Fuchsian Differential Systems: Arithmetic Theory
We record here all the properties of p-adic analytic functions (and prove some of them) which will be used in the sequel of the book. We refer the reader to the general introduction [41], or to the more advanced booklet [26]. Everything becomes simpler if
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Yves André
G-functions and geometry A Publication of the Max-PlanckInstitut für Mathematik, Bonn
Yves Andre
G-Functions and Geometry
Aspects of Matherratics
Aspekte der Mathematik Editor: Klas Oiederich
All volumes of the series are listed on pages 230-231.
Yves Andre
G-Functions and Geometry
A Publication of the Max-Planck-Institut für Mathematik, Bonn Adviser: Friedrich Hirzebruch
Springer Fachmedien Wiesbaden GmbH
Dr. Yves Andre Institut H. Poincani, UA 763 du C~RS 11 rue P. et M. Curie 75231 Paris 5 AMS Subject classification: 11 G xx, 11 J xx
All righ ts reserved © Springer Fachmedien Wiesbaden 1989
Originally published by Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig in 1989.
No part of this publication may be reproduced, stored in a retrieval system or transmitted, mechanical, photocopying or otherwise, without prior permission of the copyright holder.
Produced by Wilhelm + Adam, Heusenstamm
ISSN
0179-2156
ISBN 978-3-528-06317-7 ISBN 978-3-663-14108-2 (eBook) DOI 10.1007/978-3-663-14108-2
v
Foreword
This is an introduction to some geometrie aspects of G-function theory. Most of the results presented here appear in print for the flrst time; hence this text is something intermediate between a standard monograph and a research artic1e; it is not a complete survey of the topic. Except for geometrie chapters (I.3.3, II, IX, X), I have tried to keep it reasonably selfcontained; for instance, the second part may be used as an introduction to p-adic analysis, starting from a few basic facts wh ich are recalled in IV.l.l. I have inc1uded about forty exercises, most of them giving some complements to the main text. Acknowledgements
This book was written during a stay at the Max-Planck-Institut in Bonn. I should like here to express my special gratitude to this institute and its director, F. Hirzebruch, for their generous hospitality. G. Wüstholz has suggested the whole project and made its realization possible, and this book would not exist without his help; I thank him heartily. I also thank D. Bertrand, E. Bombieri, K. Diederich, and S. Lang for their encouragements, and D. Bertrand, G. Christo I and H Esnault for stimulating conversations and their help in removing some inaccuracies after a careful reading of parts of the text (any remaining error is however my sole responsibility). It is a pleasure to acknowledge the influence of previous work of Bombieri; Christol, and G. V. Chudnovsky on this book. Finally, I wish to thank Miss Grau for her patience in deciphering and typing the whole manuscript. May 1988
Yves Andre
VI
Contents
Logical dependence of the chapters .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
IX X 1
Part One: What are G-functions? ..............................
11
Chapter I: G-functions ...... ; . . . . . . . . . . . . .
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