éloge Historique d’Henri Poincaré, printed in Poincaré, Oeuvres, II, vii-lxxi.Google Scholar
- 6.7 letters between Poincare and Fuchs were printed in Acta Mathematica (38) 1921 175-187, and reprinted in Poincaré Oeuvres, XI, 13-25. An eighth letter is given in photographs in Oeuvres, XI, 275-276.Google Scholar
- 7.When the essay was published in 1923 this figure was incorrectly printed as figure 6, before the one depicting the situation which Fuchs had shown to be impossible. The text refers to the annular case as the second one, which it is in Poincaré’s original essay as deposited in the Académie.Google Scholar
- 8.Accounts of the supplements appear in Gray [1982a,b] and Dieudonné [1983]. I would like to thank J. Dieudonne for his help during my work on these supplements.Google Scholar
- 10.See [Bottazzini 1977, 32].Google Scholar
- 11.c.f. Riemann, “Vorlesungen über die hypergeometrische Reihe”, Nachträge, 77.Google Scholar
- 12.The correspondence is printed in Klein, Gesammelte Mathematische Abhandlungen, III, 1923, 587-626, in Acta Mathematica (38) 1922, and again in Poincaré Oeuvres, XI. The simplest way to refer to the letters is to give their date.Google Scholar
- 13.Published somewhat accidentally in Poincaré’s Oeuvres XI, 275-276.Google Scholar
- 14.The same approach was taken by Picard in proving his celebrated ‘little’ theorem that an entire function which fails to take two values is constant, Picard [1879a]. He argued that, if there is an entire function which is never equal to a, b, or ∞ (being entire), then a Möbius transformation produces an entire function, G, say, which is never 0, 1, or ∞. So G never maps a loop in the complex plane onto a loop enclosing 0, 1, or ∞. Now, G maps a small patch into a region of, say, the upper half plane, and if k
^{2}: H → Ȼ is the familiar modular function, then composing G with a branch of the inverse of k^{2}maps the patch into half a fundamental domain for k^{2}. Analytic continuation of this composite function cannot proceed on loops enclosing 0, 1, or ∞ (by the remark just made) so the composite (k^{2})^{−1}o G is single-valued. But it is also bounded, so, by Liouville’s theorem, it is constant, and the ‘little’ theorem is proved. To see that the function is bounded replace the domain of k^{2}by the conformally equivalent unit disc; Picard missed that trick and proved it directly. For a nearly complete English translation see Birkhoff and Merzbach, 79-80.Google Scholar - 15.Many years later, Mittag-Leffier wrote in an editorial in Acta Mathematica ((39) 1923, iii) that Poincaré’s work was not at first appreciated. “Kronecker, for example, expressed his regret to me via a mutual friend that the journal seemed bound to fail without help on the publication of a work so incomplete, so immature and so obscure.”.Google Scholar
- 16.The uniformization theorem was first proved by Koebe and Poincaré independently in 1907, after an incomplete attempt by Schlesinger. The validity of the continuity method was a central question in the new theory of the topology of manifolds, and was decisively studied by Brouwer in 1911-12. For a full discussion and references see Scholz [1980, 198-222]. Hubert raised uniformization as a problem as the 22nd in his famous list, and one should also consult Bers’ article on it in Mathèmatical developments arising from Hubert Problems, AMS, PSPUM, 28, 1976, 559-609, and D. M. Johnson [1982].Google Scholar

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© Springer Science+Business Media New York 1986