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Non-Euclidean geometry in the theory of automorphic functions. Edited by Jeremy J. Gray and Abe Shenitzer. Transl. from the Russian by Abe Shenitzer. With historical introduction by Jeremy J. Gray. (English) Zbl 0949.01010

History of Mathematics (Providence) 17. Providence, RI: American Mathematical Society; London: London Mathematical Society (ISBN 0-8218-2030-3/pbk). xii, 95 p. (1999).
In the 1920s several Russian mathematicians were involved in the preparation of the collected works of N. I. Lobachevskij. By that time Hadamard had written a long article on the mathematical work of H. Poincaré [Acta Math. 38, 203-287 (1921), in: Oeuvres de Henri Poincaré, Tome XI, 152-242 (1956; Zbl 0072.24103)] and Hadamard was invited to contribute to the edition of Lobachevskij’s works as well. Hadamard wrote a small booklet demonstrating the fundamental importance of hyperbolic geometry for the theory of automorphic functions as developed by H. Poincaré. This text was translated into Russian, and since the original French text appears now to be lost the present booklet is a translation of the Russian version into English. This complicated genesis of the present text may raise doubts in its reliability and the attentive reader may find some formulations which sound doubtful. But these do not really distract from the fact that the present text is an interesting document on the early history of the theory of automorphic functions as seen by a French admirer of the genius of H. Poincaré.
The book starts with a translation of the introduction to the Russian edition. This is followed by a historical introduction by J. Gray explaining the genesis of Hadamard’s manuscript and Hadamard’s relations with Soviet mathematicians. Moreover, there is a brief (13 pages) history of automorphic function theory, 1880-1930 by J. Gray.
Hadamard’s text consists of six chapters: Chapter I contains a description of hyperbolic geometry in 2 and 3 dimensions (ball and upper half-space models and the projective model), the corresponding groups of motions and their properly discontinuous subgroups. Chapter II is concerned with discontinuous groups in the spherical, Euclidean and hyperbolic geometries, with the classification of linear fractional transformations and with Poincaré’s theorem on the existence of Fuchsian groups with prescribed fundamental region. Fuchsian functions are considered in Chapter III. In particular, the author gives Poincaré’s two convergence proofs for his theta series, the first by means of Euclidean geometry, the second by means of hyperbolic geometry, Klein’s existence proof for Fuchsian functions by means of Dirichlet’s principle and conformal mapping is also mentioned. (Nevertheless it must be said that Hadamard concentrates mainly on the work of Poincaré and neglects the work of German writers such as F. Klein. One reason for this may well be the political situation after World War I when Germany was barred from international scientific organizations.) The Riemann surface and the algebraic curve associated with a Fuchsian group are also discussed in Chapter III. The brief Chapter IV deals with Kleinian groups and functions (Picard’s group, fundamental polyhedra, groups with prescribed fundamental polyhedra, Schottky groups). The heart of the booklet is Chapter V where the fundamental problems are investigated: Every cofinite Fuchsian group has an associated algebraic curve the coordinates of whose points can be parametrized by two Fuchsian functions. Can every algebraic curve be so represented? Every Fuchsian function has an associated linear differential equation with algebraic coefficients. Can every second-order linear differential equation with algebraic coefficients be integrated in this way? This leads to a discussion of the Uniformization Theorem and the role of the universal covering surface, one of Poincaré’s truly great ideas. Hadamard lays special emphasis on the connection between the theory of automorphic functions and the theory of linear differential equations. Chapter VI on Fuchsian groups and geodesics attracts attention to the then open problem of asymptotic behaviour of the course of geodesics on surfaces of negative curvature.
It is said at the end of the introduction to the Russian edition that the footnotes to the Russian edition appear as notes at the end of each chapter but it seems that more notes have been added by the editions since there are references to more recent sources. Unfortunately, the editing does not fully satisfy this reviewer since there are gaps and some errata in the references. (By the way, the German word “Losung” (engl. “watchword” or “dung”) has a meaning quite different from “Lösung” (engl. “solution”)).

MSC:

01A55 History of mathematics in the 19th century
01A75 Collected or selected works; reprintings or translations of classics
01A60 History of mathematics in the 20th century
30-03 History of functions of a complex variable
30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization)
30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
30F10 Compact Riemann surfaces and uniformization
34-03 History of ordinary differential equations
34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
51-03 History of geometry
11F03 Modular and automorphic functions

Citations:

Zbl 0072.24103
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