# Non-Euclidean Geometry in the Theory of Automorphic Functions

Share this page *Editors and Authors: *
*Jeremy J. Gray; Abe Shenitzer; Jacques Hadamard*

A co-publication of the AMS and the London Mathematical Society

This is the English translation of a volume originally published only in
Russian and now out of print. The book was written by Jacques Hadamard on the
work of Poincaré.

Poincaré's creation of a theory of automorphic functions in
the early 1880s was one of the most significant mathematical
achievements of the nineteenth century. It directly inspired the
uniformization theorem, led to a class of functions adequate to solve
all linear ordinary differential equations, and focused attention on a
large new class of discrete groups. It was the first significant
application of non-Euclidean geometry. The implications of these
discoveries continue to be important to this day in numerous different
areas of mathematics.

Hadamard begins with hyperbolic geometry, which he compares with plane and
spherical geometry. He discusses the corresponding isometry groups, introduces
the idea of discrete subgroups, and shows that the corresponding quotient
spaces are manifolds. In Chapter 2 he presents the appropriate automorphic
functions, in particular, Fuchsian functions. He shows how to represent Fuchsian
functions as quotients, and how Fuchsian functions invariant under the same group
are related, and indicates how these functions can be used to solve
differential equations. Chapter 4 is devoted to the outlines of the more
complicated Kleinian case. Chapter 5 discusses algebraic functions and linear
algebraic differential equations, and the last chapter sketches the theory of
Fuchsian groups and geodesics.

This unique exposition by Hadamard offers a fascinating and
intuitive introduction to the subject of automorphic functions and
illuminates its connection to differential equations, a
connection not often found in other texts.

This volume is one of an informal sequence of works within the
History of Mathematics series. Volumes in this subset,
“Sources”, are classical mathematical works that served as
cornerstones for modern mathematical thought.

#### Reviews & Endorsements

Gives a fascinating and highly instructive brief exposition on Poincaré's creation of the theory of automorphic functions … The historical circumstances, sources and perspectives of Poincaré's discovery are very instructionally and vividly depicted … this beautiful thin book will please analysts as well as geometers and also all fans of the history of mathematics.

-- Mathematica Bohemica

The book is substantially enhanced by editor Gray's introduction, ‘Brief History of Automorphic Function Theory, 1880–1930’, which deftly treats both mathematics and related nontechnical matters …

Jeremy Gray and Abe Shenitzer deserve the gratitude of the entire mathematical community for bringing out this Hadamard volume. It would be remiss to fail to praise the quality of the writing found here. Gray's insightful introductory remarks and Shenitzer's translation of the Russian text are recognizably English, clearly, carefully and intelligently wrought. This is an important addition to the literature, of great interest both to mathematical historians and to anyone with even a passing acquaintance with non-Euclidian geometry and automorphic functions. For mathematicians actively engaged in research in these areas, this book is essential reading.

-- Mathematical Reviews

It is cause for joy and celebration that the “Sources” subseries of the AMS/LMS “History of Mathematics” series continues to grow. This little book should be of interest both to historians seeking to understand the evolution of the theory of automorphic functions and to mathematicians working in the area, and thus it is a valuable addition to the (rather short) list of original source material available in English translation. Keep them coming, AMS!

-- MAA Online

#### Table of Contents

# Table of Contents

## Non-Euclidean Geometry in the Theory of Automorphic Functions

- Cover Cover11
- Title page v6
- Contents vii8
- Acknowledgments ix10
- Introduction by the Publishers of the Russian Translation xi12
- Historical introduction 114
- A brief history of automorphic function theory, 1880-1930 316
- Chapter I. The group of motions of the hyperbolic plane and its properly discontinuous subgroups 1730
- Chapter II. Discontinuous groups in three geometries. Fuchsian functions 3750
- Chapter III. Fuchsian functions 5770
- Chapter IV. Kleinian groups and functions 7184
- Chapter V. Algebraic functions and linear algebraic differential equations 7992
- Chapter VI. Fuchsian groups and geodesics 87100
- References 93106
- Additional references 95108
- Back Cover Back Cover1109